C ALGORITHM 765, COLLECTED ALGORITHMS FROM ACM. C THIS WORK PUBLISHED IN TRANSACTIONS ON MATHEMATICAL SOFTWARE, C VOL. 23, NO. 1, March, 1997, P. 81--90. C #! /bin/sh # This is a shell archive, meaning: # 1. Remove everything above the #! /bin/sh line. # 2. Save the resulting text in a file. # 3. Execute the file with /bin/sh (not csh) to create the files: # Doc # Drivers # Src # This archive created: Sat Apr 19 22:23:29 1997 export PATH; PATH=/bin:$PATH if test ! -d 'Doc' then mkdir 'Doc' fi cd 'Doc' if test -f 'stenmin_user_manual.ps' then echo shar: will not over-write existing file "'stenmin_user_manual.ps'" else cat << \SHAR_EOF > 'stenmin_user_manual.ps' %!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: stenmin_2.dvi %%Pages: 19 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%EndComments %DVIPSCommandLine: dvips stenmin_2 %DVIPSParameters: dpi=300, compressed, comments removed %DVIPSSource: TeX output 1996.07.03:2046 %%BeginProcSet: texc.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale 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Fi(f)5 b Fl(\()p Fi(x)p Fl(\))18 b(at)g(the)g(curren)o(t)h(p)q(oin)o(t)g Fj(X)p Fl(.)f Fj(USH)g Fl(m)o(ust)g(b)q(e)h(declared)h Fj(EXTERNAL)d Fl(in)j(the)e(user's)h(calling)h(program)59 951 y(and)15 b(m)o(ust)g(conform)g(to)f(the)i(usage)527 1045 y Fj(CALL)23 b(USH\(N,)g(X,)h(NZ,)f(LICN,)g(HESS,)g(IRN,)h(ICN\))59 1139 y Fl(where)14 b Fj(N)g Fl(is)g(the)g(dimension)h(of)e(the)h (problem,)g Fj(X)g Fl(is)g(a)g(v)o(ector)f(of)g(length)h Fj(N)p Fl(,)f Fj(HESS)h Fl(is)g(the)g(Hessian)g(matrix)f(at)59 1196 y Fj(X)p Fl(,)g Fj(LICN)g Fl(is)h(the)f(length)h(of)f Fj(HESS)p Fl(,)g Fj(NZ)g Fl(is)h(the)f(n)o(um)o(b)q(er)h(of)f(nonzeros) h(in)g(the)f(lo)o(w)o(er)g(or)g(upp)q(er)i(triangular)e(part)59 1252 y(of)g Fj(HESS)p Fl(,)g(and)g Fj(IRN)g Fl(and)h Fj(ICN)f Fl(are)g(the)h(ro)o(w)e(and)i(column)h(indices)g(of)e(the)h (nonzeros)f(in)i Fj(HESS)p Fl(.)d Fj(USH)h Fl(m)o(ust)g(not)59 1308 y(alter)18 b(the)f(v)m(alues)i(of)e Fj(N)p Fl(,)g Fj(X)p Fl(,)g(and)h Fj(LICN)p Fl(.)f(Only)i(the)e(lo)o(w)o(er)h(or)f (upp)q(er)h(triangular)g(part)f(of)g Fj(HESS)g Fl(should)i(b)q(e)59 1365 y(giv)o(en.)27 b(When)18 b(using)g(the)g(in)o(terface)f Fj(STUMCD)p Fl(,)g(if)h(no)f(analytic)h(Hessian)g(is)g(supplied)i(\()p Fj(HSNFLG)j(=)h(0)p Fl(\),)17 b(the)59 1421 y(user)e(m)o(ust)g(use)h (the)f(dumm)o(y)g(name)g Fj(STDUSH)p Fl(.)59 1534 y Fj(*TYPX)p Ff(!)p Fl(:)32 b(An)22 b(arra)o(y)e(of)h(length)h Fj(N)g Fl(in)g(whic)o(h)g(the)g(t)o(ypical)g(size)h(of)e(the)g(comp)q(onen)o (ts)h(of)f Fj(X)g Fl(are)g(sp)q(eci-)59 1591 y(\014ed.)f(The)13 b(t)o(ypical)h(comp)q(onen)o(t)g(sizes)g(should)g(b)q(e)g(p)q(ositiv)o (e)g(real)g(scalars.)19 b(If)14 b(a)e(negativ)o(e)i(v)m(alue)g(is)g(sp) q(eci\014ed,)59 1647 y(its)j(absolute)h(v)m(alue)h(will)f(b)q(e)g (used.)27 b(When)18 b(0.0)e(is)i(sp)q(eci\014ed,)h(1.0)d(will)j(b)q(e)f (used.)27 b(The)17 b(program)f(will)j(not)59 1704 y(ab)q(ort.)g(This)14 b(v)o(ector)e(is)i(used)g(b)o(y)f(the)g(the)g(pac)o(k)m(age)h(to)e (determine)j(the)e(scaling)h(matrix)f Fi(D)1633 1711 y Fd(x)1655 1704 y Fl(.)19 b(Although)14 b(the)59 1760 y(pac)o(k)m(age)k(ma)o(y)f(w)o(ork)g(reasonably)h(w)o(ell)h(in)g(a)e (large)h(n)o(um)o(b)q(er)g(of)g(instances)h(without)e(scaling,)j(it)e (ma)o(y)f(fail)59 1817 y(when)c(the)f(comp)q(onen)o(ts)h(of)f Fi(x)573 1824 y Fm(\003)605 1817 y Fl(are)g(of)g(radically)i (di\013eren)o(t)e(magnitude)h(and)g(scaling)g(is)g(not)f(in)o(v)o(ok)o (ed.)19 b(If)13 b(the)59 1873 y(sizes)j(of)f(the)h(parameters)e(are)h (kno)o(wn)g(to)g(di\013er)h(b)o(y)f(man)o(y)g(orders)g(of)g(magnitude,) g(then)h(the)f(scale)h(v)o(ector)59 1929 y Fj(TYPX)e Fl(should)h(de\014nitely)i(b)q(e)e(used.)20 b(Mo)q(dule)c Fj(STDFLT)d Fl(returns)i Fj(TYPX)f Fl(=)h(\(1.0,)p Fi(:)8 b(:)f(:)m Fl(,)14 b(1.0\).)19 b(F)l(or)13 b(example,)j(if)f(it)59 1986 y(is)h(an)o(ticipated)g(that)e(the)i(range)f(of)f(v)m(alues)j(for) d(the)h(iterates)g Fi(x)1164 1993 y Fd(k)1201 1986 y Fl(is)512 2080 y Fi(x)538 2087 y Fq(1)595 2080 y Ff(2)36 b Fl([)p Ff(\000)p Fl(10)755 2063 y Fq(10)816 2080 y Fi(;)31 b Fl(10)906 2063 y Fq(10)943 2080 y Fl(])512 2136 y Fi(x)538 2143 y Fq(2)595 2136 y Ff(2)36 b Fl([)p Ff(\000)p Fl(10)755 2120 y Fq(2)798 2136 y Fi(;)c Fl(10)889 2120 y Fq(4)908 2136 y Fl(])512 2193 y Fi(x)538 2200 y Fq(3)595 2193 y Ff(2)k Fl([)p Ff(\000)p Fl(6)10 b Ff(\002)h Fl(10)834 2176 y Fm(\000)p Fq(6)880 2193 y Fi(;)31 b Fl(9)10 b Ff(\002)h Fl(10)1049 2176 y Fm(\000)p Fq(6)1095 2193 y Fl(])59 2286 y(then)16 b(an)f(appropriate)g(c)o(hoice)h(will)h (b)q(e)f Fj(TYPX)e Fl(=)i(\(1.0)p Fj(E)p Fl(+10,)d(1.0)p Fj(E)p Fl(+3,)h(7.0)p Fj(E)p Fl(-6\).)59 2399 y Fj(FSCALE)p Ff(!)p Fl(:)29 b(A)20 b(p)q(ositiv)o(e)h(real)f(n)o(um)o(b)q(er)g (estimating)g(the)g(magnitude)h(of)e Fi(f)5 b Fl(\()p Fi(x)p Fl(\))20 b(near)g(the)g(minimizer)h Fi(x)1916 2406 y Fm(\003)1936 2399 y Fl(.)59 2456 y(It)16 b(is)h(used)f(in)h(the) f(gradien)o(t)g(stopping)h(condition)g(giv)o(en)g(b)q(elo)o(w.)23 b(If)16 b Fi(f)5 b Fl(\()p Fi(x)1344 2463 y Fq(0)1364 2456 y Fl(\))15 b(is)i(m)o(uc)o(h)f(greater)f(than)h Fi(f)5 b Fl(\()p Fi(x)1899 2463 y Fm(\003)1919 2456 y Fl(\),)59 2512 y Fj(FSCALE)15 b Fl(should)i(b)q(e)f(appro)o(ximately)g Fi(f)5 b Fl(\()p Fi(x)796 2519 y Fm(\003)815 2512 y Fl(\).)21 b(If)16 b(a)f(negativ)o(e)h(v)m(alue)h(is)f(sp)q(eci\014ed)i(for)d Fj(FSCALE)p Fl(,)g(its)g(absolute)59 2569 y(v)m(alue)h(is)g(used.)21 b(When)15 b(0.0)f(is)i(sp)q(eci\014ed,)h(1.0)d(will)j(b)q(e)f(used.)21 b(The)15 b(program)f(will)j(not)e(ab)q(ort.)993 2758 y(4)p eop %%Page: 5 5 5 4 bop 59 199 a Fj(GRADTL)p Ff(!)p Fl(:)33 b(P)o(ositiv)o(e)22 b(scalar)f(giving)i(the)f(tolerance)g(at)f(whic)o(h)i(the)f(scaled)h (gradien)o(t)f(of)f Fi(f)5 b Fl(\()p Fi(x)p Fl(\))21 b(is)h(con-)59 256 y(sidered)15 b(close)f(enough)f(to)g(zero)g(to)g (terminate)g(the)g(algorithm.)20 b(The)13 b(scaled)h(gradien)o(t)g(is)g (a)f(measure)g(of)g(the)59 312 y(relativ)o(e)18 b(c)o(hange)g(in)h Fi(f)j Fl(in)d(eac)o(h)f(direction)h Fi(x)855 319 y Fd(i)887 312 y Fl(divided)h(b)o(y)d(the)h(relativ)o(e)g(c)o(hange)g(in)h Fi(x)1595 319 y Fd(i)1609 312 y Fl(.)27 b(More)17 b(precisely)l(,)59 369 y(the)e(test)g(used)h(b)o(y)f(the)g(program)f(is)508 484 y(max)545 513 y Fd(i)600 425 y Fc(\032)657 454 y Ff(j)f(r)p Fi(f)5 b Fl(\()p Fi(x)p Fl(\))12 b Ff(j)835 461 y Fd(i)876 454 y Fl(max)p Ff(fj)g Fi(x)1035 461 y Fd(i)1061 454 y Ff(j)p Fi(;)c Fj(TYPX)1190 461 y Fd(i)1204 454 y Ff(g)p 657 474 570 2 v 756 515 a Fl(max)p Ff(fj)k Fi(f)18 b Ff(j)p Fi(;)8 b Fj(FSCALE)n Ff(g)1252 425 y Fc(\033)1296 484 y Ff(\024)13 b Fj(GRADTL)o Fi(:)59 610 y Fl(The)19 b(mo)q(dule)h Fj(STDFLT)d Fl(returns)i(the)g(v)m(alue)h Fi(\017)863 594 y Fq(1)p Fd(=)p Fq(3)918 610 y Fl(.)30 b(If)19 b(the)g(user)g(sp)q(eci\014es)h(a)e(negativ)o(e)h(v)m(alue,)h (the)f(default)59 667 y(v)m(alue)d(is)g(used)g(instead.)59 780 y Fj(STEPTL)p Ff(!)p Fl(:)31 b(A)22 b(p)q(ositiv)o(e)g(scalar)f (pro)o(viding)h(the)f(minim)o(um)h(allo)o(w)o(able)g(relativ)o(e)g (step)f(length.)39 b Fj(STEPTL)59 836 y Fl(should)16 b(b)q(e)f(at)g(least)g(as)f(small)i(as)e(10)701 820 y Fm(\000)p Fd(d)748 836 y Fl(,)h(where)g Fi(d)f Fl(is)i(the)e(n)o(um)o (b)q(er)i(of)e(accurate)h(digits)g(the)g(user)g(desires)h(in)59 892 y(the)f(solution)h Fi(x)335 899 y Fm(\003)355 892 y Fl(.)k(The)15 b(actual)h(test)e(used)i(is)617 1015 y(max)654 1044 y Fd(i)709 943 y Fc(\()817 987 y Ff(j)p Fi(x)856 994 y Fd(i)870 970 y(k)902 987 y Ff(\000)10 b Fi(x)973 994 y Fd(i)987 970 y(k)q Fm(\000)p Fq(1)1054 987 y Ff(j)p 769 1007 347 2 v 769 1049 a Fl(max)o Ff(fj)p Fi(x)915 1056 y Fd(i)929 1035 y(k)950 1049 y Fi(;)e Fj(TYPX)1066 1056 y Fd(i)1080 1049 y Ff(jg)1141 943 y Fc(\))1187 1015 y Ff(\024)13 b Fj(STEPTL)o Fi(;)59 1147 y Fl(where)21 b Fi(x)222 1131 y Fd(k)264 1147 y Fl(and)f Fi(x)383 1131 y Fd(k)q Fm(\000)p Fq(1)470 1147 y Fl(are)g(the)h(new)f(and)h(old)g (iterates,)g(resp)q(ectiv)o(ely)l(.)38 b(The)21 b(program)e(ma)o(y)h (terminate)59 1204 y(prematurely)c(if)h Fj(STEPTL)e Fl(is)i(to)q(o)e (large.)23 b(Mo)q(dule)17 b Fj(STDFLT)e Fl(returns)h(the)g(v)m(alue)h Fi(\017)1474 1187 y Fq(2)p Fd(=)p Fq(3)1530 1204 y Fl(.)22 b(If)17 b(the)f(user)g(sp)q(eci\014es)59 1260 y(a)f(negativ)o(e)g(v)m (alue,)h(then)g(the)f(default)h(v)m(alue)g(is)g(used)g(instead.)59 1373 y Fj(ILIM)p Ff(!)p Fl(:)29 b(P)o(ositiv)o(e)20 b(in)o(teger)g(sp)q (ecifying)i(the)e(maxim)o(um)f(iterations)h(to)f(b)q(e)i(p)q(erformed)f (b)q(efore)g(the)g(pro-)59 1430 y(gram)d(is)i(terminated.)30 b(Mo)q(dule)19 b Fj(STDFLT)e Fl(returns)h Fj(ILIM)24 b(=)f(500)p Fl(.)29 b(If)19 b(the)f(user)g(sp)q(eci\014es)j Fj(ILIM)c Ff(\024)h Fl(0,)h(the)59 1486 y(default)d(v)m(alue)g(is)g (used)g(instead.)59 1599 y Fj(STEPMX)p Ff(!)p Fl(:)h(A)11 b(p)q(ositiv)o(e)g(scalar)f(pro)o(viding)i(the)e(maxim)o(um)h(allo)o(w) o(able)g(scaled)g(step)g(length)g Ff(jj)p Fi(D)1700 1606 y Fd(x)1721 1599 y Fl(\()p Fi(x)1765 1606 y Fq(+)1794 1599 y Ff(\000)q Fi(x)1856 1606 y Fd(c)1874 1599 y Fl(\))p Ff(jj)1918 1606 y Fq(2)1936 1599 y Fl(,)59 1655 y(where)19 b Fi(D)232 1662 y Fd(x)272 1655 y Fl(=)f Fi(diag)r Fl(\()p Fj(1/TYPX)575 1662 y Fq(1)592 1655 y Fj(,)p Fi(:)8 b(:)g(:)n Fj(,)24 b(1/TYPX)861 1662 y Fd(n)884 1655 y Fl(\).)29 b Fj(STEPMX)18 b Fl(is)h(used)g(to)f(prev)o(en)o(t)g(steps)h(that)f(w)o (ould)h(cause)59 1712 y(the)14 b(optimization)h(problem)f(to)g(o)o(v)o (er\015o)o(w,)e(to)h(prev)o(en)o(t)h(the)g(algorithm)g(from)f(lea)o (ving)i(the)f(area)f(of)h(in)o(terest)59 1768 y(in)19 b(parameter)e(space,)i(or)e(to)h(detect)g(div)o(ergence)h(in)g(the)f (algorithm.)28 b Fj(STEPMX)17 b Fl(should)i(b)q(e)g(c)o(hosen)f(small) 59 1825 y(enough)i(to)f(prev)o(en)o(t)h(these)g(o)q(ccurrences)h(but)e (should)i(b)q(e)g(larger)e(than)h(an)o(y)f(an)o(ticipated)i (\\reasonable")59 1881 y(step.)j(The)17 b(algorithm)f(will)i(halt)f (and)g(pro)o(vide)g(a)f(diagnostic)h(if)g(it)g(attempts)e(to)h(exceed)i Fj(STEPMX)d Fl(on)i(\014v)o(e)59 1938 y(successiv)o(e)f(iterations.)21 b(If)15 b(a)g(nonp)q(ositiv)o(e)h(v)m(alue)h(is)f(sp)q(eci\014ed)h(for) d Fj(STEPMX)p Fl(,)g(the)i(default)g(is)f(used.)21 b(Mo)q(dule)59 1994 y Fj(STDFLT)13 b Fl(returns)h(the)f(v)m(alue)i Fj(STEPMX)e Fl(=)h(max)p Ff(fjj)p Fi(x)930 2001 y Fq(0)949 1994 y Ff(jj)975 2001 y Fq(2)1001 1994 y Ff(\001)7 b Fl(10)1067 1978 y Fq(3)1086 1994 y Fi(;)h Fl(10)1153 1978 y Fq(3)1171 1994 y Ff(g)p Fl(,)13 b(where)h Fi(x)1376 2001 y Fq(0)1410 1994 y Fl(is)g(the)g(initial)i(appro)o(ximation)59 2051 y(pro)o(vided)g(b)o(y)f(the)g(user.)59 2164 y Fj(IPR)p Ff(!)p Fl(:)22 b(The)16 b(unit)h(on)f(whic)o(h)h(the)f(routine)g (outputs)g(information.)23 b Fj(STDFLT)15 b Fl(returns)h(the)g(v)m (alue)h(6,)f(whic)o(h)59 2220 y(is)g(the)f(standard)g Fj(FORTRAN)f Fl(unit)i(for)f(the)g(prin)o(ter.)59 2333 y Fj(METHOD)p Ff(!)p Fl(:)20 b(An)15 b(in)o(teger)g(\015ag)g (designating)h(whic)o(h)g(metho)q(d)g(to)e(use.)127 2386 y Ff(\017)23 b Fj(METHOD)14 b Fl(=)i(0)e(:)20 b(Use)c(Newton's)e(metho) q(d.)127 2444 y Ff(\017)23 b Fj(METHOD)14 b Fl(=)i(1)e(:)20 b(Use)c(the)f(tensor)g(metho)q(d.)59 2498 y(Mo)q(dule)k Fj(STDFLT)f Fl(returns)g(v)m(alue)i(1.)29 b(If)19 b(the)g(user)f(sp)q (eci\014es)j(an)d(illegal)j(v)m(alue,)f(mo)q(dule)f Fj(OPTCHK)f Fl(will)i(set)59 2555 y Fj(METHOD)14 b Fl(to)h(1;)f(the)i(program)e (will)j(not)d(ab)q(ort.)59 2667 y Fj(GRDFLG)p Ff(!)p Fl(:)j(In)o(teger)10 b(\015ag)h(designating)g(whether)f(or)g(not)g (analytic)i(Hessian)f(has)f(b)q(een)h(supplied)i(b)o(y)e(the)f(user.) 993 2758 y(5)p eop %%Page: 6 6 6 5 bop 127 199 a Ff(\017)23 b Fj(GRDFLG)14 b Fl(=)i(0)e(:)20 b(No)15 b(analytic)h(gradien)o(t)f(supplied.)127 260 y Ff(\017)23 b Fj(GRDFLG)10 b Fl(=)i(1)f(:)18 b(Analytic)13 b(gradien)o(t)e(supplied)j(\(will)e(b)q(e)h(c)o(hec)o(k)o(ed)e(against) g(\014nite)i(di\013erence)g(gradien)o(t.\))127 320 y Ff(\017)23 b Fj(GRDFLG)d Fl(=)h(2)g(:)31 b(Analytic)22 b(gradien)o(t)f(supplied)i(\(will)g(not)d(b)q(e)i(c)o(hec)o(k)o(ed)f (against)g(\014nite)h(di\013erence)173 376 y(gradien)o(t.\))59 437 y(When)16 b Fj(GRDFLG)e Fl(=)i(0,)f(the)g(gradien)o(t)h(is)g (obtained)g(b)o(y)f(forw)o(ard)f(\014nite)j(di\013erences.)k(When)16 b Fj(GRDFLG)e Fl(=)i(1)f(or)59 493 y(2,)h(the)g(name)h(of)e(the)i(user) f(supplied)j(routine)d(that)g(ev)m(aluates)h Ff(r)p Fi(f)5 b Fl(\()p Fi(x)p Fl(\))16 b(m)o(ust)f(b)q(e)i(supplied)i(in)e Fj(UGR)p Fl(.)e(When)59 549 y Fj(GRDFLG)g Fl(=)g(1,)g(the)h(program)e (compares)i(the)f(v)m(alue)i(of)e(the)g(user's)h(analytic)g(gradien)o (t)f(routine)h(at)f Fi(x)1825 556 y Fq(0)1860 549 y Fl(with)59 606 y(a)h(\014nite)g(di\013erence)h(estimate)f(and)g(ab)q(orts)f(if)h (the)g(relativ)o(e)g(di\013erence)h(b)q(et)o(w)o(een)g(an)o(y)e(t)o(w)o (o)f(comp)q(onen)o(ts)i(is)59 662 y(greater)d(than)g(0.01.)18 b(The)c(mo)q(dule)h Fj(STDFLT)e Fl(returns)g Fj(GRDFLG)g Fl(=)h(0.)19 b(If)14 b(the)f(user)h(sp)q(eci\014es)i(an)d(illegal)j(v)m (alue,)59 719 y(the)f(mo)q(dule)i Fj(OPTCHK)d Fl(supplies)j(the)f(v)m (alue)g(0.)59 832 y Fj(HSNFLG)p Ff(!)p Fl(:)h(In)o(teger)10 b(\015ag)h(designating)g(whether)f(or)g(not)g(analytic)i(Hessian)f(has) f(b)q(een)h(supplied)i(b)o(y)e(the)f(user.)127 892 y Ff(\017)23 b Fj(HSNFLG)14 b Fl(=)i(0)e(:)20 b(No)15 b(analytic)h (Hessian)g(supplied.)127 952 y Ff(\017)23 b Fj(HSNFLG)12 b Fl(=)i(1)f(:)19 b(Analytic)14 b(Hessian)g(supplied)i(\(will)f(b)q(e)f (c)o(hec)o(k)o(ed)g(against)f(\014nite)h(di\013erence)h(Hessian.\))127 1013 y Ff(\017)23 b Fj(HSNFLG)e Fl(=)h(2)f(:)33 b(Analytic)23 b(Hessian)g(supplied)h(\(will)f(not)f(b)q(e)g(c)o(hec)o(k)o(ed)g (against)g(\014nite)h(di\013erence)173 1069 y(Hessian.\))59 1129 y(When)13 b Fj(HSNFLG)e Fl(=)i(0,)g(the)f(Hessian)h(v)m(alues)h (are)e(computed)h(b)o(y)f(forw)o(ard)f(\014nite)j(di\013erences)f (based)g(on)g(gradi-)59 1186 y(en)o(t)g(v)m(alues.)20 b(When)13 b Fj(HSNFLG)f Fl(=)h(1)g(or)f(2,)h(the)g(name)f(of)h(the)g (user{supplied)i(routine)e(that)f(ev)m(aluates)i Ff(r)1841 1169 y Fq(2)1860 1186 y Fi(f)5 b Fl(\()p Fi(x)p Fl(\))59 1242 y(m)o(ust)14 b(b)q(e)i(supplied)h(in)e Fj(USH)p Fl(.)f(When)h Fj(HSNFLG)f Fl(=)h(1,)g(the)f(program)g(compares)g(the)h (v)m(alue)h(of)f(the)g(user's)f(ana-)59 1299 y(lytic)j(Hessian)f (routine)g(at)f Fi(x)567 1306 y Fq(0)603 1299 y Fl(with)g(a)h(\014nite) g(di\013erence)h(estimate)f(and)g(ab)q(orts)f(if)h(the)f(relativ)o(e)i (di\013erence)59 1355 y(b)q(et)o(w)o(een)f(an)o(y)f(t)o(w)o(o)g(comp)q (onen)o(ts)g(is)i(greater)d(than)i(0.01.)k(The)c(mo)q(dule)h Fj(STDFLT)d Fl(returns)i Fj(HSNFLG)f Fl(=)h(0.)21 b(If)59 1412 y(the)15 b(user)h(sp)q(eci\014es)h(an)e(illegal)i(v)m(alue,)f(the) f(mo)q(dule)i Fj(OPTCHK)d Fl(supplies)j(the)f(v)m(alue)g(0.)59 1524 y Fj(NDIGIT)p Ff(!)p Fl(:)31 b(In)o(teger)21 b(estimating)h(the)f (n)o(um)o(b)q(er)g(of)g(accurate)g(digits)h(on)f(the)g(ob)s(jectiv)o(e) g(function)h Fi(f)5 b Fl(\()p Fi(x)p Fl(\).)59 1581 y Fj(STDFLT)12 b Fl(returns)h(the)g(v)m(alue)h(-)p Fj(LOG)648 1588 y Fq(10)685 1581 y Fl(\()p Fi(\017)p Fl(\),)f(where)g Fi(\017)g Fl(is)h(mac)o(hine)f(precision.)21 b(If)13 b Fj(NDIGIT)f Ff(\024)h Fl(0)g(then)g(the)g(default)59 1637 y(v)m(alue)j(is)g(used)g(instead.)59 1750 y Fj(*MSG)p Ff( )-8 b(!)p Fl(:)34 b(An)23 b(in)o(teger)f(v)m(ariable)h(that)f(the)g (user)g(ma)o(y)f(set)h(on)g(input)h(to)f(inhibit)i(certain)e(automatic) 59 1807 y(c)o(hec)o(ks)e(or)g(o)o(v)o(erride)g(certain)g(default)h(c)o (haracteristics)f(of)g(the)g(pac)o(k)m(age.)34 b(Curren)o(tly)l(,)22 b(three)e(\\message")59 1863 y(features)15 b(can)g(b)q(e)h(used)g (individuall)q(y)i(or)d(in)h(com)o(bination.)127 1924 y Ff(\017)23 b Fj(MSG)14 b Fl(=)i(0)f(:)20 b(No)15 b(output)g(will)h(b) q(e)g(pro)q(duced.)127 1984 y Ff(\017)23 b Fj(MSG)14 b Fl(=)i(1)f(:)20 b(Prin)o(t)15 b(the)g(input)h(state,)e(the)i(\014nal) g(results,)f(and)g(the)g(stopping)h(conditions.)127 2044 y Ff(\017)23 b Fj(MSG)11 b Fl(=)h(2)g(:)18 b(Prin)o(t)12 b(the)g(in)o(termediate)h(results,)g(that)e(is,)i(the)f(input)h(state,) e(the)h(v)m(alues)h(of)f(the)g(ob)s(jectiv)o(e)173 2100 y(function)23 b(and)f(the)g(scaled)h(gradien)o(t)f(at)g(eac)o(h)g (iteration,)i(and)e(the)g(\014nal)h(results)g(including)i(the)173 2157 y(stopping)15 b(conditions)i(and)e(the)g(n)o(um)o(b)q(er)h(of)f (function,)g(gradien)o(t,)g(and)h(Hessian)f(ev)m(aluations.)59 2221 y(The)h(mo)q(dule)g Fj(STDFLT)e Fl(returns)h(a)g(v)m(alue)i(of)d (1.)20 b(On)c(output,)e(if)i(the)f(program)f(has)i(terminated)f(b)q (ecause)h(of)59 2277 y(erroneous)f(input,)h Fj(MSG)f Fl(con)o(tains)g(an)g(error)g(co)q(de)g(indicating)j(the)d(reason:)127 2338 y Ff(\017)23 b Fj(MSG)14 b Fl(=)i(-1)f(:)20 b(Illegal)d(dimension) g Fj(N)p Fl(;)d Fj(N)h Ff(\024)e Fl(0.)20 b(The)15 b(program)f(ab)q (orts.)127 2398 y Ff(\017)23 b Fj(MSG)14 b Fl(=)i(-2)f(:)20 b(Illegal)d(length)f(of)f Fj(LIRN)f Fl(or)h Fj(LICN)p Fl(;)f Fj(LIRN)h Ff(\024)e Fl(0)i(or)f Fj(LICN)h Ff(\024)e Fl(0.)19 b(The)d(program)e(ab)q(orts.)127 2458 y Ff(\017)23 b Fj(MSG)15 b Fl(=)h(-3)f(:)21 b(Illegal)d(length)e(of)g Fj(LIWRK)f Fl(or)g Fj(LWRK)p Fl(;)f Fj(LIWRK)h Fi(<)h Fj(2*LIRN+12*N+2)e Fl(or)i Fj(LWRK)f Fi(<)h Fj(7*N)p Fl(.)f(The)173 2514 y(program)f(ab)q(orts.)127 2575 y Ff(\017)23 b Fj(MSG)14 b Fl(=)i(-4)f(:)20 b(Illegal)d(n)o(um)o(b)q(er)e (of)g(nonzeros)g Fj(NZ)p Fl(;)g Fj(NZ)g Ff(\024)e Fl(0.)19 b(The)d(program)e(ab)q(orts.)993 2758 y(6)p eop %%Page: 7 7 7 6 bop 127 184 a Ff(\017)23 b Fj(MSG)15 b Fl(=)i(-5)f(:)21 b(The)c Fj(K)p Fl(-th)f(elemen)o(t)h(of)e Fj(IRN)h Fl(or)f(the)i Fj(K)p Fl(-th)f(elemen)o(t)g(of)g Fj(ICN)g Fl(is)g(not)g(an)g(in)o (teger)g(b)q(et)o(w)o(een)173 241 y(1)h(and)h Fj(N)p Fl(;)f(\()p Fj(IRN\(K\))23 b Fi(<)h Fj(1)17 b Fl(or)g Fj(IRN\(K\))23 b Fi(>)h Fj(N)p Fl(\))18 b(or)f(\()p Fj(ICN\(K\))22 b Fi(<)j Fj(1)17 b Fl(or)g Fj(ICN\(K\))23 b Fi(>)h Fj(N)p Fl(\).)17 b(The)h(program)173 297 y(ab)q(orts.)127 358 y Ff(\017)23 b Fj(MSG)15 b Fl(=)g(-6)h(:)k(The)c Fj(K)p Fl(-th)f(diagonal)h(elemen)o(t)g(is)g(not)f(in)i(the)e(sparsit)o(y)g (pattern.)20 b(This)c(is)g(c)o(hec)o(k)o(ed)g(only)173 414 y(if)i Fj(HSNFLG)23 b(=)h(0)18 b Fl(b)q(ecause)h(the)f(\014nite)g (di\013erence)i(Hessian)e(appro)o(ximation)g(require)h(that)e(diagonal) 173 470 y(elemen)o(ts)f(b)q(e)f(in)h(the)g(sparsit)o(y)f(pattern.)k (The)c(program)f(ab)q(orts.)127 531 y Ff(\017)23 b Fj(MSG)f Fl(=)g(-7)g(:)34 b(Redundan)o(t)23 b(en)o(tries)g(in)g(sparsit)o(y)f (pattern)g(w)o(as)f(encoun)o(tered.)42 b(When)23 b Fj(HSNFLG)g(=)173 587 y(1)c Fl(or)g Fj(HSNFLG)k(=)h(2)p Fl(,)c(the)g(program)e(ab)q (orts.)32 b(When)20 b Fj(HSNFLG)j(=)h(0)p Fl(,)c(the)g(program)e (eliminates)k(the)173 644 y(redundan)o(t)15 b(en)o(tries)h(and)f(con)o (tin)o(ue)h(the)f(execution)i(\(no)d(error)h(message)g(is)g(rep)q (orted)h(in)g(this)f(case\).)127 704 y Ff(\017)23 b Fj(MSG)13 b Fl(=)h(-8)g(:)19 b(Probable)14 b(co)q(ding)h(error)f(in)g(the)g (user's)g(analytic)g(gradien)o(t)g(routine)h Fj(UGR)p Fl(.)e(Analytic)i(and)173 760 y(\014nite)20 b(di\013erence)h(gradien)o (t)e(do)g(not)g(agree)g(within)i(a)e(tolerance)h(of)f(0.01.)31 b(The)20 b(program)e(ab)q(orts.)173 817 y(\(This)d(c)o(hec)o(k)h(can)f (b)q(e)h(o)o(v)o(erridden)g(b)o(y)f(setting)g Fj(GRDFLG)f Fl(=)i(2.\))127 877 y Ff(\017)23 b Fj(MSG)14 b Fl(=)h(-9)g(:)k (Probable)c(co)q(ding)h(error)e(in)i(the)e(user's)h(analytic)g(Hessian) h(routine)f Fj(USH)p Fl(.)f(Analytic)i(and)173 934 y(\014nite)21 b(di\013erence)g(Hessian)g(do)f(not)g(agree)g(within)h(a)f(tolerance)h (of)f(0.01.)33 b(The)21 b(program)e(ab)q(orts.)173 990 y(\(This)c(c)o(hec)o(k)h(can)f(b)q(e)h(o)o(v)o(erridden)g(b)o(y)f (setting)g Fj(HSNFLG)f Fl(=)i(2.\))59 1107 y Fj(*XPLS)p Ff( )p Fl(:)28 b(An)20 b(arra)o(y)e(of)h(length)h Fj(N)f Fl(con)o(taining)h(the)g(b)q(est)f(appro)o(ximation)g(to)g(the)h (minimizer)h Fi(x)1810 1114 y Fm(\003)1849 1107 y Fl(up)q(on)59 1163 y(return.)f(\(If)15 b(the)g(algorithm)g(has)h(not)e(con)o(v)o (erged,)h(the)g(last)g(iterate)g(is)h(returned.\))59 1276 y Fj(*FPLS)p Ff( )p Fl(:)k(A)15 b(scalar)g(v)m(ariable)i(that)d (con)o(tains)h(the)h(function)g(v)m(alue)g(at)f(the)g(\014nal)h (iterate)f Fj(XPLS)p Fl(.)59 1389 y Fj(*GPLS)p Ff( )p Fl(:)20 b(An)15 b(arra)o(y)f(of)h(length)h Fj(N)f Fl(con)o(taining)h (the)f(gradien)o(t)g(v)m(alue)i(at)d Fj(XPLS)p Fl(.)59 1502 y Fj(HESS)p Ff( )-8 b(!)p Fl(:)32 b(An)21 b(arra)o(y)e(that)h(is)h (used)g(to)f(store)g(the)h(Hessian)h(matrix)e(at)g(eac)o(h)h (iteration.)36 b(It)21 b(needs)g(to)59 1558 y(b)q(e)d(at)f(least)g(of)g (dimension)i Fj(LICN)p Fl(.)e(Only)h(the)f(nonzeros)h(in)g(the)f(lo)o (w)o(er)g(or)g(upp)q(er)i(triangular)e(part)g(of)g(the)59 1615 y(Hessian)f(matrix)e(is)h(stored)f(in)i Fj(HESS)p Fl(.)e(On)h(en)o(try)l(,)f(these)h(nonzeros)g(ma)o(y)f(b)q(e)i(giv)o (en)f(in)h(an)o(y)e(order.)19 b(On)d(exit,)59 1671 y Fj(HESS)h Fl(con)o(tains)g(the)g(Hessian)h(matrix)f(at)f(the)i (minimizer)h Fi(x)1128 1678 y Fm(\003)1165 1671 y Fl(with)e(the)g (nonzeros)h(sorted)e(b)o(y)h(columns)h(if)59 1728 y Fj(HSNFLG)c Fl(w)o(as)h(set)g(to)f(0.)59 1841 y Fj(*WRK)p Ff(!)p Fl(:)19 b(An)c(arra)o(y)e(of)h(length)i Fj(LWRK)p Fl(.)d(This)i(is)g (used)g(as)f(w)o(orkspace)g(b)o(y)g(the)h(pac)o(k)m(age.)20 b(Its)14 b(length)h(m)o(ust)f(b)q(e)59 1897 y(at)h(least)g Fj(8*N)g Fl(if)g(the)h Fj(STUMSD)e Fl(in)o(terface)h(is)h(used)g(and)f (at)g(least)g Fj(7*N)g Fl(if)h(the)f Fj(STUMCD)f Fl(in)o(terface)i(is)f (used.)59 2010 y Fj(*LWRK)p Ff(!)p Fl(:)29 b(An)20 b(in)o(teger)g(v)m (ariable.)35 b(It)20 b(m)o(ust)f(b)q(e)i(set)f(b)o(y)f(the)h(user)g(to) f(the)h(length)h(of)e(arra)o(y)g Fj(WRK)g Fl(and)h(is)59 2066 y(not)15 b(altered)h(b)o(y)f(the)g(pac)o(k)m(age.)59 2179 y Fj(*IWRK)p Ff(!)p Fl(:)31 b(An)21 b(in)o(teger)h(arra)o(y)d(of)i (length)g Fj(LIWRK)p Fl(.)f(This)i(is)f(used)h(as)e(w)o(orkspace)h(b)o (y)g(the)g(pac)o(k)m(age.)37 b(Its)59 2236 y(length)16 b(m)o(ust)f(b)q(e)g(at)g(least)g Fj(2*LIRN+12*N+2)p Fl(.)59 2349 y Fj(*LIWRK)p Ff(!)p Fl(:)29 b(An)20 b(in)o(teger)g(v)m(ariable.) 36 b(It)20 b(m)o(ust)f(b)q(e)i(set)f(b)o(y)g(the)g(user)g(to)f(the)h (length)h(of)f(arra)o(y)e Fj(IWRK)i Fl(and)59 2405 y(is)c(not)f (altered)g(b)o(y)g(the)h(pac)o(k)m(age.)59 2518 y Fj(*TERMCD)p Ff( )p Fl(:)j(An)d(in)o(teger)f(that)g(sp)q(eci\014es)i(the)e(reason)g (wh)o(y)g(the)g(algorithm)g(has)g(terminated.)127 2579 y Ff(\017)23 b Fj(TERMCD)14 b Fl(=)i(1)e(:)20 b(The)c(scaled)g(gradien) o(t)f(at)f(the)i(\014nal)g(iterate)f(w)o(as)f(less)i(than)f Fj(GRADTL)p Fl(.)127 2639 y Ff(\017)23 b Fj(TERMCD)14 b Fl(=)i(2)e(:)20 b(The)c(length)g(of)e(the)i(last)f(step)g(w)o(as)f (less)i(than)f Fj(STEPTL)p Fl(.)993 2758 y(7)p eop %%Page: 8 8 8 7 bop 127 199 a Ff(\017)23 b Fj(TERMCD)17 b Fl(=)h(3)f(:)26 b(Last)17 b(global)i(step)f(failed)h(to)e(lo)q(cate)h(a)g(p)q(oin)o(t)g (lo)o(w)o(er)g(than)g Fj(XPLS)p Fl(.)f(It)g(is)i(lik)o(ely)g(that)173 256 y(either)d Fj(XPLS)e Fl(is)i(an)f(appro)o(ximate)g(solution)h(of)f (the)g(function)h(or)f Fj(STEPTL)f Fl(is)i(to)q(o)e(large.)127 316 y Ff(\017)23 b Fj(TERMCD)14 b Fl(=)i(4)e(:)20 b(The)c(iteration)f (limit)i(has)e(b)q(een)h(exceeded.)127 376 y Ff(\017)23 b Fj(TERMCD)14 b 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y(CALL)g(STDFLT\(N,TYPX,FSCALE,GR)o(ADTL,ST)o(EPTL,IL)o(IM,STEPM)o (X,)178 2006 y(*)287 b(IPR,METHOD,GRDFLG,HSNFL)o(G,NDIGI)o(T,INFORM)o (,MSG\))59 2119 y(CD)95 b(GRADTL)23 b(=)h(1.0D-5)59 2175 y(CS)95 b(GRADTL)23 b(=)h(1.0E-3)202 2232 y(GRDFLG)f(=)h(2)59 2345 y(C)48 b(CALL)23 b(THE)g(SPARSE)g(OPTIMIZER.)59 2457 y(CD)95 b(CALL)24 b(STUMCD\(N,X,NZ,IRN,LIRN,)o(ICN,LIC)o(N,DODCF)o (,DODCG,)59 2514 y(CS)95 b(CALL)24 b(STUMCS\(N,X,NZ,IRN,LIRN,)o (ICN,LIC)o(N,DODCF)o(,DODCG,S)o(TDUSH,)178 2570 y(*)287 b(TYPX,FSCALE,GRADTL,STEP)o(TL,ILIM)o(,STEPMX,)o(IPR,)178 2627 y(*)g(METHOD,GRDFLG,HSNFLG,ND)o(IGIT,MS)o(G,XPLS,F)o(PLS,GPL)o(S,) 981 2758 y Fl(16)p eop %%Page: 17 17 17 16 bop 178 199 a Fj(*)287 b(HESS,WRK,LWRK,IWRK,LIWR)o(K,TERMC)o (D,HTV,IN)o(FORM\))202 312 y(STOP)202 369 y(END)209 502 y Fl(Figure)15 b(A.1:)k(Co)q(de)d(to)e(solv)o(e)i(the)f(optimal)h (design)g(with)f(comp)q(osite)h(materials)f(problem)59 616 y(If)g(w)o(e)f(use)h(the)g(double{precision)i(v)o(ersion)e(of)f 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272 a %%BeginDocument: odc_last.eps % MathWorks dictionary /MathWorks 150 dict begin % definition operators /bdef {bind def} bind def /ldef {load def} bind def /xdef {exch def} bdef /xstore {exch store} bdef % operator abbreviations /c /clip ldef /cc /concat ldef /cp /closepath ldef /gr /grestore ldef /gs /gsave ldef /mt /moveto ldef /np /newpath ldef /cm /currentmatrix ldef /sm /setmatrix ldef /rc {rectclip} bdef /rf {rectfill} bdef /rm /rmoveto ldef /rl /rlineto ldef /s /show ldef /sc {setcmykcolor} bdef /sr /setrgbcolor ldef /w /setlinewidth ldef /cap /setlinecap ldef % page state control /pgsv () def /bpage {/pgsv save def} bdef /epage {pgsv restore} bdef /bplot /gsave ldef /eplot {stroke grestore} bdef % orientation switch /portraitMode 0 def /landscapeMode 1 def % coordinate system mappings /dpi2point 0 def % font control /FontSize 0 def /FMS { /FontSize xstore %save size off stack findfont [FontSize 0 0 FontSize neg 0 0] makefont setfont }bdef /reencode { exch dup where {pop load} {pop StandardEncoding} ifelse exch dup 3 1 roll findfont dup length dict begin { 1 index /FID ne {def}{pop pop} ifelse } forall /Encoding exch def currentdict end definefont pop } bdef /isroman { findfont /CharStrings get /Agrave known } bdef /FMSR { 3 1 roll 1 index dup isroman {reencode} {pop pop} ifelse exch FMS } bdef /csm { 1 dpi2point div -1 dpi2point div scale neg translate landscapeMode eq {90 rotate} if } bdef % line types: solid, dotted, dashed, dotdash /SO { [] 0 setdash } bdef /DO { [.5 dpi2point mul 4 dpi2point mul] 0 setdash } bdef /DA { [6 dpi2point mul] 0 setdash } bdef /DD { [.5 dpi2point mul 4 dpi2point mul 6 dpi2point mul 4 dpi2point mul] 0 setdash } bdef % macros for lines and objects /L { lineto stroke } bdef /MP { 3 1 roll moveto 1 sub {rlineto} repeat } bdef /AP { {rlineto} repeat } bdef /PP { closepath fill } bdef /DP { closepath stroke } bdef /MR { 4 -2 roll moveto dup 0 exch rlineto exch 0 rlineto neg 0 exch rlineto closepath } bdef /FR { MR stroke } bdef /PR { MR fill } bdef /L1i { { currentfile picstr readhexstring pop } image } bdef /tMatrix matrix def /MakeOval { newpath tMatrix currentmatrix pop translate scale 0 0 1 0 360 arc tMatrix setmatrix } bdef /FO { MakeOval stroke } bdef /PO { MakeOval fill } bdef /PD { 2 copy moveto lineto stroke } bdef currentdict end def MathWorks begin 0 cap 0 setlinejoin end MathWorks begin bpage bplot /dpi2point 12 def portraitMode 0216 7344 csm 513 340 6135 4486 MR c np 76 dict begin %Colortable dictionary /c0 { 0 0 0 sr} bdef /c1 { 1 1 1 sr} bdef /c2 { 1 0 0 sr} bdef /c3 { 0 1 0 sr} bdef /c4 { 0 0 1 sr} bdef /c5 { 1 1 0 sr} bdef /c6 { 1 0 1 sr} bdef /c7 { 0 1 1 sr} bdef /Helvetica /ISOLatin1Encoding 144 FMSR c1 0 0 6914 5184 PR 6 w DO 4 w SO 6 w c0 3224 4612 mt 6256 3792 L 3224 4612 mt 898 3543 L 898 3543 mt 898 1208 L 3224 4612 mt 3296 4645 L 3327 4793 mt (0) s 3730 4475 mt 3801 4508 L 3832 4656 mt (20) s 4235 4339 mt 4306 4371 L 4338 4520 mt (40) s 4740 4202 mt 4812 4235 L 4843 4383 mt (60) s 5245 4065 mt 5317 4098 L 5348 4246 mt (80) s 5751 3928 mt 5822 3961 L 5853 4110 mt (100) s 6256 3792 mt 6327 3824 L 6359 3973 mt (120) s 3224 4612 mt 3149 4633 L 3035 4776 mt (0) s 2449 4256 mt 2373 4276 L 2180 4419 mt (50) s 1673 3899 mt 1598 3920 L 1324 4063 mt (100) s 898 3543 mt 822 3563 L 549 3706 mt (150) s 898 3543 mt 827 3510 L 715 3549 mt (0) s 898 3209 mt 827 3177 L 515 3215 mt (0.05) s 898 2876 mt 827 2843 L 595 2882 mt (0.1) s 898 2542 mt 827 2510 L 515 2548 mt (0.15) s 898 2209 mt 827 2176 L 595 2215 mt (0.2) s 898 1875 mt 827 1843 L 515 1881 mt (0.25) s 898 1542 mt 827 1509 L 595 1548 mt (0.3) s 898 1208 mt 827 1176 L 515 1214 mt (0.35) s gs 898 388 5359 4225 MR c np 16 -110 25 111 -16 29 4210 3048 4 MP gs /c8 { 1.000000 0.375000 0.000000 sr} bdef c8 PP gr c0 DP 16 -29 25 94 -15 2 4184 2981 4 MP gs /c9 { 1.000000 0.562500 0.000000 sr} bdef c9 PP gr c0 DP 15 -94 25 30 -15 67 4225 2981 4 MP gs c9 PP gr c0 DP 15 -67 26 67 -16 29 4200 2952 4 MP gs /c10 { 1.000000 0.656250 0.000000 sr} bdef c10 PP gr c0 DP 15 -2 25 84 -15 -8 4159 2907 4 MP gs /c11 { 1.000000 0.843750 0.000000 sr} bdef c11 PP gr c0 DP 16 -84 25 3 -16 73 4241 2908 4 MP gs c11 PP gr c0 DP 16 -29 25 74 -16 5 4175 2902 4 MP gs c11 PP gr c0 DP 16 -73 25 29 -15 50 4215 2902 4 MP gs c11 PP gr c0 DP 15 -50 25 50 -15 27 4190 2875 4 MP gs /c12 { 1.000000 0.937500 0.000000 sr} bdef c12 PP gr c0 DP 15 8 26 79 -16 -14 4134 2834 4 MP gs /c13 { 0.968750 1.000000 0.000000 sr} bdef c13 PP gr c0 DP 15 -78 25 -8 -15 73 4256 2835 4 MP gs c13 PP gr c0 DP 16 -5 25 73 -15 -10 4149 2844 4 MP gs c13 PP gr c0 DP 15 -73 26 6 -16 57 4231 2845 4 MP gs c13 PP gr c0 DP 15 -27 26 58 -16 8 4165 2836 4 MP gs c13 PP gr c0 DP 16 -57 25 27 -16 39 4206 2836 4 MP gs c13 PP gr c0 DP 16 14 25 78 -16 -19 4109 2761 4 MP gs /c14 { 0.781250 1.000000 0.000000 sr} bdef c14 PP gr c0 DP 16 -78 25 -13 -16 73 4272 2762 4 MP gs c14 PP gr c0 DP 16 -39 25 39 -15 22 4180 2814 4 MP gs /c15 { 0.875000 1.000000 0.000000 sr} bdef c15 PP gr c0 DP 15 10 25 73 -15 -23 4124 2784 4 MP gs c14 PP gr c0 DP 16 -73 25 -10 -15 60 4246 2785 4 MP gs c14 PP gr c0 DP 16 -8 25 60 -16 -7 4140 2791 4 MP gs c14 PP gr c0 DP 15 -60 25 9 -15 45 4221 2791 4 MP gs c14 PP gr c0 DP 15 -22 25 45 -15 7 4155 2784 4 MP gs c14 PP gr c0 DP 15 -45 26 22 -16 29 4196 2785 4 MP gs c14 PP gr c0 DP 16 19 25 85 -15 -26 4083 2683 4 MP gs /c16 { 0.500000 1.000000 0.000000 sr} bdef c16 PP gr c0 DP 15 23 26 78 -16 -42 4099 2725 4 MP gs /c17 { 0.593750 1.000000 0.000000 sr} bdef c17 PP gr c0 DP 15 -85 25 -18 -15 77 4287 2685 4 MP gs c16 PP gr c0 DP 15 -77 26 -23 -16 59 4262 2726 4 MP gs c17 PP gr c0 DP 16 7 25 59 -15 -21 4114 2746 4 MP gs /c18 { 0.687500 1.000000 0.000000 sr} bdef c18 PP gr c0 DP 16 -59 25 -6 -16 44 4237 2747 4 MP gs c18 PP gr c0 DP 16 -29 25 30 -16 15 4171 2769 4 MP gs c18 PP gr c0 DP 15 -7 26 45 -16 -6 4130 2752 4 MP gs c18 PP gr c0 DP 16 -44 25 6 -15 33 4211 2752 4 MP gs c18 PP gr c0 DP 16 -15 25 32 -15 5 4145 2747 4 MP gs c17 PP gr c0 DP 15 -33 25 16 -15 21 4186 2748 4 MP gs c17 PP gr c0 DP 15 21 25 44 -15 -25 4089 2706 4 MP gs c16 PP gr c0 DP 15 -43 25 -21 -15 40 4252 2707 4 MP gs c16 PP gr c0 DP 16 6 25 40 -16 -14 4105 2720 4 MP gs c16 PP gr c0 DP 16 42 25 84 -16 -82 4074 2681 4 MP gs c16 PP gr c0 DP 15 -40 26 -5 -16 31 4227 2721 4 MP gs c16 PP gr c0 DP 16 -84 25 -41 -15 43 4277 2683 4 MP gs c16 PP gr c0 DP 15 -21 26 22 -16 10 4161 2737 4 MP gs c17 PP gr c0 DP 15 -5 25 32 -15 -4 4120 2724 4 MP gs c16 PP gr c0 DP 16 -31 25 4 -16 23 4202 2725 4 MP gs c16 PP gr c0 DP 15 26 25 122 -15 -64 4058 2599 4 MP gs /c19 { 0.218750 1.000000 0.000000 sr} bdef c19 PP gr c0 DP 16 -122 25 -26 -16 84 4303 2601 4 MP gs c19 PP gr c0 DP 15 25 26 38 -16 -30 4064 2673 4 MP gs /c20 { 0.406250 1.000000 0.000000 sr} bdef c20 PP gr c0 DP 16 -10 25 23 -16 2 4136 2722 4 MP gs c16 PP gr c0 DP 16 -23 25 11 -15 14 4176 2723 4 MP gs c16 PP gr c0 DP 16 -38 25 -24 -16 32 4268 2675 4 MP gs c20 PP gr c0 DP 16 14 25 33 -15 -17 4079 2690 4 MP gs c20 PP gr c0 DP 16 -32 25 -14 -15 30 4242 2691 4 MP gs c20 PP gr c0 DP 15 4 26 30 -16 -10 4095 2700 4 MP gs c20 PP gr c0 DP 15 -30 25 -4 -15 24 4217 2701 4 MP gs c20 PP gr c0 DP 16 82 25 86 -15 -130 4048 2643 4 MP gs /c21 { 0.312500 1.000000 0.000000 sr} bdef c21 PP gr c0 DP 15 -14 25 15 -15 7 4151 2715 4 MP gs c16 PP gr c0 DP 15 -86 26 -82 -16 38 4293 2645 4 MP gs c21 PP gr c0 DP 16 -2 25 24 -15 -4 4110 2704 4 MP gs c20 PP gr c0 DP 15 -24 26 2 -16 18 4192 2705 4 MP gs c20 PP gr c0 DP 16 30 25 42 -16 -36 4039 2637 4 MP gs c21 PP gr c0 DP 15 17 25 36 -15 -24 4054 2661 4 MP gs c21 PP gr c0 DP 15 -42 25 -30 -15 37 4283 2638 4 MP gs c21 PP gr c0 DP 15 -37 26 -16 -16 29 4258 2662 4 MP gs c21 PP gr c0 DP 16 10 25 29 -16 -15 4070 2676 4 MP gs c20 PP gr c0 DP 15 -7 26 18 -16 4 4126 2700 4 MP gs c20 PP gr c0 DP 16 -18 25 8 -16 15 4167 2700 4 MP gs c20 PP gr c0 DP 16 -29 25 -10 -16 23 4233 2678 4 MP gs c20 PP gr c0 DP 15 4 25 24 -15 -5 4085 2681 4 MP gs c21 PP gr c0 DP 16 -23 25 -4 -15 23 4207 2682 4 MP gs c21 PP gr c0 DP 15 64 26 133 -16 -111 4033 2513 4 MP gs /c22 { 0.031250 1.000000 0.000000 sr} bdef c22 PP gr c0 DP 15 -132 25 -64 -15 86 4318 2515 4 MP gs c22 PP gr c0 DP 15 24 26 37 -16 -32 4029 2632 4 MP gs c19 PP gr c0 DP 15 130 26 82 -16 -170 4023 2601 4 MP gs c19 PP gr c0 DP 16 -36 25 -24 -15 29 4273 2633 4 MP gs c19 PP gr c0 DP 16 36 25 50 -15 -49 4013 2600 4 MP gs /c23 { 0.125000 1.000000 0.000000 sr} bdef c23 PP gr c0 DP 16 -82 25 -130 -15 42 4308 2603 4 MP gs c19 PP gr c0 DP 16 15 25 29 -15 -14 4044 2646 4 MP gs c19 PP gr c0 DP 16 -15 25 15 -15 77 4141 2623 4 MP gs c19 PP gr c0 DP 15 -29 25 -16 -15 30 4248 2648 4 MP gs c19 PP gr c0 DP 16 -49 25 -35 -16 36 4299 2602 4 MP gs c23 PP gr c0 DP 16 -4 25 23 -16 63 4101 2618 4 MP gs c23 PP gr c0 DP 15 -23 25 5 -15 81 4182 2619 4 MP gs c23 PP gr c0 DP 15 5 26 30 -16 42 4060 2604 4 MP gs c23 PP gr c0 DP 15 -30 26 -4 -16 77 4223 2605 4 MP gs c23 PP gr c0 DP 16 32 25 40 -15 -36 4003 2596 4 MP gs c23 PP gr c0 DP 15 -40 26 -31 -16 35 4289 2598 4 MP gs c23 PP gr c0 DP 15 14 26 36 -16 14 4019 2582 4 MP gs c22 PP gr c0 DP 16 -35 25 -15 -16 64 4264 2584 4 MP gs c22 PP gr c0 DP 15 49 26 62 -16 -71 3988 2560 4 MP gs c22 PP gr c0 DP 15 -62 25 -48 -15 40 4314 2562 4 MP gs c22 PP gr c0 DP 16 170 25 80 -16 -200 3998 2551 4 MP gs c22 PP gr c0 DP 15 -79 26 -170 -16 49 4324 2554 4 MP gs c22 PP gr c0 DP 16 111 25 122 -15 -151 4007 2431 4 MP gs /c24 { 0.000000 1.000000 0.250000 sr} bdef c24 PP gr c0 DP 16 -123 25 -110 -16 82 4334 2433 4 MP gs c24 PP gr c0 DP 15 -77 25 82 -15 173 4116 2445 4 MP gs /c25 { 0.000000 1.000000 0.343750 sr} bdef c25 PP gr c0 DP 15 -81 26 77 -16 177 4157 2446 4 MP gs c25 PP gr c0 DP 16 -63 25 77 -15 149 4075 2455 4 MP gs c25 PP gr c0 DP 16 -77 25 63 -16 163 4198 2456 4 MP gs c25 PP gr c0 DP 15 36 25 45 -15 -26 3978 2541 4 MP gs /c26 { 0.000000 1.000000 0.062500 sr} bdef c26 PP gr c0 DP 16 -44 25 -36 -15 55 4304 2543 4 MP gs c26 PP gr c0 DP 16 -42 25 64 -16 112 4035 2470 4 MP gs c24 PP gr c0 DP 16 -64 25 43 -15 133 4238 2472 4 MP gs c24 PP gr c0 DP 16 71 25 79 -16 -105 3963 2515 4 MP gs c26 PP gr c0 DP 16 -14 25 55 -16 62 3994 2479 4 MP gs c24 PP gr c0 DP 15 -55 25 14 -15 103 4279 2481 4 MP gs c24 PP gr c0 DP 16 -79 25 -70 -16 44 4330 2518 4 MP gs c26 PP gr c0 DP 16 200 25 78 -15 -216 3972 2489 4 MP gs /c27 { 0.000000 1.000000 0.156250 sr} bdef c27 PP gr c0 DP 16 -78 25 -200 -15 62 4339 2492 4 MP gs c27 PP gr c0 DP 15 26 26 58 -16 -3 3953 2460 4 MP gs c24 PP gr c0 DP 15 -58 26 -25 -16 80 4320 2463 4 MP gs c24 PP gr c0 DP 15 151 25 107 -15 -178 3982 2351 4 MP gs /c28 { 0.000000 1.000000 0.437500 sr} bdef c28 PP gr c0 DP 15 -106 25 -151 -15 79 4349 2354 4 MP gs c28 PP gr c0 DP 16 105 25 87 -15 -134 3937 2457 4 MP gs c24 PP gr c0 DP 15 -86 25 -105 -15 58 4345 2460 4 MP gs c24 PP gr c0 DP 16 -112 25 103 -15 138 4009 2341 4 MP gs /c29 { 0.000000 1.000000 0.625000 sr} bdef c29 PP gr c0 DP 15 -103 26 112 -16 130 4254 2342 4 MP gs c29 PP gr c0 DP 16 -62 25 81 -15 94 3968 2366 4 MP gs /c30 { 0.000000 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gs c35 PP gr c0 DP 15 20 26 -12 -16 -21 3789 3125 4 MP gs c18 PP gr c0 DP 16 21 25 -11 -16 -22 3733 3095 4 MP gs c18 PP gr c0 DP 15 3 26 -3 -16 -6 4083 2250 4 MP gs c34 PP gr c0 DP 16 16 25 -16 -16 -17 3927 3155 4 MP gs c14 PP gr c0 DP 15 17 26 -15 -16 -18 3886 3154 4 MP gs c14 PP gr c0 DP 16 16 25 14 -15 -17 3929 2185 4 MP gs c35 PP gr c0 DP 16 11 25 -22 -16 -11 4166 3076 4 MP gs c17 PP gr c0 DP 16 15 25 -17 -15 -16 3967 3154 4 MP gs c14 PP gr c0 DP 16 23 25 -11 -15 -22 3676 3061 4 MP gs c17 PP gr c0 DP 15 6 25 1 -15 -9 4042 2246 4 MP gs c34 PP gr c0 DP 15 19 25 -14 -15 -19 3845 3150 4 MP gs c14 PP gr c0 DP 16 0 25 -6 -16 -3 4124 2247 4 MP gs c34 PP gr c0 DP 15 9 26 -23 -16 -10 4257 3020 4 MP gs c16 PP gr c0 DP 16 13 25 -19 -15 -14 4074 3127 4 MP gs c18 PP gr c0 DP 15 21 25 -12 -15 -21 3748 3116 4 MP gs c18 PP gr c0 DP 16 15 25 -18 -15 -15 4008 3151 4 MP gs c14 PP gr c0 DP 16 19 25 -13 -15 -20 3804 3145 4 MP gs c14 PP gr c0 DP 16 11 25 -21 -15 -12 4140 3098 4 MP gs c18 PP gr c0 DP 16 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c34 PP gr c0 DP 16 8 25 12 -15 -7 2851 3516 4 MP gs c16 PP gr c0 DP 16 3 25 10 -16 -4 3271 3696 4 MP gs c14 PP gr c0 DP 15 13 26 -69 -16 -14 5018 1713 4 MP gs c62 PP gr c0 DP 15 -157 25 -61 -15 179 4865 2950 4 MP gs c32 PP gr c0 DP 16 39 25 -9 -16 -40 2871 1438 4 MP gs c68 PP gr c0 DP 15 47 25 -53 -15 -45 5318 2346 4 MP gs c42 PP gr c0 DP 15 16 25 -68 -15 -15 5039 1825 4 MP gs c60 PP gr c0 DP 16 64 25 -38 -15 -64 5481 2352 4 MP gs c42 PP gr c0 DP 16 31 25 -57 -16 -30 5175 2223 4 MP gs c46 PP gr c0 DP 16 27 25 -59 -16 -26 5144 2165 4 MP gs c53 PP gr c0 DP 16 7 25 13 -16 -7 2908 3543 4 MP gs c16 PP gr c0 DP 16 3 25 -9 -15 -2 2310 2748 4 MP gs c34 PP gr c0 DP 15 29 25 -58 -15 -28 5159 2193 4 MP gs c53 PP gr c0 DP 16 5 25 11 -16 -5 3174 3662 4 MP gs c14 PP gr c0 DP 16 66 25 -55 -15 -65 2229 2210 4 MP gs c46 PP gr c0 DP 15 66 25 -23 -15 -65 2604 1924 4 MP gs c56 PP gr c0 DP 16 -11 25 -4 -15 10 3907 3683 4 MP gs c14 PP gr c0 DP 15 69 25 -17 -15 -68 2715 1757 4 MP gs c61 PP gr c0 DP 15 40 26 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c17 PP gr c0 DP 16 -12 25 -6 -15 12 4146 3598 4 MP gs c17 PP gr c0 DP 16 -12 25 -7 -15 12 4278 3546 4 MP gs c16 PP gr c0 DP 15 11 25 -70 -15 -12 5012 1618 4 MP gs c64 PP gr c0 DP 16 -20 25 -45 -15 35 2071 2813 4 MP gs c7 PP gr c0 DP 15 63 25 -29 -15 -63 2503 2028 4 MP gs c55 PP gr c0 DP 16 -13 25 -7 -15 13 4344 3519 4 MP gs c16 PP gr c0 DP 16 65 25 -28 -15 -63 2528 1999 4 MP gs c57 PP gr c0 DP 16 62 25 -40 -15 -61 2361 2140 4 MP gs c50 PP gr c0 DP 15 2 25 -11 -15 0 2351 2740 4 MP gs c35 PP gr c0 DP 16 -256 25 -15 -15 266 4587 3176 4 MP gs c25 PP gr c0 DP 15 14 25 -10 -15 -9 5077 2737 4 MP gs c35 PP gr c0 DP 15 -45 25 -68 -15 73 4976 2828 4 MP gs c7 PP gr c0 DP 15 10 25 276 -15 -2 2702 3167 4 MP gs c25 PP gr c0 DP 16 -12 25 -6 -15 12 4080 3623 4 MP gs c18 PP gr c0 DP 16 62 25 -65 -16 -62 5441 2353 4 MP gs c42 PP gr c0 DP 16 6 25 12 -15 -6 3076 3622 4 MP gs c18 PP gr c0 DP 16 -10 25 -3 -15 9 3841 3700 4 MP gs c14 PP gr c0 DP 15 4 26 10 -16 -4 3230 3682 4 MP gs c14 PP gr c0 DP 16 21 25 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0 3424 3729 4 MP gs c15 PP gr c0 DP 16 -80 25 82 -15 93 1935 2897 4 MP gs c33 PP gr c0 DP 16 66 25 -50 -16 -66 2255 2156 4 MP gs c53 PP gr c0 DP 16 13 25 -5 -16 -11 5093 2748 4 MP gs c35 PP gr c0 DP 15 9 26 -19 -16 2 5052 2745 4 MP gs c35 PP gr c0 DP 15 19 26 -1 -16 -12 5312 2779 4 MP gs c34 PP gr c0 DP 15 15 26 -70 -16 -15 5049 1742 4 MP gs c62 PP gr c0 DP 15 -126 25 79 -15 154 5533 2909 4 MP gs c33 PP gr c0 DP 16 7 25 12 -16 -7 3036 3604 4 MP gs c17 PP gr c0 DP 15 0 26 -8 -16 0 2326 2748 4 MP gs c35 PP gr c0 DP 15 26 26 -63 -16 -25 5138 2054 4 MP gs c55 PP gr c0 DP 15 68 25 -20 -15 -68 2639 1814 4 MP gs c60 PP gr c0 DP 15 -266 25 -12 -15 275 4562 3179 4 MP gs c25 PP gr c0 DP 15 -13 25 -7 -15 13 4187 3579 4 MP gs c17 PP gr c0 DP 16 64 25 -44 -16 -63 2321 2122 4 MP gs c50 PP gr c0 DP 15 42 25 -57 -15 -40 5287 2258 4 MP gs c45 PP gr c0 DP 15 -12 25 -7 -15 12 4253 3553 4 MP gs c16 PP gr c0 DP 16 -20 25 -52 -15 45 5001 2788 4 MP gs c34 PP gr c0 DP 15 -13 25 -7 -15 13 4319 3526 4 MP gs c16 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DP 15 3 25 -4 -15 -1 2178 2759 4 MP gs c35 PP gr c0 DP 16 5 25 11 -15 -5 3148 3651 4 MP gs c18 PP gr c0 DP 16 68 25 -23 -16 -68 2589 1859 4 MP gs c59 PP gr c0 DP 16 1 25 9 -15 -2 3342 3712 4 MP gs c15 PP gr c0 DP 16 17 25 -70 -15 -16 5064 1758 4 MP gs c61 PP gr c0 DP 15 -10 26 -5 -16 11 3923 3672 4 MP gs c14 PP gr c0 DP 16 40 25 -58 -16 -39 5272 2218 4 MP gs c46 PP gr c0 DP 16 7 25 13 -15 -7 2882 3530 4 MP gs c16 PP gr c0 DP 15 2 26 -3 -16 -1 2219 2758 4 MP gs c35 PP gr c0 DP 15 14 25 -71 -15 -13 5043 1643 4 MP gs c64 PP gr c0 DP 15 20 25 -67 -15 -20 5101 1899 4 MP gs c58 PP gr c0 DP 15 58 25 -65 -15 -58 5425 2291 4 MP gs c45 PP gr c0 DP 16 65 25 -31 -15 -65 2462 1996 4 MP gs c57 PP gr c0 DP 15 58 26 70 -16 -53 5370 3003 4 MP gs c29 PP gr c0 DP 15 12 25 -3 -15 -11 5108 2759 4 MP gs c35 PP gr c0 DP 16 71 25 -17 -16 -71 2700 1689 4 MP gs c63 PP gr c0 DP 16 7 25 13 -16 -8 2939 3558 4 MP gs c16 PP gr c0 DP 16 68 25 -25 -15 -67 2563 1883 4 MP gs c59 PP gr c0 DP 15 2 26 -4 -16 0 2260 2756 4 MP gs c35 PP gr c0 DP 15 46 26 -59 -16 -45 5328 2248 4 MP gs c46 PP gr c0 DP 15 65 26 -41 -16 -64 2346 2079 4 MP gs c54 PP gr c0 DP 16 10 25 -40 -16 -9 5032 1453 4 MP gs c68 PP gr c0 DP 16 66 25 -30 -16 -65 2488 1965 4 MP gs c56 PP gr c0 DP 15 -102 26 -64 -16 130 4916 2873 4 MP gs c33 PP gr c0 DP 15 67 25 -26 -15 -66 2538 1908 4 MP gs c58 PP gr c0 DP 15 7 26 12 -16 -6 2995 3584 4 MP gs c17 PP gr c0 DP 15 66 26 -28 -16 -66 2513 1936 4 MP gs c58 PP gr c0 DP 15 67 26 -48 -16 -66 2280 2106 4 MP gs c54 PP gr c0 DP 15 39 26 -12 -16 -40 2795 1472 4 MP gs c68 PP gr c0 DP 15 -8 25 -1 -15 8 3750 3715 4 MP gs c15 PP gr c0 DP 16 0 25 -6 -16 2 2301 2752 4 MP gs c35 PP gr c0 DP 16 4 25 11 -16 -5 3205 3672 4 MP gs c14 PP gr c0 DP 15 -9 26 -4 -16 10 3857 3690 4 MP gs c14 PP gr c0 DP 15 -275 26 -10 -16 284 4537 3180 4 MP gs c25 PP gr c0 DP 16 11 25 -8 -15 -6 5067 2751 4 MP gs c35 PP gr c0 DP 16 22 25 -67 -16 -21 5117 1920 4 MP gs c58 PP gr c0 DP 16 2 25 9 -16 -3 3302 3702 4 MP gs c14 PP gr c0 DP 15 -12 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c0 DP 16 6 25 12 -15 -6 3107 3634 4 MP gs c18 PP gr c0 DP 15 23 25 -66 -15 -22 5132 1942 4 MP gs c56 PP gr c0 DP 15 6 25 -5 -15 -5 5205 2775 4 MP gs c34 PP gr c0 DP 16 14 25 -70 -16 -14 5059 1657 4 MP gs c64 PP gr c0 DP 15 9 25 294 -15 -2 2733 3169 4 MP gs c28 PP gr c0 DP 15 66 25 -39 -15 -65 2371 2039 4 MP gs c55 PP gr c0 DP 16 -25 25 -47 -16 40 2062 2820 4 MP gs c7 PP gr c0 DP 15 7 26 -6 -16 -5 5246 2776 4 MP gs c34 PP gr c0 DP 16 1 25 -8 -15 1 2341 2747 4 MP gs c35 PP gr c0 DP 16 7 25 -4 -15 -6 5164 2773 4 MP gs c34 PP gr c0 DP 15 51 26 -62 -16 -50 5369 2240 4 MP gs c46 PP gr c0 DP 16 35 25 -60 -16 -34 5241 2143 4 MP gs c50 PP gr c0 DP 16 -11 25 -5 -16 11 3964 3657 4 MP gs c18 PP gr c0 DP 16 -5 25 1 -15 5 3643 3730 4 MP gs c15 PP gr c0 DP 16 43 25 -60 -15 -42 5312 2203 4 MP gs c53 PP gr c0 DP 16 0 25 7 -16 0 3399 3722 4 MP gs c15 PP gr c0 DP 15 10 25 -4 -15 -1 2382 2742 4 MP gs c35 PP gr c0 DP 15 -8 26 -3 -16 9 3791 3705 4 MP gs c14 PP gr c0 DP 16 65 25 36 -16 -48 5349 2884 4 MP gs c33 PP gr c0 DP 15 -12 26 -30 -16 25 2087 2788 4 MP gs c34 PP gr c0 DP 15 3 26 10 -16 -3 3261 3689 4 MP gs c14 PP gr c0 DP 16 24 25 -65 -16 -24 5148 1966 4 MP gs c56 PP gr c0 DP 16 10 25 -2 -16 -9 5124 2768 4 MP gs c34 PP gr c0 DP 15 70 26 -19 -16 -70 2649 1726 4 MP gs c62 PP gr c0 DP 15 7 26 13 -16 -7 2898 3537 4 MP gs c16 PP gr c0 DP 15 33 25 -61 -15 -32 5225 2109 4 MP gs c54 PP gr c0 DP 16 71 25 -15 -16 -71 2735 1585 4 MP gs c65 PP gr c0 DP 15 5 26 11 -16 -5 3164 3656 4 MP gs c18 PP gr c0 DP 16 18 25 -68 -15 -19 5095 1793 4 MP gs c61 PP gr c0 DP 16 40 25 -13 -16 -40 2770 1485 4 MP gs c68 PP gr c0 DP 15 11 25 -39 -15 -11 5047 1464 4 MP gs c68 PP gr c0 DP 16 -76 25 155 -15 84 1838 2798 4 MP gs c35 PP gr c0 DP 15 1 26 -4 -16 2 2153 2760 4 MP gs c35 PP gr c0 DP 16 -11 25 -4 -16 10 3898 3677 4 MP gs c14 PP gr c0 DP 15 26 25 -65 -15 -26 5163 1992 4 MP gs c57 PP gr c0 DP 15 8 25 12 -15 -7 2954 3565 4 MP gs c16 PP gr c0 DP 15 -72 25 -60 -15 102 4941 2837 4 MP gs c7 PP gr c0 DP 16 32 25 -63 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26 -27 -16 22 5017 2766 4 MP gs c35 PP gr c0 DP 16 -3 25 3 -15 3 3536 3733 4 MP gs c15 PP gr c0 DP 16 -2 25 5 -16 1 3496 3732 4 MP gs c15 PP gr c0 DP 16 20 25 -68 -15 -20 5126 1832 4 MP gs c60 PP gr c0 DP 16 68 25 -41 -15 -68 2330 2015 4 MP gs c57 PP gr c0 DP 15 6 25 -1 -15 -6 5139 2774 4 MP gs c35 PP gr c0 DP 16 17 25 -70 -16 -17 5090 1689 4 MP gs c63 PP gr c0 DP 15 69 26 -25 -16 -69 2548 1816 4 MP gs c60 PP gr c0 DP 15 68 25 -30 -15 -68 2472 1900 4 MP gs c59 PP gr c0 DP 16 38 25 -62 -15 -37 5281 2121 4 MP gs c54 PP gr c0 DP 16 -4 25 2 -15 4 3577 3732 4 MP gs c15 PP gr c0 DP 16 8 25 13 -15 -7 2913 3544 4 MP gs c16 PP gr c0 DP 15 -1 26 5 -16 1 3455 3728 4 MP gs c15 PP gr c0 DP 16 69 25 -27 -16 -68 2523 1842 4 MP gs c60 PP gr c0 DP 16 68 25 -28 -15 -68 2497 1870 4 MP gs c59 PP gr c0 DP 16 -22 25 -40 -16 45 4992 2783 4 MP gs c34 PP gr c0 DP 16 13 25 -71 -15 -14 5068 1573 4 MP gs c66 PP gr c0 DP 16 -9 25 -1 -16 8 3766 3707 4 MP gs c14 PP gr c0 DP 15 71 26 -17 -16 -71 2684 1618 4 MP gs c65 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gs c17 PP gr c0 DP 15 -7 26 1 -16 5 3659 3725 4 MP gs c14 PP gr c0 DP 15 51 25 -66 -15 -51 5394 2179 4 MP gs c53 PP gr c0 DP 15 54 26 -37 -16 -55 5435 2168 4 MP gs c50 PP gr c0 DP 16 -1 25 -1 -15 3 2168 2757 4 MP gs c35 PP gr c0 DP 16 -294 25 -9 -15 302 4486 3181 4 MP gs c28 PP gr c0 DP 16 9 25 -5 -16 -4 5058 2760 4 MP gs c35 PP gr c0 DP 16 44 25 -64 -16 -43 5338 2144 4 MP gs c50 PP gr c0 DP 16 0 25 8 -15 -2 3373 3716 4 MP gs c14 PP gr c0 DP 15 17 25 -70 -15 -17 5105 1706 4 MP gs c63 PP gr c0 DP 15 6 25 11 -15 -6 3082 3623 4 MP gs c17 PP gr c0 DP 16 23 25 -67 -15 -23 5157 1877 4 MP gs c59 PP gr c0 DP 16 69 25 -40 -16 -68 2356 1974 4 MP gs c56 PP gr c0 DP 16 71 25 -18 -16 -72 2659 1637 4 MP gs c65 PP gr c0 DP 15 24 25 19 -15 -6 2413 2752 4 MP gs c35 PP gr c0 DP 15 -11 25 -5 -15 11 3979 3646 4 MP gs c18 PP gr c0 DP 15 -2 25 0 -15 4 2209 2754 4 MP gs c35 PP gr c0 DP 16 34 25 -64 -15 -33 5250 2049 4 MP gs c55 PP gr c0 DP 15 13 25 -40 -15 -12 5078 1488 4 MP gs c68 PP gr c0 DP 16 -9 25 -2 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DP 16 -8 25 -17 -15 16 2102 2772 4 MP gs c35 PP gr c0 DP 15 4 25 -2 -15 -2 5236 2780 4 MP gs c35 PP gr c0 DP 16 3 25 -2 -15 -2 5195 2779 4 MP gs c35 PP gr c0 DP 15 7 26 14 -16 -8 2929 3552 4 MP gs c16 PP gr c0 DP 16 19 25 -70 -16 -18 5121 1724 4 MP gs c63 PP gr c0 DP 15 69 26 -36 -16 -69 2381 1935 4 MP gs c58 PP gr c0 DP 16 4 25 -1 -16 -2 5155 2776 4 MP gs c35 PP gr c0 DP 15 39 26 -42 -16 -38 2315 1947 4 MP gs c58 PP gr c0 DP 15 65 26 41 -16 -48 5339 2891 4 MP gs c33 PP gr c0 DP 16 30 25 37 -16 -9 2429 2761 4 MP gs c35 PP gr c0 DP 15 7 25 12 -15 -7 2985 3579 4 MP gs c16 PP gr c0 DP 15 42 25 -65 -15 -41 5322 2101 4 MP gs c54 PP gr c0 DP 15 -31 26 -47 -16 47 2052 2820 4 MP gs c34 PP gr c0 DP 15 6 26 0 -16 -5 5114 2773 4 MP gs c35 PP gr c0 DP 16 -2 25 -3 -16 4 2332 2746 4 MP gs c35 PP gr c0 DP 15 71 25 -21 -15 -71 2608 1677 4 MP gs c64 PP gr c0 DP 16 -8 25 -2 -15 8 3740 3709 4 MP gs c14 PP gr c0 DP 15 4 26 11 -16 -5 3195 3666 4 MP gs c18 PP gr c0 DP 15 -9 25 -4 -15 10 3847 3684 4 MP gs c18 PP gr c0 DP 15 -13 25 -7 -15 13 4152 3580 4 MP gs c16 PP gr c0 DP 15 -13 25 -7 -15 13 4218 3554 4 MP gs c16 PP gr c0 DP 16 7 25 12 -15 -6 3041 3604 4 MP gs c17 PP gr c0 DP 15 2 26 9 -16 -3 3292 3696 4 MP gs c14 PP gr c0 DP 15 -3 25 4 -15 2 3511 3730 4 MP gs c14 PP gr c0 DP 15 -12 25 -6 -15 11 4086 3606 4 MP gs c17 PP gr c0 DP 16 40 25 -18 -15 -39 2668 1547 4 MP gs c67 PP gr c0 DP 15 -4 26 3 -16 3 3552 3730 4 MP gs c14 PP gr c0 DP 16 13 25 -39 -16 -13 5094 1501 4 MP gs c68 PP gr c0 DP 15 -16 25 -32 -15 31 2077 2789 4 MP gs c34 PP gr c0 DP 16 -1 25 5 -15 1 3470 3727 4 MP gs c14 PP gr c0 DP 16 15 25 -71 -15 -15 5099 1602 4 MP gs c65 PP gr c0 DP 15 70 25 -35 -15 -69 2406 1899 4 MP gs c59 PP gr c0 DP 16 4 25 -3 -15 2 2372 2743 4 MP gs c35 PP gr c0 DP 16 48 25 -67 -16 -47 5379 2128 4 MP gs c54 PP gr c0 DP 15 36 25 58 -15 -10 2444 2771 4 MP gs c35 PP gr c0 DP 15 19 25 -69 -15 -20 5136 1744 4 MP gs c62 PP gr c0 DP 15 -11 26 -6 -16 12 4020 3629 4 MP gs c17 PP gr c0 DP 15 8 25 0 -15 -6 5073 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gs c35 PP gr c0 DP 15 -10 26 -3 -16 9 3822 3688 4 MP gs c18 PP gr c0 DP 16 35 25 -67 -16 -34 5276 1986 4 MP gs c56 PP gr c0 DP 16 4 25 10 -15 -4 3210 3670 4 MP gs c18 PP gr c0 DP 16 23 25 -69 -16 -23 5183 1810 4 MP gs c61 PP gr c0 DP 15 -5 25 -17 -15 18 5007 2765 4 MP gs c35 PP gr c0 DP 16 -12 25 -5 -16 11 3995 3635 4 MP gs c17 PP gr c0 DP 15 -8 25 -1 -15 7 3715 3711 4 MP gs c14 PP gr c0 DP 16 -4 25 1 -15 6 2306 2743 4 MP gs c38 PP gr c0 DP 16 39 25 -20 -16 -40 2593 1606 4 MP gs c66 PP gr c0 DP 16 2 25 8 -15 -2 3307 3698 4 MP gs c14 PP gr c0 DP 16 -307 25 -7 -16 310 4436 3188 4 MP gs c28 PP gr c0 DP 16 18 25 -72 -15 -17 5130 1636 4 MP gs c65 PP gr c0 DP 16 5 25 2 -16 -4 5089 2770 4 MP gs c35 PP gr c0 DP 16 39 25 -35 -16 -39 2391 1830 4 MP gs c61 PP gr c0 DP 15 48 26 -38 -16 -48 5404 2062 4 MP gs c55 PP gr c0 DP 15 26 25 -69 -15 -25 5198 1835 4 MP gs c60 PP gr c0 DP 15 32 25 -67 -15 -32 5260 1952 4 MP gs c58 PP gr c0 DP 15 28 25 205 -15 -10 2537 2845 4 MP gs c34 PP gr c0 DP 16 5 25 11 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DP 15 39 26 -32 -16 -39 2416 1795 4 MP gs c61 PP gr c0 DP 15 -1 25 5 -15 0 3445 3723 4 MP gs c14 PP gr c0 DP 16 42 25 -68 -15 -42 5347 2037 4 MP gs c57 PP gr c0 DP 15 7 25 313 -15 -6 2795 3187 4 MP gs c28 PP gr c0 DP 15 -8 26 -2 -16 7 3756 3702 4 MP gs c14 PP gr c0 DP 15 39 25 -23 -15 -40 2542 1652 4 MP gs c65 PP gr c0 DP 16 3 25 10 -16 -4 3267 3687 4 MP gs c18 PP gr c0 DP 15 7 26 13 -16 -7 2960 3566 4 MP gs c16 PP gr c0 DP 15 19 26 -71 -16 -19 5146 1655 4 MP gs c65 PP gr c0 DP 16 5 25 11 -16 -5 3170 3655 4 MP gs c17 PP gr c0 DP 16 -5 25 1 -15 4 3608 3724 4 MP gs c14 PP gr c0 DP 16 -10 25 -3 -16 9 3863 3675 4 MP gs c18 PP gr c0 DP 15 39 25 -31 -15 -39 2441 1763 4 MP gs c62 PP gr c0 DP 16 0 25 6 -15 0 3404 3717 4 MP gs c14 PP gr c0 DP 15 40 26 -26 -16 -39 2517 1677 4 MP gs c64 PP gr c0 DP 16 25 25 223 -16 -8 2553 2853 4 MP gs c34 PP gr c0 DP 15 -18 26 -24 -16 36 4982 2771 4 MP gs c35 PP gr c0 DP 15 6 25 13 -15 -7 3016 3592 4 MP gs c16 PP gr c0 DP 16 39 25 -28 -15 -39 2466 1732 4 MP gs c63 PP gr c0 DP 16 39 25 -27 -16 -39 2492 1704 4 MP gs c64 PP gr c0 DP 16 -12 25 -7 -16 13 4168 3567 4 MP gs c16 PP gr c0 DP 16 -12 25 -7 -16 13 4102 3593 4 MP gs c16 PP gr c0 DP 15 9 25 3 -15 -4 5267 2786 4 MP gs c35 PP gr c0 DP 15 16 25 -39 -15 -16 5140 1547 4 MP gs c67 PP gr c0 DP 16 -6 25 1 -16 8 2159 2748 4 MP gs c38 PP gr c0 DP 16 -12 25 -5 -16 11 4036 3618 4 MP gs c17 PP gr c0 DP 15 -6 25 1 -15 5 3649 3719 4 MP gs c14 PP gr c0 DP 16 1 25 8 -16 -2 3364 3710 4 MP gs c14 PP gr c0 DP 16 6 25 12 -15 -6 3072 3617 4 MP gs c17 PP gr c0 DP 16 45 25 -38 -15 -45 5388 2014 4 MP gs c56 PP gr c0 DP 16 19 25 -71 -15 -19 5161 1674 4 MP gs c64 PP gr c0 DP 16 -11 25 -5 -15 11 3969 3640 4 MP gs c17 PP gr c0 DP 15 39 26 -68 -16 -39 5332 1995 4 MP gs c56 PP gr c0 DP 16 -310 25 -7 -15 312 4410 3193 4 MP gs c28 PP gr c0 DP 16 2 25 1 -15 0 5226 2779 4 MP gs c35 PP gr c0 DP 16 -9 25 -2 -16 8 3797 3691 4 MP gs c18 PP gr c0 DP 15 3 26 10 -16 -4 3226 3674 4 MP gs c18 PP gr c0 DP 16 -7 25 3 -15 8 2199 2743 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c64 PP gr c0 DP 15 65 26 27 -16 -62 2986 3133 4 MP gs c41 PP gr c0 DP 16 56 25 46 -15 -58 2974 2954 4 MP gs c45 PP gr c0 DP 15 -49 26 -29 -16 49 3930 2628 4 MP gs c56 PP gr c0 DP 15 -50 25 -26 -15 49 3955 2599 4 MP gs c58 PP gr c0 DP 16 11 25 57 -16 -11 2587 2210 4 MP gs c66 PP gr c0 DP 15 17 25 55 -15 -17 2705 2355 4 MP gs c63 PP gr c0 DP 16 -57 25 -9 -15 57 4390 2201 4 MP gs c66 PP gr c0 DP 15 68 25 51 -15 -66 2980 3031 4 MP gs c42 PP gr c0 DP 16 42 25 46 -16 -43 2938 2794 4 MP gs c50 PP gr c0 DP 16 8 25 58 -15 -8 2499 2112 4 MP gs c69 PP gr c0 DP 15 -58 25 -6 -15 58 4507 2111 4 MP gs c69 PP gr c0 DP 16 -55 25 -15 -15 55 4223 2333 4 MP gs c64 PP gr c0 DP 16 -45 25 -44 -16 43 3769 2846 4 MP gs c53 PP gr c0 DP 16 -27 25 -64 -15 29 3638 3138 4 MP gs c41 PP gr c0 DP 16 -26 25 -3 -15 26 4598 2070 4 MP gs c69 PP gr c0 DP 16 18 25 55 -16 -19 2721 2374 4 MP gs c63 PP gr c0 DP 16 5 25 26 -16 -6 2428 2065 4 MP gs c69 PP gr c0 DP 16 -45 25 -56 -15 44 3708 2959 4 MP gs c45 PP gr c0 DP 15 12 25 57 -15 -13 2602 2223 4 MP gs c66 PP gr c0 DP 16 -50 25 -68 -15 53 3673 3036 4 MP gs c42 PP gr c0 DP 15 -57 25 -10 -15 56 4365 2212 4 MP gs c66 PP gr c0 DP 15 39 25 48 -15 -41 2922 2751 4 MP gs c54 PP gr c0 DP 15 -55 25 -16 -15 55 4198 2349 4 MP gs c63 PP gr c0 DP 15 20 25 54 -15 -19 2736 2393 4 MP gs c62 PP gr c0 DP 15 53 26 47 -16 -54 2959 2896 4 MP gs c46 PP gr c0 DP 15 -46 26 -42 -16 45 3794 2800 4 MP gs c50 PP gr c0 DP 15 8 26 58 -16 -9 2515 2121 4 MP gs c68 PP gr c0 DP 15 -58 26 -7 -16 58 4482 2118 4 MP gs c69 PP gr c0 DP 16 20 25 55 -16 -21 2752 2414 4 MP gs c62 PP gr c0 DP 16 14 25 56 -16 -13 2618 2236 4 MP gs c66 PP gr c0 DP 16 38 25 48 -16 -38 2907 2710 4 MP gs c55 PP gr c0 DP 15 -55 26 -17 -16 55 4173 2366 4 MP gs c63 PP gr c0 DP 16 67 25 25 -15 -65 2970 3071 4 MP gs c44 PP gr c0 DP 15 -26 25 -4 -15 25 4573 2075 4 MP gs c69 PP gr c0 DP 15 7 25 25 -15 -6 2443 2071 4 MP gs c69 PP gr c0 DP 15 -56 26 -11 -16 56 4340 2223 4 MP gs c66 PP gr c0 DP 15 22 25 54 -15 -22 2767 2436 4 MP gs c61 PP gr c0 DP 15 35 25 50 -15 -37 2891 2672 4 MP gs c57 PP gr c0 DP 15 -47 25 -39 -15 46 3819 2757 4 MP gs c54 PP gr c0 DP 16 -47 25 -52 -16 45 3734 2902 4 MP gs c46 PP gr c0 DP 16 62 25 51 -16 -62 2965 2965 4 MP gs c45 PP gr c0 DP 16 -55 25 -17 -16 54 4148 2384 4 MP gs c63 PP gr c0 DP 16 23 25 54 -16 -24 2783 2460 4 MP gs c61 PP gr c0 DP 16 33 25 51 -16 -34 2876 2635 4 MP gs c56 PP gr c0 DP 15 14 25 57 -15 -14 2633 2250 4 MP gs c66 PP gr c0 DP 16 49 25 48 -15 -50 2943 2842 4 MP gs c53 PP gr c0 DP 16 10 25 57 -15 -9 2530 2130 4 MP gs c68 PP gr c0 DP 16 -25 25 -67 -16 27 3664 3076 4 MP gs c44 PP gr c0 DP 15 24 25 53 -15 -25 2798 2485 4 MP gs c60 PP gr c0 DP 15 31 25 51 -15 -31 2860 2601 4 MP gs c58 PP gr c0 DP 16 -58 25 -7 -15 57 4456 2126 4 MP gs c69 PP gr c0 DP 16 26 25 52 -16 -26 2814 2511 4 MP gs c60 PP gr c0 DP 16 -48 25 -37 -15 47 3844 2717 4 MP gs c55 PP gr c0 DP 16 -54 25 -19 -15 54 4122 2403 4 MP gs c62 PP gr c0 DP 16 30 25 52 -16 -31 2845 2570 4 MP gs c59 PP gr c0 DP 16 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3920 2609 4 MP gs c58 PP gr c0 DP 16 16 25 57 -15 -17 2664 2282 4 MP gs c65 PP gr c0 DP 16 -53 25 -24 -15 53 4021 2493 4 MP gs c60 PP gr c0 DP 16 -51 25 -29 -15 51 3945 2577 4 MP gs c59 PP gr c0 DP 15 -53 26 -26 -16 52 3996 2520 4 MP gs c60 PP gr c0 DP 16 -52 25 -27 -16 51 3971 2548 4 MP gs c59 PP gr c0 DP 15 8 25 26 -15 -8 2474 2086 4 MP gs c69 PP gr c0 DP 16 -25 25 -6 -16 25 4523 2086 4 MP gs c69 PP gr c0 DP 15 -56 25 -14 -15 56 4264 2262 4 MP gs c66 PP gr c0 DP 16 43 25 51 -15 -45 2912 2745 4 MP gs c55 PP gr c0 DP 16 11 25 58 -15 -11 2561 2152 4 MP gs c68 PP gr c0 DP 15 17 26 56 -16 -17 2680 2299 4 MP gs c65 PP gr c0 DP 15 -49 25 -46 -15 48 3784 2798 4 MP gs c54 PP gr c0 DP 15 -24 26 -65 -16 25 3689 3011 4 MP gs c48 PP gr c0 DP 15 -58 26 -9 -16 57 4406 2144 4 MP gs c68 PP gr c0 DP 15 -51 26 -57 -16 50 3724 2909 4 MP gs c46 PP gr c0 DP 15 -56 26 -15 -16 56 4239 2277 4 MP gs c65 PP gr c0 DP 16 19 25 56 -15 -19 2695 2318 4 MP gs c65 PP gr c0 DP 15 41 26 51 -16 -41 2897 2700 4 MP gs c57 PP gr c0 DP 16 9 25 26 -16 -9 2490 2095 4 MP gs c69 PP gr c0 DP 16 54 25 51 -16 -54 2934 2845 4 MP gs c50 PP gr c0 DP 15 13 26 58 -16 -13 2577 2165 4 MP gs c68 PP gr c0 DP 16 -25 25 -7 -15 25 4497 2093 4 MP gs c69 PP gr c0 DP 16 -50 25 -43 -15 49 3809 2751 4 MP gs c55 PP gr c0 DP 15 19 26 56 -16 -19 2711 2337 4 MP gs c64 PP gr c0 DP 16 62 25 23 -15 -62 2939 2942 4 MP gs c45 PP gr c0 DP 16 -57 25 -11 -16 58 4381 2154 4 MP gs c68 PP gr c0 DP 16 -56 25 -16 -16 56 4214 2293 4 MP gs c65 PP gr c0 DP 16 38 25 52 -15 -39 2881 2659 4 MP gs c56 PP gr c0 DP 16 21 25 56 -15 -22 2726 2359 4 MP gs c64 PP gr c0 DP 16 13 25 58 -15 -13 2592 2178 4 MP gs c68 PP gr c0 DP 16 -56 25 -17 -15 56 4188 2310 4 MP gs c65 PP gr c0 DP 15 9 25 26 -15 -10 2505 2105 4 MP gs c69 PP gr c0 DP 16 -51 25 -40 -16 50 3835 2707 4 MP gs c57 PP gr c0 DP 15 -51 25 -54 -15 51 3749 2851 4 MP gs c50 PP gr c0 DP 15 37 26 52 -16 -37 2866 2620 4 MP gs c58 PP gr c0 DP 15 22 26 55 -16 -22 2742 2381 4 MP gs c63 PP gr c0 DP 15 -25 26 -8 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gs c58 PP gr c0 DP 16 -26 25 -8 -16 25 4447 2109 4 MP gs c69 PP gr c0 DP 16 -52 25 -50 -15 51 3774 2797 4 MP gs c54 PP gr c0 DP 16 -55 25 -21 -16 56 4113 2368 4 MP gs c64 PP gr c0 DP 16 15 25 57 -15 -15 2623 2208 4 MP gs c67 PP gr c0 DP 16 47 25 53 -16 -47 2903 2739 4 MP gs c55 PP gr c0 DP 16 -53 25 -34 -15 52 3910 2591 4 MP gs c59 PP gr c0 DP 16 -56 25 -21 -15 54 4087 2391 4 MP gs c63 PP gr c0 DP 15 -58 26 -13 -16 58 4305 2190 4 MP gs c68 PP gr c0 DP 16 -53 25 -32 -16 53 3936 2556 4 MP gs c60 PP gr c0 DP 15 11 25 26 -15 -12 2536 2127 4 MP gs c69 PP gr c0 DP 15 -54 25 -24 -15 55 4062 2414 4 MP gs c63 PP gr c0 DP 16 -23 25 -58 -15 23 3739 2886 4 MP gs c53 PP gr c0 DP 15 -54 26 -29 -16 53 3961 2524 4 MP gs c60 PP gr c0 DP 15 -55 26 -24 -16 54 4037 2439 4 MP gs c62 PP gr c0 DP 15 17 26 57 -16 -16 2639 2224 4 MP gs c67 PP gr c0 DP 16 -25 25 -10 -15 26 4421 2118 4 MP gs c69 PP gr c0 DP 16 -54 25 -27 -15 54 4011 2466 4 MP gs c62 PP gr c0 DP 15 -54 25 -28 -15 54 3986 2494 4 MP gs c61 PP gr c0 DP 15 45 25 53 -15 -45 2887 2692 4 MP gs c56 PP gr c0 DP 16 -53 25 -47 -16 52 3800 2746 4 MP gs c55 PP gr c0 DP 16 54 25 24 -15 -54 2908 2821 4 MP gs c54 PP gr c0 DP 16 -58 25 -14 -16 57 4280 2205 4 MP gs c68 PP gr c0 DP 16 17 25 58 -15 -18 2654 2242 4 MP gs c67 PP gr c0 DP 16 13 25 25 -16 -12 2552 2139 4 MP gs c69 PP gr c0 DP 15 -26 25 -10 -15 25 4396 2129 4 MP gs c69 PP gr c0 DP 16 41 25 54 -16 -42 2872 2647 4 MP gs c58 PP gr c0 DP 15 19 26 57 -16 -18 2670 2260 4 MP gs c66 PP gr c0 DP 16 -57 25 -15 -15 57 4254 2220 4 MP gs c67 PP gr c0 DP 15 -53 26 -44 -16 53 3825 2698 4 MP gs c56 PP gr c0 DP 16 -24 25 -54 -16 23 3765 2828 4 MP gs c54 PP gr c0 DP 15 13 25 26 -15 -13 2567 2152 4 MP gs c69 PP gr c0 DP 16 19 25 58 -16 -20 2686 2280 4 MP gs c66 PP gr c0 DP 15 39 25 55 -15 -40 2856 2605 4 MP gs c59 PP gr c0 DP 15 52 26 24 -16 -52 2893 2767 4 MP gs c55 PP gr c0 DP 15 -57 25 -16 -15 57 4229 2236 4 MP gs c67 PP gr c0 DP 15 -25 26 -11 -16 25 4371 2140 4 MP gs c69 PP gr c0 DP 15 22 25 57 -15 -22 2701 2302 4 MP gs c66 PP gr c0 DP 15 -54 25 -41 -15 53 3850 2654 4 MP gs c58 PP gr c0 DP 16 37 25 55 -16 -37 2841 2565 4 MP gs c60 PP gr c0 DP 16 15 25 26 -16 -15 2583 2167 4 MP gs c69 PP gr c0 DP 15 -57 26 -17 -16 57 4204 2253 4 MP gs c67 PP gr c0 DP 16 22 25 57 -16 -22 2717 2324 4 MP gs c65 PP gr c0 DP 15 34 25 56 -15 -35 2825 2528 4 MP gs c61 PP gr c0 DP 16 -25 25 -12 -16 25 4346 2152 4 MP gs c69 PP gr c0 DP 15 24 25 57 -15 -24 2732 2348 4 MP gs c65 PP gr c0 DP 15 -24 26 -51 -16 24 3790 2773 4 MP gs c55 PP gr c0 DP 16 47 25 25 -15 -48 2877 2715 4 MP gs c57 PP gr c0 DP 16 -54 25 -39 -15 54 3875 2612 4 MP gs c59 PP gr c0 DP 16 33 25 56 -16 -33 2810 2493 4 MP gs c61 PP gr c0 DP 16 -57 25 -18 -16 57 4179 2271 4 MP gs c66 PP gr c0 DP 16 25 25 57 -16 -26 2748 2374 4 MP gs c64 PP gr c0 DP 15 15 25 26 -15 -15 2598 2182 4 MP gs c68 PP gr c0 DP 15 30 25 56 -15 -30 2794 2460 4 MP gs c62 PP gr c0 DP 15 27 25 56 -15 -27 2763 2401 4 MP gs c64 PP gr c0 DP 16 29 25 56 -16 -29 2779 2430 4 MP gs c63 PP gr c0 DP 16 -57 25 -20 -15 57 4153 2291 4 MP gs c66 PP gr c0 DP 16 -55 25 -36 -16 54 3901 2573 4 MP gs c60 PP gr c0 DP 16 -25 25 -13 -15 25 4320 2165 4 MP gs c69 PP gr c0 DP 16 16 25 26 -16 -17 2614 2199 4 MP gs c68 PP gr c0 DP 15 -57 25 -20 -15 56 4128 2312 4 MP gs c66 PP gr c0 DP 15 45 26 25 -16 -45 2862 2667 4 MP gs c58 PP gr c0 DP 15 -55 26 -35 -16 55 3926 2536 4 MP gs c61 PP gr c0 DP 15 -24 25 -48 -15 24 3815 2722 4 MP gs c57 PP gr c0 DP 15 -56 26 -23 -16 57 4103 2334 4 MP gs c65 PP gr c0 DP 15 -55 25 -32 -15 55 3951 2501 4 MP gs c61 PP gr c0 DP 15 -25 25 -15 -15 26 4295 2179 4 MP gs c69 PP gr c0 DP 16 -57 25 -23 -16 56 4078 2358 4 MP gs c65 PP gr c0 DP 15 18 25 25 -15 -17 2629 2216 4 MP gs c68 PP gr c0 DP 16 -56 25 -30 -15 55 3976 2469 4 MP gs c62 PP gr c0 DP 16 -56 25 -25 -15 56 4052 2383 4 MP gs c64 PP gr c0 DP 16 -56 25 -28 -16 56 4002 2438 4 MP gs c63 PP gr c0 DP 15 -56 26 -27 -16 56 4027 2410 4 MP gs c64 PP gr c0 DP 16 42 25 25 -15 -42 2846 2622 4 MP gs c59 PP gr c0 DP 16 18 25 26 -16 -19 2645 2235 4 MP gs c67 PP gr c0 DP 15 -26 26 -15 -16 26 4270 2194 4 MP gs c68 PP gr c0 DP 16 -24 25 -44 -15 24 3840 2674 4 MP gs c58 PP gr c0 DP 15 40 26 25 -16 -40 2831 2580 4 MP gs c60 PP gr c0 DP 16 20 25 25 -15 -20 2660 2255 4 MP gs c67 PP gr c0 DP 16 -26 25 -16 -16 26 4245 2210 4 MP gs c68 PP gr c0 DP 15 22 26 25 -16 -21 2676 2276 4 MP gs c67 PP gr c0 DP 16 -24 25 -42 -16 24 3866 2630 4 MP gs c59 PP gr c0 DP 16 37 25 25 -15 -37 2815 2540 4 MP gs c61 PP gr c0 DP 16 -26 25 -17 -15 25 4219 2228 4 MP gs c68 PP gr c0 DP 16 22 25 26 -15 -23 2691 2299 4 MP gs c66 PP gr c0 DP 15 35 26 25 -16 -35 2800 2503 4 MP gs c62 PP gr c0 DP 15 24 26 25 -16 -24 2707 2323 4 MP gs c66 PP gr c0 DP 16 33 25 25 -15 -33 2784 2468 4 MP gs c62 PP gr c0 DP 15 -25 26 -39 -16 24 3891 2588 4 MP gs c60 PP gr c0 DP 16 26 25 25 -15 -25 2722 2348 4 MP gs c65 PP gr c0 DP 15 -25 25 -18 -15 25 4194 2246 4 MP gs c67 PP gr c0 DP 15 30 26 25 -16 -30 2769 2435 4 MP gs c63 PP gr c0 DP 15 27 26 26 -16 -28 2738 2376 4 MP gs c64 PP gr c0 DP 16 29 25 25 -15 -29 2753 2405 4 MP gs c64 PP gr c0 DP 15 -25 25 -37 -15 25 3916 2548 4 MP gs c61 PP gr c0 DP 15 -25 26 -20 -16 25 4169 2266 4 MP gs c67 PP gr c0 DP 16 -25 25 -21 -16 25 4144 2287 4 MP gs c67 PP gr c0 DP 16 -24 25 -35 -15 25 3941 2511 4 MP gs c62 PP gr c0 DP 16 -25 25 -22 -15 25 4118 2309 4 MP gs c66 PP gr c0 DP 16 -25 25 -32 -16 24 3967 2477 4 MP gs c62 PP gr c0 DP 15 -25 25 -24 -15 25 4093 2333 4 MP gs c66 PP gr c0 DP 15 -24 26 -31 -16 25 3992 2444 4 MP gs c63 PP gr c0 DP 15 -25 26 -25 -16 25 4068 2358 4 MP gs c65 PP gr c0 DP 16 -25 25 -28 -15 24 4017 2414 4 MP gs c64 PP gr c0 DP 16 -25 25 -27 -16 25 4043 2385 4 MP gs c64 PP gr c0 DP gr end eplot epage end showpage %%EndDocument endTexFig 151 1315 a Fl(Figure)15 b(A.2:)k(Norm)c(of)g Ff(jjr)p Fi(v)r Ff(jj)e Fl(for)h(the)i(stress)e(\014eld)j Fi(v)g Fl(in)f(a)f(design)h(with)f(comp)q(osite)h(materials)59 1455 y(A)21 b(plot)f(of)h(the)f(norm)g Ff(jjr)p Fi(v)r Ff(jj)f Fl(of)h(the)h(gradien)o(t)f(of)g(the)h(stress)f(\014eld)i Fi(v)g Fl(in)f(the)g(b)q(ounded)h(domain)f Fi(D)h Fl(=)59 1511 y(\(0)p Fi(;)8 b Fl(1\))h Ff(\002)i Fl(\(0)p Fi(;)d Fl(1\))14 b(where)i Fj(LAMBDA)f Fl(=)h(0.008)f(is)h(giv)o(en)h(in)f (Figure)h(A.2.)k(Figure)16 b(A.3)g(sho)o(ws)f(the)h(con)o(tour)g(plot) 59 1568 y(for)f(this)g(surface.)59 1694 y Fk(References)82 1796 y Fl([1])21 b(B.)c(M.)g(Av)o(eric)o(k,)h(R.)f(G.)f(Carter,)h(J.)g (J.)g(Mor)o(\023)-21 b(e,)15 b(and)j(G.)e(L.)h(Xue.)27 b(The)17 b(MINP)l(A)o(CK-2)g(test)g(problem)152 1852 y(collection.)22 b(T)l(ec)o(hnical)17 b(Rep)q(ort)f(ANL/MCS-P153-0692,) d(Argonne)i(National)h(Lab)q(oratory)l(,)e(1992.)82 1946 y([2])21 b(I.)h(Bongartz,)h(A.)e(R.)h(Conn,)i(N.)d(I.)h(M.)f(Gould,)j (and)e(Ph.)g(L.)g(T)l(oin)o(t.)40 b Fb(CUTE)p Fl(:)21 b(Constrained)i(and)152 2002 y(Unconstrained)17 b(Testing)e(En)o (vironmen)o(t.)20 b Fa(A)o(CM)15 b(T)m(r)n(ans.)g(Math.)h(Softwar)n(e)p Fl(,)f(21\(1\):123{16)o(0,)d(1995.)82 2096 y([3])21 b(W.)14 b(J.)g(Co)q(dy)l(.)j Fb(MA)o(CHAR)p Fl(:)e(A)f(subroutine)g(to)g (dynamically)h(determine)g(mac)o(hine)g(parameters.)i Fa(A)o(CM)152 2153 y(T)m(r)n(ans.)e(Math.)h(Softw.)p Fl(,)f(14:303{311,)d(1988.)82 2246 y([4])21 b(T.)c(F.)f(Coleman,)i(B.)e (S.)h(Garb)q(o)o(w,)f(and)i(J.)f(J.)g(Mor)o(\023)-21 b(e.)23 b(F)l(ortran)16 b(subroutines)i(for)f(estimating)g(sparse)152 2303 y(Hessian)f(matrices.)k Fa(A)o(CM)15 b(T)m(r)n(ans.)g(Math.)h (Softwar)n(e)p Fl(,)f(11:378,)e(1985.)82 2397 y([5])21 b(T.)f(F.)g(Coleman,)i(B.)e(S.)g(Garb)q(o)o(w,)h(and)f(J.)h(J.)f(Mor)o (\023)-21 b(e.)34 b(Soft)o(w)o(are)19 b(for)h(estimating)h(sparse)f (Hessian)152 2453 y(matrices.)g Fa(A)o(CM)15 b(T)m(r)n(ans.)g(Math.)h (Softwar)n(e)p Fl(,)f(11:363{377,)d(1985.)82 2547 y([6])21 b(T.)16 b(F.)g(Coleman)h(and)f(J.)g(J.)h(Mor)o(\023)-21 b(e.)21 b(Estimation)c(of)f(sparse)g(Hessian)h(matrices)f(and)h(graph)f (coloring)152 2603 y(problems.)21 b Fa(Math.)16 b(Pr)n(o)n(gr)n(amming) p Fl(,)f(28:243{270,)d(1984.)981 2758 y(18)p eop %%Page: 19 19 19 18 bop 266 272 a 23308497 14917438 4538941 13814169 35785277 38350766 startTexFig 266 272 a %%BeginDocument: osc2.eps % MathWorks dictionary /MathWorks 150 dict begin % definition operators /bdef {bind def} bind def /ldef {load def} bind def /xdef {exch def} bdef /xstore {exch store} bdef % operator abbreviations /c /clip ldef /cc /concat ldef /cp /closepath ldef /gr /grestore ldef /gs /gsave ldef /mt /moveto ldef /np /newpath ldef /cm /currentmatrix ldef /sm /setmatrix ldef /rc {rectclip} bdef /rf {rectfill} bdef /rm /rmoveto ldef /rl /rlineto ldef /s /show ldef /sc {setcmykcolor} bdef /sr /setrgbcolor ldef /w /setlinewidth ldef /cap /setlinecap ldef % page state control /pgsv () def /bpage {/pgsv save def} bdef /epage {pgsv restore} bdef /bplot /gsave ldef /eplot {stroke grestore} bdef % orientation switch /portraitMode 0 def /landscapeMode 1 def % coordinate system mappings /dpi2point 0 def % font control /FontSize 0 def /FMS { /FontSize xstore %save size 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23 2 41 3 42 3 42 4 42 6 42 5 41 7 42 8 42 8 42 6 24 4 18 10 42 12 41 13 42 14 42 30 42 23 29 898 1739 42 MP stroke 23 29 30 42 14 42 13 42 11 41 11 42 4 19 5 23 9 42 8 42 6 42 6 41 5 42 5 42 3 42 3 42 2 41 1 26 1 16 0 42 0 42 0 42 -1 17 -1 25 -2 41 -3 42 -3 42 -4 42 -6 42 -5 41 -7 42 -8 42 -8 42 -6 24 -4 18 -10 42 -12 41 -13 42 -14 42 0 1 -30 41 -23 29 6256 1739 43 MP stroke gr end eplot epage end showpage %%EndDocument endTexFig 116 1315 a Fl(Figure)16 b(A.3:)j(Con)o(tours)14 b(of)h Ff(jjr)p Fi(v)r Ff(jj)e Fl(for)i(the)g(stress)g(\014eld)h Fi(v)h Fl(in)f(a)f(design)h(with)g(comp)q(osite)f(materials)82 1455 y([7])21 b(A.)c(R.)g(Conn,)g(N.)g(I.)g(M.)f(Gould,)i(and)f(Ph.)g (L.)g(T)l(oin)o(t.)25 b(An)18 b(in)o(tro)q(duction)g(to)e(the)h (structure)g(of)g(large)152 1511 y(scale)f(nonlinear)f(optimization)h (problems)f(and)g(the)f(LANCELOT)h(pro)s(ject.)j(Rep)q(ort)d(89-19,)e (Nam)o(ur)152 1568 y(Univ)o(ersit)o(y)l(,)j(Nam)o(ur,)e(Belgium,)j (1989.)82 1661 y([8])k(A.)16 b(R.)g(Conn,)g(N.)g(I.)g(M.)g(Gould,)g (and)h(Ph.)f(L.)g(T)l(oin)o(t.)22 b Fb(LANCELOT)p Fl(.)h(Springer)17 b(Series)g(in)g(Computa-)152 1718 y(tional)f(Mathematics.)f(Springer-V) l(erlag,)h(1992.)82 1812 y([9])21 b(I.)14 b(S.)f(Du\013)f(and)i(J.)f (K.)g(Reid.)18 b(MA27:)g(A)13 b(set)g(of)f(Fortran)g(subroutines)i(for) f(solving)h(sparse)f(symmetric)152 1868 y(sets)19 b(of)f(linear)h (equations.)30 b(T)l(ec)o(hnical)20 b(Rep)q(ort)f(R-10533,)f(AERE)g (Harw)o(ell)h(Lab)q(oratory)l(,)g(Harw)o(ell,)152 1925 y(UK,)d(1983.)59 2018 y([10])21 b(J.)11 b(J.)g(Mor)o(\023)-21 b(e,)10 b(B.)g(S.)h(Garb)q(o)o(w,)f(and)h(K.)g(E.)f(Hillstrom.)k(T)l (esting)d(unconstrained)h(optimization)g(soft)o(w)o(are.)152 2075 y Fa(A)o(CM)k(T)m(r)n(ans.)e(Math.)j(Softwar)n(e)p Fl(,)d(7\(1\):17{41,)e(1981.)981 2758 y(19)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF SHAR_EOF fi # end of overwriting check cd .. if test ! -d 'Drivers' then mkdir 'Drivers' fi cd 'Drivers' if test ! -d 'Sp' then mkdir 'Sp' fi cd 'Sp' if test -f 'driver2.f' then echo shar: will not over-write existing file "'driver2.f'" else cat << \SHAR_EOF > 'driver2.f' C C STENMIN MINIMIZES AN UNCONSTRAINED NONLINEAR FUNCTION IN N C UNKNOWNS WHERE THE HESSIAN IS LARGE AND SPARSE, USING TENSOR C METHODS. C C EXAMPLE OF USE FOR STENMIN. THE TEST PROBLEM IS THE C OPTIMAL DESIGN WITH COMPOSITE MATERIALS PROBLEM FROM C THE MINPACK-2 TEST PROBLEM COLLECTION. C C*****WARNING***** C C THE SINGLE-PRECISION FINITE DIFFERENCE HESSIAN APPROXIMATION C MAY NOT BE ACCURATE ENOUGH. THIS MAY LEAD TO A CONVERGENCE C DETERIORATION. THE USER IS ADVISED TO PROVIDE AN ANALYTIC C HESSIAN WHEN THE SINGLE-PRECISION VERSION OF STENMIN IS USED. C C***************** C C ALI BOUARICHA, OCTOBER 1994. C MCS DIVISION, ARGONNE NATIONAL LAB. C INTEGER NMAX, N, NZ, LIRN, LICN, ILIM, IPR, METHOD INTEGER GRDFLG, HSNFLG, NDIGIT, MSG, LWRK, LIWRK INTEGER TERMCD, INFORM, I, J, K, NX, NY REAL FSCALE, GRADTL, STEPTL, FPLS, STEPMX REAL LAMBDA, HX, HY, TEMP, ONE PARAMETER ( NMAX = 10000, LIRN = 50000, LICN = 500000 ) PARAMETER ( LIWRK = 2 * LIRN + 12 * NMAX + 2 ) PARAMETER ( LWRK = 7 * NMAX ) INTEGER IRN ( LIRN ), ICN ( LICN ) INTEGER IWRK( LIWRK ) REAL X ( NMAX ), TYPX( NMAX ), XPLS( NMAX ) REAL GPLS( NMAX ), HESS( LICN ), WRK ( LWRK ) REAL HTV ( NMAX ) COMMON / PARAM / NX, NY COMMON / OTHER / LAMBDA EXTERNAL DODCF, DODCG, STDUSH INTRINSIC FLOAT, MIN DATA ONE / 1.0E0 / C RUN OPTIMAL DESIGN PROBLEM WITH NX = 50, NY = 50, AND LAMBDA = 0.008 NX = 50 NY = 50 LAMBDA = 0.008E0 N = NX * NY C COMPUTE THE STANDARD STARTING POINT. HX = ONE/FLOAT(NX+1) HY = ONE/FLOAT(NY+1) DO 20 J = 1, NY TEMP = FLOAT(MIN(J,NY-J+1))*HY DO 10 I = 1, NX K = NX*(J-1) + I X(K) = -(MIN(FLOAT(MIN(I,NX-I+1))*HX,TEMP))**2 10 CONTINUE 20 CONTINUE C DEFINE THE SPARSITY STRUCTURE OF THE HESSIAN. CALL DODCSP(NX,NY,NZ,IRN,ICN) C SET THE DEFAULT VALUES OF THE PACKAGE. CALL STDFLT(N, TYPX, FSCALE, GRADTL, STEPTL, ILIM, STEPMX, * IPR, METHOD, GRDFLG, HSNFLG, NDIGIT, INFORM, MSG) GRADTL = 1.0E-3 GRDFLG = 2 C CALL THE SPARSE OPTIMIZER. CALL STUMCS(N,X,NZ,IRN,LIRN,ICN,LICN,DODCF,DODCG, * STDUSH,TYPX,FSCALE,GRADTL,STEPTL,ILIM,STEPMX,IPR, * METHOD,GRDFLG,HSNFLG,NDIGIT,MSG,XPLS,FPLS,GPLS, * HESS,WRK,LWRK,IWRK,LIWRK,TERMCD,HTV,INFORM) STOP END SUBROUTINE DODCF(N,X,F) C ********** C C PURPOSE C ------- C C THIS SUBROUTINE COMPUTES THE FUNCTION OF THE C OPTIMAL DESIGN WITH COMPOSITE MATERIALS PROBLEM. C C PARAMETERS C ---------- C C N IS THE DIMENSION OF THE PROBLEM C C NX IS AN INTEGER VARIABLE. C ON ENTRY NX IS THE NUMBER OF GRID POINTS IN THE FIRST C COORDINATE DIRECTION. C ON EXIT NX IS UNCHANGED. C C NY IS AN INTEGER VARIABLE. C ON ENTRY NY IS THE NUMBER OF GRID POINTS IN THE SECOND C COORDINATE DIRECTION. C ON EXIT NY IS UNCHANGED. C C X IS A DOUBLE PRECISION (REAL) ARRAY OF DIMENSION N = NX*NY. C C F IS A DOUBLE PRECISION (REAL) VARIABLE. C ON EXIT F IS SET TO THE FUNCTION EVALUATED AT X. C C LAMBDA IS A DOUBLE PRECISION (REAL) VARIABLE. C ON ENTRY LAMBDA IS THE LAGRANGE MULTIPLIER. C ON EXIT LAMBDA IS UNCHANGED. C C SUBPROGRAMS CALLED C C MINPACK-SUPPLIED ... DODCPS C C MINPACK-2 PROJECT. NOVEMBER 1993. C ARGONNE NATIONAL LABORATORY AND UNIVERSITY OF MINNESOTA. C BRETT M. AVERICK. MODIFIED BY ALI BOUARICHA ON OCTOBER 1994. C C ********** C INTEGER N, NX, NY REAL F, LAMBDA REAL X(N) COMMON/PARAM/NX,NY COMMON/OTHER/LAMBDA C C LOCAL VARIABLES C INTEGER I, J, K REAL MU1, MU2, ONE, P5, TWO, ZERO REAL AREA, DPSI, DVDX, DVDY, GRADV, HX, HXHY REAL HY, TEMP, T1, T2, V, VB, VL, VR, VT PARAMETER (ZERO=0.0E0,P5=0.5E0,ONE=1.0E0,TWO=2.0E0) PARAMETER (MU1=ONE,MU2=TWO) INTRINSIC FLOAT, SQRT EXTERNAL DODCPS C C INITIALIZATION. C HX = ONE/FLOAT(NX+1) HY = ONE/FLOAT(NY+1) HXHY = HX*HY AREA = P5*HXHY C C COMPUTE THE BREAK POINTS. C T1 = SQRT(TWO*LAMBDA*MU1/MU2) T2 = SQRT(TWO*LAMBDA*MU2/MU1) F = ZERO C C COMPUTATION OF THE FUNCTION OVER THE LOWER C TRIANGULAR ELEMENTS. C DO 50 J = 0, NY DO 40 I = 0, NX K = NX*(J-1) + I V = ZERO VR = ZERO VT = ZERO IF (J .GE. 1 .AND. I .GE. 1) V = X(K) IF (I .LT. NX .AND. J .GT. 0) VR = X(K+1) IF (I .GT. 0 .AND. J .LT. NY) VT = X(K+NX) DVDX = (VR-V)/HX DVDY = (VT-V)/HY GRADV = DVDX**2 + DVDY**2 CALL DODCPS(GRADV,MU1,MU2,T1,T2,DPSI,0,LAMBDA) F = F + DPSI 40 CONTINUE 50 CONTINUE C C COMPUTATION OF THE FUNCTION OVER THE UPPER C TRIANGULAR ELEMENTS. C DO 70 J = 1, NY + 1 DO 60 I = 1, NX + 1 K = NX*(J-1) + I VB = ZERO VL = ZERO V = ZERO IF (I .LE. NX .AND. J .GT. 1) VB = X(K-NX) IF (I .GT. 1 .AND. J .LE. NY) VL = X(K-1) IF (I .LE. NX .AND. J .LE. NY) V = X(K) DVDX = (V-VL)/HX DVDY = (V-VB)/HY GRADV = DVDX**2 + DVDY**2 CALL DODCPS(GRADV,MU1,MU2,T1,T2,DPSI,0,LAMBDA) F = F + DPSI 60 CONTINUE 70 CONTINUE C C SCALE THE FUNCTION. C F = AREA*F C C INTEGRATE V OVER THE DOMAIN. C TEMP = ZERO DO 80 K = 1, NX*NY TEMP = TEMP + X(K) 80 CONTINUE F = F + HXHY*TEMP END C SUBROUTINE DODCG(N,X,FGRAD) C ********** C C PURPOSE C ------- C C THIS SUBROUTINE COMPUTES THE GRADIENT OF THE C OPTIMAL DESIGN WITH COMPOSITE MATERIALS PROBLEM. C C PARAMETERS C ---------- C C N IS THE DIMENSION OF THE PROBLEM C C NX IS AN INTEGER VARIABLE. C ON ENTRY NX IS THE NUMBER OF GRID POINTS IN THE FIRST C COORDINATE DIRECTION. C ON EXIT NX IS UNCHANGED. C C NY IS AN INTEGER VARIABLE. C ON ENTRY NY IS THE NUMBER OF GRID POINTS IN THE SECOND C COORDINATE DIRECTION. C ON EXIT NY IS UNCHANGED. C C X IS A DOUBLE PRECISION (REAL) ARRAY OF DIMENSION N = NX*NY. C C FGRAD IS A DOUBLE PRECISION (REAL) ARRAY OF DIMENSION N = NX*NY. C ON ENTRY FGRAD NEED NOT BE SPECIFIED. C ON EXIT FGRAD CONTAINS THE GRADIENT EVALUATED AT X. C C LAMBDA IS A DOUBLE PRECISION (REAL) VARIABLE. C ON ENTRY LAMBDA IS THE LAGRANGE MULTIPLIER. C ON EXIT LAMBDA IS UNCHANGED. C C SUBPROGRAMS CALLED C C MINPACK-SUPPLIED ... DODCPS C C MINPACK-2 PROJECT. NOVEMBER 1993. C ARGONNE NATIONAL LABORATORY AND UNIVERSITY OF MINNESOTA. C BRETT M. AVERICK. MODIFIED BY ALI BOUARICHA ON OCTOBER 1994. C C ********** C INTEGER N, NX, NY REAL X(N), FGRAD(N), LAMBDA COMMON/PARAM/NX,NY COMMON/OTHER/LAMBDA C C LOCAL VARIABLES C INTEGER I, J, K REAL MU1, MU2, ONE, P5, TWO, ZERO REAL AREA, DPSIP, DVDX, DVDY, GRADV, HX, HXHY REAL HY, T1, T2, V, VB, VL, VR, VT PARAMETER (ZERO=0.0E0,P5=0.5E0,ONE=1.0E0,TWO=2.0E0) PARAMETER (MU1=ONE,MU2=TWO) INTRINSIC FLOAT, SQRT EXTERNAL DODCPS C INITIALIZATION. HX = ONE/FLOAT(NX+1) HY = ONE/FLOAT(NY+1) HXHY = HX*HY AREA = P5*HXHY C C COMPUTE THE BREAK POINTS. C T1 = SQRT(TWO*LAMBDA*MU1/MU2) T2 = SQRT(TWO*LAMBDA*MU2/MU1) DO 30 K = 1, NX*NY FGRAD(K) = ZERO 30 CONTINUE C C COMPUTATION OF THE THE GRADIENT OVER THE LOWER C TRIANGULAR ELEMENTS. C DO 50 J = 0, NY DO 40 I = 0, NX K = NX*(J-1) + I V = ZERO VR = ZERO VT = ZERO IF (J .GE. 1 .AND. I .GE. 1) V = X(K) IF (I .LT. NX .AND. J .GT. 0) VR = X(K+1) IF (I .GT. 0 .AND. J .LT. NY) VT = X(K+NX) DVDX = (VR-V)/HX DVDY = (VT-V)/HY GRADV = DVDX**2 + DVDY**2 CALL DODCPS(GRADV,MU1,MU2,T1,T2,DPSIP,1,LAMBDA) IF (I .GE. 1 .AND. J .GE. 1) + FGRAD(K) = FGRAD(K) - TWO*(DVDX/HX+DVDY/HY)*DPSIP IF (I .LT. NX .AND. J .GT. 0) + FGRAD(K+1) = FGRAD(K+1) + TWO*(DVDX/HX)*DPSIP IF (I .GT. 0 .AND. J .LT. NY) + FGRAD(K+NX) = FGRAD(K+NX) + TWO*(DVDY/HY)*DPSIP 40 CONTINUE 50 CONTINUE C C COMPUTATION OF THE GRADIENT OVER THE UPPER C TRIANGULAR ELEMENTS. C DO 70 J = 1, NY + 1 DO 60 I = 1, NX + 1 K = NX*(J-1) + I VB = ZERO VL = ZERO V = ZERO IF (I .LE. NX .AND. J .GT. 1) VB = X(K-NX) IF (I .GT. 1 .AND. J .LE. NY) VL = X(K-1) IF (I .LE. NX .AND. J .LE. NY) V = X(K) DVDX = (V-VL)/HX DVDY = (V-VB)/HY GRADV = DVDX**2 + DVDY**2 CALL DODCPS(GRADV,MU1,MU2,T1,T2,DPSIP,1,LAMBDA) IF (I .LE. NX .AND. J .GT. 1) + FGRAD(K-NX) = FGRAD(K-NX) - TWO*(DVDY/HY)*DPSIP IF (I .GT. 1 .AND. J .LE. NY) + FGRAD(K-1) = FGRAD(K-1) - TWO*(DVDX/HX)*DPSIP IF (I .LE. NX .AND. J .LE. NY) + FGRAD(K) = FGRAD(K) + TWO*(DVDX/HX+DVDY/HY)*DPSIP 60 CONTINUE 70 CONTINUE C C INTEGRATE V OVER THE DOMAIN. C DO 90 K = 1, NX*NY FGRAD(K) = AREA*FGRAD(K) + HXHY 90 CONTINUE END C SUBROUTINE DODCPS(T,MU1,MU2,T1,T2,RESULT,OPTION,LAMBDA) C C ********** C C THIS SUBROUTINE COMPUTES THE FUNCTION PSI(T) AND THE SCALED C FUNCTIONS PSI'(T)/T AND PSI''(T)/T FOR THE OPTIMAL DESIGN C WITH COMPOSITE MATERIALS PROBLEM. C C THE SUBROUTINE STATEMENT IS C C SUBROUTINE DODCPS(T,MU1,MU2,T1,T2,RESULT,OPTION,LAMBDA) C C WHERE C C T IS A DOUBLE PRECISION (REAL) VARIABLE. C ON ENTRY T IS THE VARIABLE T C ON EXIT T IS UNCHANGED C C MU1 IS A DOUBLE PRECISION (REAL) VARIABLE. C ON ENTRY MU1 IS THE RECIPROCAL SHEAR MODULUS OF MATERIAL 1. C ON EXIT MU1 IS UNCHANGED. C C MU2 IS A DOUBLE PRECISION (REAL) VARIABLE. C ON ENTRY MU2 IS THE RECIPROCAL SHEAR MODULUS OF MATERIAL 2. C ON EXIT MU2 IS UNCHANGED. C C T1 IS A DOUBLE PRECISION (REAL) VARIABLE. C ON ENTRY T1 IS THE FIRST BREAKPOINT. C ON EXIT T1 IS UNCHANGED. C C T2 IS A DOUBLE PRECISION (REAL) VARIABLE. C ON ENTRY T2 IS THE SECOND BREAKPOINT. C ON EXIT T2 IS UNCHANGED. C C RESULT IS A DOUBLE PRECISION (REAL) VARIABLE. C ON ENTRY RESULT NEED NOT BE SPECIFIED. C ON EXIT RESULT IS SET ACCORDING TO TASK. C C OPTION IS AN INTEGER VARIABLE. C ON ENTRY OPTION SPECIFIES THE ACTION OF THE SUBROUTINE: C C IF OPTION = 0 THEN EVALUATE THE FUNCTION PSI(T). C IF OPTION = 1 THEN EVALUATE THE SCALED FUNCTION PSI'(T)/T. C IF OPTION = 2 THEN EVALUATE THE SCALED FUNCTION PSI''(T)/T. C C ON OPTION TASK IS UNCHANGED. C C LAMBDA IS A DOUBLE PRECISION (REAL) VARIABLE C ON ENTRY LAMBDA IS THE LAGRANGE MULTIPLIER. C ON EXIT LAMBDA IS UNCHANGED. C C MINPACK-2 PROJECT. NOVEMBER 1993. C ARGONNE NATIONAL LABORATORY AND UNIVERSITY OF MINNESOTA. C BRETT M. AVERICK. MODIFIED BY ALI BOUARICHA ON OCTOBER 1994. C C ********** C INTEGER OPTION REAL T, MU1, MU2, T1, T2, RESULT, LAMBDA C C LOCAL VARIABLES C REAL P25, P5, ZERO PARAMETER (ZERO=0.0E0,P25=0.25E0,P5=0.5E0) REAL SQRTT INTRINSIC SQRT C SQRTT = SQRT(T) IF (OPTION .EQ. 0) THEN IF (SQRTT .LE. T1) THEN RESULT = P5*MU2*T ELSE IF (SQRTT .GT. T1 .AND. SQRTT .LT. T2) THEN RESULT = MU2*T1*SQRTT - LAMBDA*MU1 ELSE IF (SQRTT .GE. T2) THEN RESULT = P5*MU1*T + LAMBDA*(MU2-MU1) END IF ELSE IF (OPTION .EQ. 1) THEN IF (SQRTT .LE. T1) THEN RESULT = P5*MU2 ELSE IF (SQRTT .GT. T1 .AND. SQRTT .LT. T2) THEN RESULT = P5*MU2*T1/SQRTT ELSE IF (SQRTT .GE. T2) THEN RESULT = P5*MU1 END IF ELSE IF (OPTION .EQ. 2) THEN IF (SQRTT .LE. T1) THEN RESULT = ZERO ELSE IF (SQRTT .GT. T1 .AND. SQRTT .LT. T2) THEN RESULT = -P25*MU2*T1/(SQRTT*T) ELSE IF (SQRTT .GE. T2) THEN RESULT = ZERO END IF END IF END C SUBROUTINE DODCSP(NX,NY,NNZ,INDROW,INDCOL) C C ********** C C SUBROUTINE DODCSP C C THIS SUBROUTINE DEFINES THE SPARSITY STRUCTURE OF THE HESSIAN C MATRIX FOR THE OPTIMAL DESIGN WITH COMPOSITES PROBLEM. C C THE SUBROUTINE STATEMENT IS C C SUBROUTINE DODCSP(NX,NY,NNZ,INDROW,INDCOL) C C WHERE C C NX IS AN INTEGER VARIABLE. C ON ENTRY NX IS THE NUMBER OF GRID POINTS IN THE FIRST C COORDINATE DIRECTION. C ON EXIT NX IS UNCHANGED. C C NY IS AN INTEGER VARIABLE. C ON ENTRY NY IS THE NUMBER OF GRID POINTS IN THE SECOND C COORDINATE DIRECTION. C ON EXIT NY IS UNCHANGED. C C NNZ IS AN INTEGER VARIABLE. C ON ENTRY NNZ NEED NOT BE SPECIFIED. C ON EXIT NNZ IS SET TO THE NUMBER OF NONZEROS IN THE C LOWER TRIANGLE OF THE HESSIAN MATRIX. C C INDROW IS AN INTEGER ARRAY OF DIMENSION AT LEAST NNZ. C ON ENTRY INDROW NEED NOT BE SPECIFIED. C ON EXIT INDROW CONTAINS THE ROW INDICES OF THE NONZEROS C IN THE LOWER TRIANGLE OF THE HESSIAN MATRIX. C C INDCOL IS AN INTEGER ARRAY OF DIMENSION AT LEAST NNZ. C ON ENTRY INDCOL NEED NOT BE SPECIFIED. C ON EXIT INDCOL CONTAINS THE COLUMN INDICES OF THE NONZEROS C IN THE LOWER TRIANGLE OF THE HESSIAN MATRIX. C C MINPACK-2 PROJECT. NOVEMBER 1993. C ARGONNE NATIONAL LABORATORY AND UNIVERSITY OF MINNESOTA. C BRETT M. AVERICK. MODIFIED BY ALI BOUARICHA ON OCTOBER 1994. C C ********** C INTEGER NX, NY, NNZ INTEGER INDROW(*), INDCOL(*) C C LOCAL VARIABLES C INTEGER I, J C C COMPUTE THE SPARSITY STRUCTURE. C NNZ = 0 DO 20 J = 1, NY DO 10 I = 1, NX NNZ = NNZ + 1 INDROW(NNZ) = (J-1)*NX + I INDCOL(NNZ) = (J-1)*NX + I IF (I .NE. NX) THEN NNZ = NNZ + 1 INDROW(NNZ) = (J-1)*NX + I + 1 INDCOL(NNZ) = (J-1)*NX + I END IF IF (J .NE. NY) THEN NNZ = NNZ + 1 INDROW(NNZ) = (J-1)*NX + I + NX INDCOL(NNZ) = (J-1)*NX + I IF (I .NE. 1) THEN NNZ = NNZ + 1 INDROW(NNZ) = (J-1)*NX + I + NX - 1 INDCOL(NNZ) = (J-1)*NX + I END IF END IF 10 CONTINUE 20 CONTINUE END SHAR_EOF fi # end of overwriting check if test -f 'driver1.f' then echo shar: will not over-write existing file "'driver1.f'" else cat << \SHAR_EOF > 'driver1.f' C C STENMIN MINIMIZES AN UNCONSTRAINED NONLINEAR FUNCTION IN N C UNKNOWNS WHERE THE HESSIAN IS LARGE AND SPARSE, USING TENSOR C METHODS. C C EXAMPLE OF USE FOR STENMIN. THE TEST PROBLEM IS C THE BROYDEN TRIDIAGONAL (SOURCE: BUCKLEY#78 (P. 42)). C C ALI BOUARICHA, OCTOBER 1994. C MCS DIVISION, ARGONNE NATIONAL LAB. C INTEGER NMAX, N, NZ, LIRN, LICN, ILIM, IPR, METHOD INTEGER GRDFLG, HSNFLG, NDIGIT, MSG, LWRK, LIWRK INTEGER TERMCD, INFORM, I REAL FSCALE, GRADTL, STEPTL, FPLS, STEPMX, ONE PARAMETER ( NMAX = 10000, LIRN = 50000, LICN = 500000 ) PARAMETER ( LIWRK = 2 * LIRN + 12 * NMAX + 2 ) PARAMETER ( LWRK = 7 * NMAX ) INTEGER IRN ( LIRN ), ICN ( LICN ) INTEGER IWRK( LIWRK ) REAL X ( NMAX ), TYPX( NMAX ), XPLS( NMAX ) REAL GPLS( NMAX ), HESS( LICN ), WRK ( LWRK ) REAL HTV ( NMAX ) EXTERNAL FCN, UGRAD, UHESS DATA ONE / 1.0E0 / C RUN BROYDEN PROBLEM WITH N = 10000. N = 10000 C COMPUTE THE STANDARD STARTING POINT. DO 10 I = 1, N X(I) = -ONE 10 CONTINUE C SET THE DEFAULT VALUES. CALL STDFLT(N, TYPX, FSCALE, GRADTL, STEPTL, ILIM, STEPMX, * IPR, METHOD, GRDFLG, HSNFLG, NDIGIT, INFORM, MSG) GRADTL = 1.0E-3 GRDFLG = 2 HSNFLG = 2 C CALL THE SPARSE OPTIMIZER. CALL STUMCS(N,X,NZ,IRN,LIRN,ICN,LICN,FCN,UGRAD, * UHESS,TYPX,FSCALE,GRADTL,STEPTL,ILIM,STEPMX,IPR, * METHOD,GRDFLG,HSNFLG,NDIGIT,MSG,XPLS,FPLS,GPLS, * HESS,WRK,LWRK,IWRK,LIWRK,TERMCD,HTV,INFORM) STOP END C C THE FOLLOWING IS A SUBROUTINE FOR THE BROYDEN TRIDIAGONAL C PROBLEM (SOURCE: BUCKLEY#78 (P. 42)) C SUBROUTINE FCN(N, X, F) INTEGER N REAL X(N), F C C LOCAL VARIABLES C INTEGER I REAL ONE, TWO, THREE DATA ONE, TWO, THREE / 1.0E0, 2.0E0, 3.0E0 / C F = ((THREE - TWO * X(1)) * X(1) - TWO * X(2) + ONE) ** 2 DO 10 I = 2, N-1 F = F + ((THREE - TWO * X(I)) * X(I) - X(I-1) - * TWO * X(I+1) + ONE) ** 2 10 CONTINUE F = F + ((THREE - TWO * X(N)) * X(N) - X(N-1) + ONE) ** 2 RETURN END C C THE FOLLOWING IS A SUBROUTINE FOR THE GRADIENT OF THE BROYDEN C TRIDIAGONAL PROBLEM C SUBROUTINE UGRAD(N, X, G) INTEGER N REAL X(N), G(N) C C LOCAL VARIABLES C INTEGER I REAL RL, RM, RR, ONE, TWO, THREE, FOUR DATA ONE, TWO, THREE, FOUR/ 1.0E0, 2.0E0, 3.0E0, 4.0E0/ C RL = (THREE - TWO * X(1)) * X(1) - TWO * X(2) + ONE RR = (THREE - TWO * X(2)) * X(2) - X(1) - TWO * X(3) + ONE G(1) = TWO * (RL * (THREE - FOUR * X(1)) - RR) DO 10 I = 2, N-1 IF(I .NE. 2) THEN RL = (THREE - TWO * X(I-1)) * X(I-1) - X(I-2) - * TWO * X(I) + ONE ENDIF RM = (THREE - TWO * X(I)) * X(I) - X(I-1) - * TWO * X(I+1) + ONE IF(I .EQ. N-1) THEN RR = (THREE - TWO * X(N)) * X(N) - X(N-1) + ONE ELSE RR = (THREE - TWO * X(I+1)) * X(I+1) - X(I) - * TWO * X(I+2) + ONE ENDIF G(I) = -TWO * (TWO * RL - RM * (THREE - FOUR * X(I)) + RR) 10 CONTINUE G(N) = -TWO * (TWO * RM - RR * (THREE - FOUR * X(N))) RETURN END C C THE FOLLOWING IS A SUBROUTINE FOR THE HESSIAN OF THE BROYDEN C TRIDIAGONAL PROBLEM C SUBROUTINE UHESS(N,X,NZ,LICN,HESS,IRN,ICN) INTEGER N, NZ, LICN INTEGER IRN(*), ICN(LICN) REAL X(N), HESS(LICN) C C LOCAL VARIABLES C INTEGER I REAL RL,RM,RR REAL ONE,TWO,THREE,FOUR DATA ONE, TWO, THREE, FOUR/1.0E0, 2.0E0, 3.0E0, 4.0E0/ C NZ = 1 RL = (THREE - TWO * X(1)) * X(1) - TWO * X(2) + ONE HESS(NZ) = TWO * ((THREE - FOUR * X(1))**2 - * FOUR * RL + ONE) IRN(NZ) = 1 ICN(NZ) = 1 DO 10 I = 2, N-1 IF(I .NE. 2) THEN NZ = NZ + 1 HESS(NZ) = FOUR IRN(NZ) = I ICN(NZ) = I-2 ENDIF NZ = NZ + 1 HESS(NZ) = -TWO * (TWO * (THREE - FOUR * X(I-1)) + * ONE * (THREE - FOUR * X(I))) IRN(NZ) = I ICN(NZ) = I-1 RM = (THREE - TWO * X(I)) * X(I) - X(I-1) - * TWO * X(I+1) + ONE NZ = NZ + 1 HESS(NZ) = -TWO * (-FOUR - (THREE - FOUR * X(I))**2 + * FOUR * RM - ONE) IRN(NZ) = I ICN(NZ) = I 10 CONTINUE RR = (THREE - TWO * X(N)) * X(N) - X(N-1) + ONE NZ = NZ + 1 HESS(NZ) = FOUR IRN(NZ) = N ICN(NZ) = N-2 NZ = NZ + 1 HESS(NZ) = -TWO * (TWO * (THREE - FOUR * X(N-1)) + * THREE - FOUR * X(N)) IRN(NZ) = N ICN(NZ) = N-1 NZ = NZ + 1 HESS(NZ) = TWO * (FOUR + (THREE - FOUR * X(N))**2 - FOUR * RR) IRN(NZ) = N ICN(NZ) = N RETURN END SHAR_EOF fi # end of overwriting check if test -f 'RES2' then echo shar: will not over-write existing file "'RES2'" else cat << \SHAR_EOF > 'RES2' STDRUO GRADIENT FLAG = 2 STDRUO HESSIAN FLAG = 0 STDRUO METHOD = 1 STDRUO ITERATION LIMIT = 500 STDRUO MACHINE EPSILON = 0.1192092895508E-06 STDRUO STEP TOLERANCE = 0.2422181023576E-04 STDRUO GRADIENT TOLERANCE = 0.1000000047497E-02 STDRUO MAXIMUM STEP SIZE = 0.3295197753906E+04 --------------------------------------------- STRSLT ITERATION K = 0 STRSLT FUNCTION AT X(K) STRSLT 0.4808234050870E-01 STRSLT SCALED GRADIENT AT X(K) STRSLT 0.3729332610965E-01 --------------------------------------------- STCHKS SUCCESSIVE ITERATES WITHIN TOLERANCE STCHKS CURRENT ITERATE IS PROBABLY SOLUTION --------------------------------------------- STRSLT ITERATION K = 496 STRSLT FUNCTION AT X(K) STRSLT -0.1037819869816E-01 STRSLT SCALED GRADIENT AT X(K) STRSLT 0.5580618511885E-02 --------------------------------------------- STRSLT NUMBER OF FUNCTION EVALUATIONS 3273 STRSLT NUMBER OF GRADIENT EVALUATIONS 497 STRSLT NUMBER OF HESSIAN EVALUATIONS 496 SHAR_EOF fi # end of overwriting check if test -f 'RES1' then echo shar: will not over-write existing file "'RES1'" else cat << \SHAR_EOF > 'RES1' STDRUO GRADIENT FLAG = 2 STDRUO HESSIAN FLAG = 2 STDRUO METHOD = 1 STDRUO ITERATION LIMIT = 500 STDRUO MACHINE EPSILON = 0.1192092895508E-06 STDRUO STEP TOLERANCE = 0.2422181023576E-04 STDRUO GRADIENT TOLERANCE = 0.1000000047497E-02 STDRUO MAXIMUM STEP SIZE = 0.1000000000000E+06 --------------------------------------------- STRSLT ITERATION K = 0 STRSLT FUNCTION AT X(K) STRSLT 0.1001100000000E+05 STRSLT SCALED GRADIENT AT X(K) STRSLT 0.3800000000000E+02 --------------------------------------------- STCHKS RELATIVE GRADIENT CLOSE TO ZERO STCHKS CURRENT ITERATE IS PROBABLY SOLUTION --------------------------------------------- STRSLT ITERATION K = 4 STRSLT FUNCTION AT X(K) STRSLT 0.1446949227102E-09 STRSLT SCALED GRADIENT AT X(K) STRSLT 0.8123669431370E-05 --------------------------------------------- STRSLT NUMBER OF FUNCTION EVALUATIONS 5 STRSLT NUMBER OF GRADIENT EVALUATIONS 5 STRSLT NUMBER OF HESSIAN EVALUATIONS 4 SHAR_EOF fi # end of overwriting check cd .. if test ! -d 'Dp' then mkdir 'Dp' fi cd 'Dp' if test -f 'driver2.f' then echo shar: will not over-write existing file "'driver2.f'" else cat << \SHAR_EOF > 'driver2.f' C C STENMIN MINIMIZES AN UNCONSTRAINED NONLINEAR FUNCTION IN N C UNKNOWNS WHERE THE HESSIAN IS LARGE AND SPARSE, USING TENSOR C METHODS. C C EXAMPLE OF USE FOR STENMIN. THE TEST PROBLEM IS THE C OPTIMAL DESIGN WITH COMPOSITE MATERIALS PROBLEM FROM C THE MINPACK-2 TEST PROBLEM COLLECTION. C C ALI BOUARICHA, OCTOBER 1994. C MCS DIVISION, ARGONNE NATIONAL LAB. C INTEGER NMAX, N, NZ, LIRN, LICN, ILIM, IPR, METHOD INTEGER GRDFLG, HSNFLG, NDIGIT, MSG, LWRK, LIWRK INTEGER TERMCD, INFORM, I, J, K, NX, NY DOUBLE PRECISION FSCALE, GRADTL, STEPTL, FPLS, STEPMX DOUBLE PRECISION LAMBDA, HX, HY, TEMP, ONE PARAMETER ( NMAX = 10000, LIRN = 50000, LICN = 500000 ) PARAMETER ( LIWRK = 2 * LIRN + 12 * NMAX + 2 ) PARAMETER ( LWRK = 7 * NMAX ) INTEGER IRN ( LIRN ), ICN ( LICN ) INTEGER IWRK( LIWRK ) DOUBLE PRECISION X ( NMAX ), TYPX( NMAX ), XPLS( NMAX ) DOUBLE PRECISION GPLS( NMAX ), HESS( LICN ), WRK ( LWRK ) DOUBLE PRECISION HTV ( NMAX ) COMMON / PARAM / NX, NY COMMON / OTHER / LAMBDA EXTERNAL DODCF, DODCG, STDUSH INTRINSIC DBLE, MIN DATA ONE / 1.0D0 / C RUN OPTIMAL DESIGN PROBLEM WITH NX = 50, NY = 50, AND LAMBDA = 0.008 NX = 50 NY = 50 LAMBDA = 0.008D0 N = NX * NY C COMPUTE THE STANDARD STARTING POINT. HX = ONE/DBLE(NX+1) HY = ONE/DBLE(NY+1) DO 20 J = 1, NY TEMP = DBLE(MIN(J,NY-J+1))*HY DO 10 I = 1, NX K = NX*(J-1) + I X(K) = -(MIN(DBLE(MIN(I,NX-I+1))*HX,TEMP))**2 10 CONTINUE 20 CONTINUE C DEFINE THE SPARSITY STRUCTURE OF THE HESSIAN. CALL DODCSP(NX,NY,NZ,IRN,ICN) C SET THE DEFAULT VALUES OF THE PACKAGE. CALL STDFLT(N, TYPX, FSCALE, GRADTL, STEPTL, ILIM, STEPMX, * IPR, METHOD, GRDFLG, HSNFLG, NDIGIT, INFORM, MSG) GRADTL = 1.0D-5 GRDFLG = 2 C CALL THE SPARSE OPTIMIZER. CALL STUMCD(N,X,NZ,IRN,LIRN,ICN,LICN,DODCF,DODCG, * STDUSH,TYPX,FSCALE,GRADTL,STEPTL,ILIM,STEPMX,IPR, * METHOD,GRDFLG,HSNFLG,NDIGIT,MSG,XPLS,FPLS,GPLS, * HESS,WRK,LWRK,IWRK,LIWRK,TERMCD,HTV,INFORM) STOP END SUBROUTINE DODCF(N,X,F) C ********** C C PURPOSE C ------- C C THIS SUBROUTINE COMPUTES THE FUNCTION OF THE C OPTIMAL DESIGN WITH COMPOSITE MATERIALS PROBLEM. C C PARAMETERS C ---------- C C N IS THE DIMENSION OF THE PROBLEM C C NX IS AN INTEGER VARIABLE. C ON ENTRY NX IS THE NUMBER OF GRID POINTS IN THE FIRST C COORDINATE DIRECTION. C ON EXIT NX IS UNCHANGED. C C NY IS AN INTEGER VARIABLE. C ON ENTRY NY IS THE NUMBER OF GRID POINTS IN THE SECOND C COORDINATE DIRECTION. C ON EXIT NY IS UNCHANGED. C C X IS A DOUBLE PRECISION (REAL) ARRAY OF DIMENSION N = NX*NY. C C F IS A DOUBLE PRECISION (REAL) VARIABLE. C ON EXIT F IS SET TO THE FUNCTION EVALUATED AT X. C C LAMBDA IS A DOUBLE PRECISION (REAL) VARIABLE. C ON ENTRY LAMBDA IS THE LAGRANGE MULTIPLIER. C ON EXIT LAMBDA IS UNCHANGED. C C SUBPROGRAMS CALLED C C MINPACK-SUPPLIED ... DODCPS C C MINPACK-2 PROJECT. NOVEMBER 1993. C ARGONNE NATIONAL LABORATORY AND UNIVERSITY OF MINNESOTA. C BRETT M. AVERICK. MODIFIED BY ALI BOUARICHA ON OCTOBER 1994. C C ********** C INTEGER N, NX, NY DOUBLE PRECISION F, LAMBDA DOUBLE PRECISION X(N) COMMON/PARAM/NX,NY COMMON/OTHER/LAMBDA C C LOCAL VARIABLES C INTEGER I, J, K DOUBLE PRECISION MU1, MU2, ONE, P5, TWO, ZERO DOUBLE PRECISION AREA, DPSI, DVDX, DVDY, GRADV, HX, HXHY DOUBLE PRECISION HY, TEMP, T1, T2, V, VB, VL, VR, VT PARAMETER (ZERO=0.0D0,P5=0.5D0,ONE=1.0D0,TWO=2.0D0) PARAMETER (MU1=ONE,MU2=TWO) INTRINSIC DBLE, SQRT EXTERNAL DODCPS C C INITIALIZATION. C HX = ONE/DBLE(NX+1) HY = ONE/DBLE(NY+1) HXHY = HX*HY AREA = P5*HXHY C C COMPUTE THE BREAK POINTS. C T1 = SQRT(TWO*LAMBDA*MU1/MU2) T2 = SQRT(TWO*LAMBDA*MU2/MU1) F = ZERO C C COMPUTATION OF THE FUNCTION OVER THE LOWER C TRIANGULAR ELEMENTS. C DO 50 J = 0, NY DO 40 I = 0, NX K = NX*(J-1) + I V = ZERO VR = ZERO VT = ZERO IF (J .GE. 1 .AND. I .GE. 1) V = X(K) IF (I .LT. NX .AND. J .GT. 0) VR = X(K+1) IF (I .GT. 0 .AND. J .LT. NY) VT = X(K+NX) DVDX = (VR-V)/HX DVDY = (VT-V)/HY GRADV = DVDX**2 + DVDY**2 CALL DODCPS(GRADV,MU1,MU2,T1,T2,DPSI,0,LAMBDA) F = F + DPSI 40 CONTINUE 50 CONTINUE C C COMPUTATION OF THE FUNCTION OVER THE UPPER C TRIANGULAR ELEMENTS. C DO 70 J = 1, NY + 1 DO 60 I = 1, NX + 1 K = NX*(J-1) + I VB = ZERO VL = ZERO V = ZERO IF (I .LE. NX .AND. J .GT. 1) VB = X(K-NX) IF (I .GT. 1 .AND. J .LE. NY) VL = X(K-1) IF (I .LE. NX .AND. J .LE. NY) V = X(K) DVDX = (V-VL)/HX DVDY = (V-VB)/HY GRADV = DVDX**2 + DVDY**2 CALL DODCPS(GRADV,MU1,MU2,T1,T2,DPSI,0,LAMBDA) F = F + DPSI 60 CONTINUE 70 CONTINUE C C SCALE THE FUNCTION. C F = AREA*F C C INTEGRATE V OVER THE DOMAIN. C TEMP = ZERO DO 80 K = 1, NX*NY TEMP = TEMP + X(K) 80 CONTINUE F = F + HXHY*TEMP END C SUBROUTINE DODCG(N,X,FGRAD) C ********** C C PURPOSE C ------- C C THIS SUBROUTINE COMPUTES THE GRADIENT OF THE C OPTIMAL DESIGN WITH COMPOSITE MATERIALS PROBLEM. C C PARAMETERS C ---------- C C N IS THE DIMENSION OF THE PROBLEM C C NX IS AN INTEGER VARIABLE. C ON ENTRY NX IS THE NUMBER OF GRID POINTS IN THE FIRST C COORDINATE DIRECTION. C ON EXIT NX IS UNCHANGED. C C NY IS AN INTEGER VARIABLE. C ON ENTRY NY IS THE NUMBER OF GRID POINTS IN THE SECOND C COORDINATE DIRECTION. C ON EXIT NY IS UNCHANGED. C C X IS A DOUBLE PRECISION (REAL) ARRAY OF DIMENSION N = NX*NY. C C FGRAD IS A DOUBLE PRECISION (REAL) ARRAY OF DIMENSION N = NX*NY. C ON ENTRY FGRAD NEED NOT BE SPECIFIED. C ON EXIT FGRAD CONTAINS THE GRADIENT EVALUATED AT X. C C LAMBDA IS A DOUBLE PRECISION (REAL) VARIABLE. C ON ENTRY LAMBDA IS THE LAGRANGE MULTIPLIER. C ON EXIT LAMBDA IS UNCHANGED. C C SUBPROGRAMS CALLED C C MINPACK-SUPPLIED ... DODCPS C C MINPACK-2 PROJECT. NOVEMBER 1993. C ARGONNE NATIONAL LABORATORY AND UNIVERSITY OF MINNESOTA. C BRETT M. AVERICK. MODIFIED BY ALI BOUARICHA ON OCTOBER 1994. C C ********** C INTEGER N, NX, NY DOUBLE PRECISION X(N), FGRAD(N), LAMBDA COMMON/PARAM/NX,NY COMMON/OTHER/LAMBDA C C LOCAL VARIABLES C INTEGER I, J, K DOUBLE PRECISION MU1, MU2, ONE, P5, TWO, ZERO DOUBLE PRECISION AREA, DPSIP, DVDX, DVDY, GRADV, HX, HXHY DOUBLE PRECISION HY, T1, T2, V, VB, VL, VR, VT PARAMETER (ZERO=0.0D0,P5=0.5D0,ONE=1.0D0,TWO=2.0D0) PARAMETER (MU1=ONE,MU2=TWO) INTRINSIC DBLE, SQRT EXTERNAL DODCPS C INITIALIZATION. HX = ONE/DBLE(NX+1) HY = ONE/DBLE(NY+1) HXHY = HX*HY AREA = P5*HXHY C C COMPUTE THE BREAK POINTS. C T1 = SQRT(TWO*LAMBDA*MU1/MU2) T2 = SQRT(TWO*LAMBDA*MU2/MU1) DO 30 K = 1, NX*NY FGRAD(K) = ZERO 30 CONTINUE C C COMPUTATION OF THE THE GRADIENT OVER THE LOWER C TRIANGULAR ELEMENTS. C DO 50 J = 0, NY DO 40 I = 0, NX K = NX*(J-1) + I V = ZERO VR = ZERO VT = ZERO IF (J .GE. 1 .AND. I .GE. 1) V = X(K) IF (I .LT. NX .AND. J .GT. 0) VR = X(K+1) IF (I .GT. 0 .AND. J .LT. NY) VT = X(K+NX) DVDX = (VR-V)/HX DVDY = (VT-V)/HY GRADV = DVDX**2 + DVDY**2 CALL DODCPS(GRADV,MU1,MU2,T1,T2,DPSIP,1,LAMBDA) IF (I .GE. 1 .AND. J .GE. 1) + FGRAD(K) = FGRAD(K) - TWO*(DVDX/HX+DVDY/HY)*DPSIP IF (I .LT. NX .AND. J .GT. 0) + FGRAD(K+1) = FGRAD(K+1) + TWO*(DVDX/HX)*DPSIP IF (I .GT. 0 .AND. J .LT. NY) + FGRAD(K+NX) = FGRAD(K+NX) + TWO*(DVDY/HY)*DPSIP 40 CONTINUE 50 CONTINUE C C COMPUTATION OF THE GRADIENT OVER THE UPPER C TRIANGULAR ELEMENTS. C DO 70 J = 1, NY + 1 DO 60 I = 1, NX + 1 K = NX*(J-1) + I VB = ZERO VL = ZERO V = ZERO IF (I .LE. NX .AND. J .GT. 1) VB = X(K-NX) IF (I .GT. 1 .AND. J .LE. NY) VL = X(K-1) IF (I .LE. NX .AND. J .LE. NY) V = X(K) DVDX = (V-VL)/HX DVDY = (V-VB)/HY GRADV = DVDX**2 + DVDY**2 CALL DODCPS(GRADV,MU1,MU2,T1,T2,DPSIP,1,LAMBDA) IF (I .LE. NX .AND. J .GT. 1) + FGRAD(K-NX) = FGRAD(K-NX) - TWO*(DVDY/HY)*DPSIP IF (I .GT. 1 .AND. J .LE. NY) + FGRAD(K-1) = FGRAD(K-1) - TWO*(DVDX/HX)*DPSIP IF (I .LE. NX .AND. J .LE. NY) + FGRAD(K) = FGRAD(K) + TWO*(DVDX/HX+DVDY/HY)*DPSIP 60 CONTINUE 70 CONTINUE C C INTEGRATE V OVER THE DOMAIN. C DO 90 K = 1, NX*NY FGRAD(K) = AREA*FGRAD(K) + HXHY 90 CONTINUE END C SUBROUTINE DODCPS(T,MU1,MU2,T1,T2,RESULT,OPTION,LAMBDA) C C ********** C C THIS SUBROUTINE COMPUTES THE FUNCTION PSI(T) AND THE SCALED C FUNCTIONS PSI'(T)/T AND PSI''(T)/T FOR THE OPTIMAL DESIGN C WITH COMPOSITE MATERIALS PROBLEM. C C THE SUBROUTINE STATEMENT IS C C SUBROUTINE DODCPS(T,MU1,MU2,T1,T2,RESULT,OPTION,LAMBDA) C C WHERE C C T IS A DOUBLE PRECISION (REAL) VARIABLE. C ON ENTRY T IS THE VARIABLE T C ON EXIT T IS UNCHANGED C C MU1 IS A DOUBLE PRECISION (REAL) VARIABLE. C ON ENTRY MU1 IS THE RECIPROCAL SHEAR MODULUS OF MATERIAL 1. C ON EXIT MU1 IS UNCHANGED. C C MU2 IS A DOUBLE PRECISION (REAL) VARIABLE. C ON ENTRY MU2 IS THE RECIPROCAL SHEAR MODULUS OF MATERIAL 2. C ON EXIT MU2 IS UNCHANGED. C C T1 IS A DOUBLE PRECISION (REAL) VARIABLE. C ON ENTRY T1 IS THE FIRST BREAKPOINT. C ON EXIT T1 IS UNCHANGED. C C T2 IS A DOUBLE PRECISION (REAL) VARIABLE. C ON ENTRY T2 IS THE SECOND BREAKPOINT. C ON EXIT T2 IS UNCHANGED. C C RESULT IS A DOUBLE PRECISION (REAL) VARIABLE. C ON ENTRY RESULT NEED NOT BE SPECIFIED. C ON EXIT RESULT IS SET ACCORDING TO TASK. C C OPTION IS AN INTEGER VARIABLE. C ON ENTRY OPTION SPECIFIES THE ACTION OF THE SUBROUTINE: C C IF OPTION = 0 THEN EVALUATE THE FUNCTION PSI(T). C IF OPTION = 1 THEN EVALUATE THE SCALED FUNCTION PSI'(T)/T. C IF OPTION = 2 THEN EVALUATE THE SCALED FUNCTION PSI''(T)/T. C C ON OPTION TASK IS UNCHANGED. C C LAMBDA IS A DOUBLE PRECISION (REAL) VARIABLE C ON ENTRY LAMBDA IS THE LAGRANGE MULTIPLIER. C ON EXIT LAMBDA IS UNCHANGED. C C MINPACK-2 PROJECT. NOVEMBER 1993. C ARGONNE NATIONAL LABORATORY AND UNIVERSITY OF MINNESOTA. C BRETT M. AVERICK. MODIFIED BY ALI BOUARICHA ON OCTOBER 1994. C C ********** C INTEGER OPTION DOUBLE PRECISION T, MU1, MU2, T1, T2, RESULT, LAMBDA C C LOCAL VARIABLES C DOUBLE PRECISION P25, P5, ZERO PARAMETER (ZERO=0.0D0,P25=0.25D0,P5=0.5D0) DOUBLE PRECISION SQRTT INTRINSIC SQRT C SQRTT = SQRT(T) IF (OPTION .EQ. 0) THEN IF (SQRTT .LE. T1) THEN RESULT = P5*MU2*T ELSE IF (SQRTT .GT. T1 .AND. SQRTT .LT. T2) THEN RESULT = MU2*T1*SQRTT - LAMBDA*MU1 ELSE IF (SQRTT .GE. T2) THEN RESULT = P5*MU1*T + LAMBDA*(MU2-MU1) END IF ELSE IF (OPTION .EQ. 1) THEN IF (SQRTT .LE. T1) THEN RESULT = P5*MU2 ELSE IF (SQRTT .GT. T1 .AND. SQRTT .LT. T2) THEN RESULT = P5*MU2*T1/SQRTT ELSE IF (SQRTT .GE. T2) THEN RESULT = P5*MU1 END IF ELSE IF (OPTION .EQ. 2) THEN IF (SQRTT .LE. T1) THEN RESULT = ZERO ELSE IF (SQRTT .GT. T1 .AND. SQRTT .LT. T2) THEN RESULT = -P25*MU2*T1/(SQRTT*T) ELSE IF (SQRTT .GE. T2) THEN RESULT = ZERO END IF END IF END C SUBROUTINE DODCSP(NX,NY,NNZ,INDROW,INDCOL) C C ********** C C SUBROUTINE DODCSP C C THIS SUBROUTINE DEFINES THE SPARSITY STRUCTURE OF THE HESSIAN C MATRIX FOR THE OPTIMAL DESIGN WITH COMPOSITES PROBLEM. C C THE SUBROUTINE STATEMENT IS C C SUBROUTINE DODCSP(NX,NY,NNZ,INDROW,INDCOL) C C WHERE C C NX IS AN INTEGER VARIABLE. C ON ENTRY NX IS THE NUMBER OF GRID POINTS IN THE FIRST C COORDINATE DIRECTION. C ON EXIT NX IS UNCHANGED. C C NY IS AN INTEGER VARIABLE. C ON ENTRY NY IS THE NUMBER OF GRID POINTS IN THE SECOND C COORDINATE DIRECTION. C ON EXIT NY IS UNCHANGED. C C NNZ IS AN INTEGER VARIABLE. C ON ENTRY NNZ NEED NOT BE SPECIFIED. C ON EXIT NNZ IS SET TO THE NUMBER OF NONZEROS IN THE C LOWER TRIANGLE OF THE HESSIAN MATRIX. C C INDROW IS AN INTEGER ARRAY OF DIMENSION AT LEAST NNZ. C ON ENTRY INDROW NEED NOT BE SPECIFIED. C ON EXIT INDROW CONTAINS THE ROW INDICES OF THE NONZEROS C IN THE LOWER TRIANGLE OF THE HESSIAN MATRIX. C C INDCOL IS AN INTEGER ARRAY OF DIMENSION AT LEAST NNZ. C ON ENTRY INDCOL NEED NOT BE SPECIFIED. C ON EXIT INDCOL CONTAINS THE COLUMN INDICES OF THE NONZEROS C IN THE LOWER TRIANGLE OF THE HESSIAN MATRIX. C C MINPACK-2 PROJECT. NOVEMBER 1993. C ARGONNE NATIONAL LABORATORY AND UNIVERSITY OF MINNESOTA. C BRETT M. AVERICK. MODIFIED BY ALI BOUARICHA ON OCTOBER 1994. C C ********** C INTEGER NX, NY, NNZ INTEGER INDROW(*), INDCOL(*) C C LOCAL VARIABLES C INTEGER I, J C C COMPUTE THE SPARSITY STRUCTURE. C NNZ = 0 DO 20 J = 1, NY DO 10 I = 1, NX NNZ = NNZ + 1 INDROW(NNZ) = (J-1)*NX + I INDCOL(NNZ) = (J-1)*NX + I IF (I .NE. NX) THEN NNZ = NNZ + 1 INDROW(NNZ) = (J-1)*NX + I + 1 INDCOL(NNZ) = (J-1)*NX + I END IF IF (J .NE. NY) THEN NNZ = NNZ + 1 INDROW(NNZ) = (J-1)*NX + I + NX INDCOL(NNZ) = (J-1)*NX + I IF (I .NE. 1) THEN NNZ = NNZ + 1 INDROW(NNZ) = (J-1)*NX + I + NX - 1 INDCOL(NNZ) = (J-1)*NX + I END IF END IF 10 CONTINUE 20 CONTINUE END SHAR_EOF fi # end of overwriting check if test -f 'driver1.f' then echo shar: will not over-write existing file "'driver1.f'" else cat << \SHAR_EOF > 'driver1.f' C C STENMIN MINIMIZES AN UNCONSTRAINED NONLINEAR FUNCTION IN N C UNKNOWNS WHERE THE HESSIAN IS LARGE AND SPARSE, USING TENSOR C METHODS. C C EXAMPLE OF USE FOR STENMIN. THE TEST PROBLEM IS C THE BROYDEN TRIDIAGONAL (SOURCE: BUCKLEY#78 (P. 42)). C C ALI BOUARICHA, OCTOBER 1994. C MCS DIVISION, ARGONNE NATIONAL LAB. C INTEGER NMAX, N, NZ, LIRN, LICN, ILIM, IPR, METHOD INTEGER GRDFLG, HSNFLG, NDIGIT, MSG, LWRK, LIWRK INTEGER TERMCD, INFORM, I DOUBLE PRECISION FSCALE, GRADTL, STEPTL, FPLS, STEPMX, ONE PARAMETER ( NMAX = 10000, LIRN = 50000, LICN = 500000 ) PARAMETER ( LIWRK = 2 * LIRN + 12 * NMAX + 2 ) PARAMETER ( LWRK = 7 * NMAX ) INTEGER IRN ( LIRN ), ICN ( LICN ) INTEGER IWRK( LIWRK ) DOUBLE PRECISION X ( NMAX ), TYPX( NMAX ), XPLS( NMAX ) DOUBLE PRECISION GPLS( NMAX ), HESS( LICN ), WRK ( LWRK ) DOUBLE PRECISION HTV ( NMAX ) EXTERNAL FCN, UGRAD, UHESS DATA ONE / 1.0D0 / C RUN BROYDEN PROBLEM WITH N = 10000. N = 10000 C COMPUTE THE STANDARD STARTING POINT. DO 10 I = 1, N X(I) = -ONE 10 CONTINUE C SET THE DEFAULT VALUES. CALL STDFLT(N, TYPX, FSCALE, GRADTL, STEPTL, ILIM, STEPMX, * IPR, METHOD, GRDFLG, HSNFLG, NDIGIT, INFORM, MSG) GRADTL = 1.0D-5 GRDFLG = 2 HSNFLG = 2 C CALL THE SPARSE OPTIMIZER. CALL STUMCD(N,X,NZ,IRN,LIRN,ICN,LICN,FCN,UGRAD, * UHESS,TYPX,FSCALE,GRADTL,STEPTL,ILIM,STEPMX,IPR, * METHOD,GRDFLG,HSNFLG,NDIGIT,MSG,XPLS,FPLS,GPLS, * HESS,WRK,LWRK,IWRK,LIWRK,TERMCD,HTV,INFORM) STOP END C C THE FOLLOWING IS A SUBROUTINE FOR THE BROYDEN TRIDIAGONAL C PROBLEM (SOURCE: BUCKLEY#78 (P. 42)) C SUBROUTINE FCN(N, X, F) INTEGER N DOUBLE PRECISION X(N), F C C LOCAL VARIABLES C INTEGER I DOUBLE PRECISION ONE, TWO, THREE DATA ONE, TWO, THREE / 1.0D0, 2.0D0, 3.0D0 / C F = ((THREE - TWO * X(1)) * X(1) - TWO * X(2) + ONE) ** 2 DO 10 I = 2, N-1 F = F + ((THREE - TWO * X(I)) * X(I) - X(I-1) - * TWO * X(I+1) + ONE) ** 2 10 CONTINUE F = F + ((THREE - TWO * X(N)) * X(N) - X(N-1) + ONE) ** 2 RETURN END C C THE FOLLOWING IS A SUBROUTINE FOR THE GRADIENT OF THE BROYDEN C TRIDIAGONAL PROBLEM C SUBROUTINE UGRAD(N, X, G) INTEGER N DOUBLE PRECISION X(N), G(N) C C LOCAL VARIABLES C INTEGER I DOUBLE PRECISION RL, RM, RR, ONE, TWO, THREE, FOUR DATA ONE, TWO, THREE, FOUR/1.0D0, 2.0D0, 3.0D0, 4.0D0/ C RL = (THREE - TWO * X(1)) * X(1) - TWO * X(2) + ONE RR = (THREE - TWO * X(2)) * X(2) - X(1) - TWO * X(3) + ONE G(1) = TWO * (RL * (THREE - FOUR * X(1)) - RR) DO 10 I = 2, N-1 IF(I .NE. 2) THEN RL = (THREE - TWO * X(I-1)) * X(I-1) - X(I-2) - * TWO * X(I) + ONE ENDIF RM = (THREE - TWO * X(I)) * X(I) - X(I-1) - * TWO * X(I+1) + ONE IF(I .EQ. N-1) THEN RR = (THREE - TWO * X(N)) * X(N) - X(N-1) + ONE ELSE RR = (THREE - TWO * X(I+1)) * X(I+1) - X(I) - * TWO * X(I+2) + ONE ENDIF G(I) = -TWO * (TWO * RL - RM * (THREE - FOUR * X(I)) + RR) 10 CONTINUE G(N) = -TWO * (TWO * RM - RR * (THREE - FOUR * X(N))) RETURN END C C THE FOLLOWING IS A SUBROUTINE FOR THE HESSIAN OF THE BROYDEN C TRIDIAGONAL PROBLEM C SUBROUTINE UHESS(N,X,NZ,LICN,HESS,IRN,ICN) INTEGER N, NZ, LICN INTEGER IRN(*), ICN(LICN) DOUBLE PRECISION X(N), HESS(LICN) C C LOCAL VARIABLES C INTEGER I DOUBLE PRECISION RL,RM,RR DOUBLE PRECISION ONE,TWO,THREE,FOUR DATA ONE, TWO, THREE, FOUR/1.0D0, 2.0D0, 3.0D0, 4.0D0/ C NZ = 1 RL = (THREE - TWO * X(1)) * X(1) - TWO * X(2) + ONE HESS(NZ) = TWO * ((THREE - FOUR * X(1))**2 - * FOUR * RL + ONE) IRN(NZ) = 1 ICN(NZ) = 1 DO 10 I = 2, N-1 IF(I .NE. 2) THEN NZ = NZ + 1 HESS(NZ) = FOUR IRN(NZ) = I ICN(NZ) = I-2 ENDIF NZ = NZ + 1 HESS(NZ) = -TWO * (TWO * (THREE - FOUR * X(I-1)) + * ONE * (THREE - FOUR * X(I))) IRN(NZ) = I ICN(NZ) = I-1 RM = (THREE - TWO * X(I)) * X(I) - X(I-1) - * TWO * X(I+1) + ONE NZ = NZ + 1 HESS(NZ) = -TWO * (-FOUR - (THREE - FOUR * X(I))**2 + * FOUR * RM - ONE) IRN(NZ) = I ICN(NZ) = I 10 CONTINUE RR = (THREE - TWO * X(N)) * X(N) - X(N-1) + ONE NZ = NZ + 1 HESS(NZ) = FOUR IRN(NZ) = N ICN(NZ) = N-2 NZ = NZ + 1 HESS(NZ) = -TWO * (TWO * (THREE - FOUR * X(N-1)) + * THREE - FOUR * X(N)) IRN(NZ) = N ICN(NZ) = N-1 NZ = NZ + 1 HESS(NZ) = TWO * (FOUR + (THREE - FOUR * X(N))**2 - FOUR * RR) IRN(NZ) = N ICN(NZ) = N RETURN END SHAR_EOF fi # end of overwriting check if test -f 'RES2' then echo shar: will not over-write existing file "'RES2'" else cat << \SHAR_EOF > 'RES2' STDRUO GRADIENT FLAG = 2 STDRUO HESSIAN FLAG = 0 STDRUO METHOD = 1 STDRUO ITERATION LIMIT = 500 STDRUO MACHINE EPSILON = 0.2220446049250E-15 STDRUO STEP TOLERANCE = 0.3666852862501E-10 STDRUO GRADIENT TOLERANCE = 0.1000000000000E-04 STDRUO MAXIMUM STEP SIZE = 0.3295193424014E+04 --------------------------------------------- STRSLT ITERATION K = 0 STRSLT FUNCTION AT X(K) STRSLT 0.4808237011427E-01 STRSLT SCALED GRADIENT AT X(K) STRSLT 0.3729334871203E-01 --------------------------------------------- STCHKS RELATIVE GRADIENT CLOSE TO ZERO STCHKS CURRENT ITERATE IS PROBABLY SOLUTION --------------------------------------------- STRSLT ITERATION K = 20 STRSLT FUNCTION AT X(K) STRSLT -0.1135947433813E-01 STRSLT SCALED GRADIENT AT X(K) STRSLT 0.2847027609286E-06 --------------------------------------------- STRSLT NUMBER OF FUNCTION EVALUATIONS 78 STRSLT NUMBER OF GRADIENT EVALUATIONS 21 STRSLT NUMBER OF HESSIAN EVALUATIONS 20 SHAR_EOF fi # end of overwriting check if test -f 'RES1' then echo shar: will not over-write existing file "'RES1'" else cat << \SHAR_EOF > 'RES1' STDRUO GRADIENT FLAG = 2 STDRUO HESSIAN FLAG = 2 STDRUO METHOD = 1 STDRUO ITERATION LIMIT = 500 STDRUO MACHINE EPSILON = 0.2220446049250E-15 STDRUO STEP TOLERANCE = 0.3666852862501E-10 STDRUO GRADIENT TOLERANCE = 0.1000000000000E-04 STDRUO MAXIMUM STEP SIZE = 0.1000000000000E+06 --------------------------------------------- STRSLT ITERATION K = 0 STRSLT FUNCTION AT X(K) STRSLT 0.1001100000000E+05 STRSLT SCALED GRADIENT AT X(K) STRSLT 0.3800000000000E+02 --------------------------------------------- STCHKS RELATIVE GRADIENT CLOSE TO ZERO STCHKS CURRENT ITERATE IS PROBABLY SOLUTION --------------------------------------------- STRSLT ITERATION K = 4 STRSLT FUNCTION AT X(K) STRSLT 0.1884575877651E-13 STRSLT SCALED GRADIENT AT X(K) STRSLT 0.1113397090705E-05 --------------------------------------------- STRSLT NUMBER OF FUNCTION EVALUATIONS 5 STRSLT NUMBER OF GRADIENT EVALUATIONS 5 STRSLT NUMBER OF HESSIAN EVALUATIONS 4 SHAR_EOF fi # end of overwriting check cd .. cd .. if test ! -d 'Src' then mkdir 'Src' fi cd 'Src' if test ! -d 'Sp' then mkdir 'Sp' fi cd 'Sp' if test -f 'stenmin.f' then echo shar: will not over-write existing file "'stenmin.f'" else cat << \SHAR_EOF > 'stenmin.f' * * STENMIN: A SOFTWARE PACKAGE FOR LARGE, SPARSE UNCONSTRAINED * OPTIMIZATION USING TENSOR METHODS. * * * AUTHOR: * ------ * ALI BOUARICHA * ARGONNE NATIONAL LABORATORY * MCS DIVISION * 9700 SOUTH CASS AVENUE * ARGONNE, IL 60439 * E-MAIL: BOUARICH@MCS.ANL.GOV * * DATE: OCTOBER, 1994 * ---- * * PURPOSE: * ------- * * STENMIN MINIMIZES AN UNCONSTRAINED NONLINEAR FUNCTION IN N UNKNOWNS * WHERE THE HESSIAN IS LARGE AND SPARSE USING A NEW CLASS OF METHODS * CALLED TENSOR METHODS. THE SOFTWARE ALLOWS THE USER TO SELECT BETWEEN * A TENSOR METHOD AND A STANDARD METHOD BASED UPON A QUADRATIC MODEL. * THE TENSOR METHOD MODELS THE OBJECTIVE FUNCTION BY A FOURTH-ORDER * MODEL, WHERE THE THIRD- AND FOURTH-ORDER TERMS ARE CHOSEN SUCH THAT * THE EXTRA COST OF FORMING AND SOLVING THE MODEL IS SMALL. THE TEST * RESULTS INDICATE THAT, IN GENERAL, THE TENSOR METHOD IS OFTEN MORE * EFFICIENT AND MORE RELIABLE THAN THE STANDARD nEWTON METHOD FOR * SOLVING LARGE, SPARSE UNCONSTRAINED OPTIMIZATION PROBLEMS. IT IS * ESPECIALLY USEFUL ON PROBLEMS WHERE THE HESSIAN AT THE SOLUTION IS * SINGULAR. * * TWO INTERFACES ARE PROVIDED WITH STENMIN. IF THE USER WISHES TO USE * ALL THE DEFAULTS OPTIONS PROVIDED BY THE PACKAGE, THEN HE (OR SHE) * SHOULD CALL STUMSD (STUMSS IF SINGLE-PRECISION IS USED). THE OTHER * INTERFACE, STUMCD (STUMCS IF SINGLE-PRECISION IS USED), REQUIRES THE * USER TO SUPPLY ALL PARAMETERS. THE USER MAY SPECIFY SELECTED * PARAMETERS ONLY BY FIRST INVOKING THE SUBROUTINE STDFLT, WHICH SETS * ALL PARAMETERS TO THEIR DEFAULT VALUES, AND THEN OVERRIDING ONLY THE * DESIRED VALUES. * * * LIST OF SUBROUTINE AND FUNCTION NAMES CALLED BY STENMIN: * * STCHKG,STCHKH,STCHKI,STCHKS,STCZ3P,STDFLT,STDRUO,STDUGR,STDUSH, * STFDGR,STFDHS,STFTSM,STHMUV,STLSCH,STMA27,STMNEC,STMSLV,STRSLT, * STSORT,STUMCD,STUMSD. * * PACKAGES CALLED BY STENMIN: * * * MA27 (I. S. DUFF AND J. K. REID, "MA27: A SET OF FORTRAN SUBROUTINES * FOR SOLVING SPARSE SYMMETRIC SETS OF LINEAR EQUATIONS", TECH. REP. * R-10533, AERE HARWELL LABORATORY, HARWELL, UK, 1983.) * * DSSM (T. F. COLEMAN, B. S. GARBOW, and J. J. MORE', "SOFTWARE FOR * ESTIMATING SPARSE HESSIAN MATRICES", ACM TRANS. MATH. SOFTW., 11: * 363-377,1985.) * * DSYPRC FROM LANCELOT (A. R. CONN, N. I. M. GOULD, AND C PH. L. TOINT, * "LANCELOT", SPRINGER SERIES IN COMPUTATIONAL MATHEMATICS, * SPRINGER-VERLAG, 1992.) * * R1MACH (FOX P.A., HALL A.D., SCHRYER N.L., FRAMEWORK FOR A PORTABLE * LIBRARY, ACM TRANS. MATH. SOFTW., 4:177-188, 1978.) * * BLAS CALLED BY STENMIN: * * LEVEL 1 BLAS: DCOPY,DDOT,DNRM2,DSCAL * SUBROUTINE STCHKG(N,X,FCN,F,G,TYPX,RNF,ANALTL,WRK1,MSG,IPR) * * PURPOSE: * ------- * * THIS ROUTINE CHECKS THE USER'S ANALYTIC GRADIENT AGAINST A * FINITE-DIFFERENCE GRADIENT. * * PARAMETERS: * ---------- * * N --> DIMENSION OF PROBLEM * X --> CURRENT ITERATE * FCN --> NAME OF SUBROUTINE THAT EVALUATES THE * OPTIMIZATION FUNCTION * F --> FUNCTION VALUE AT X * G --> GRADIENT VALUE AT X * TYPX --> TYPICAL SIZE FOR EACH COMPONENT OF X * RNF --> RELATIVE NOISE IN FCN * ANALTL --> TOLERANCE FOR COMPARISON OF ESTIMATED AND ANALYTICAL * GRADIENTS * WRK1 --> WORKSPACE. IT IS USED TO HOLD THE FINITE-DIFFERENCE * APPROXIMATION OF THE GRADIENT AT X * MSG <-- MESSAGE OR ERROR CODE. IF PROBABLE CODING ERROR OF * GRADIENT IS DETECTED THEN MSG = -8 * IPR --> DEVICE TO WHICH TO SEND OUTPUT * INTEGER N,MSG,IPR REAL X(N),F,G(N),TYPX(N),RNF,ANALTL,WRK1(N) * * LOCAL VARIABLES * INTEGER KER,I REAL GS,ONE EXTERNAL FCN INTRINSIC ABS,MAX DATA ONE/1.0E0/ * * COMPUTE FIRST-ORDER FINITE DIFFERENCE GRADIENT AND COMPARE TO * ANALYTIC GRADIENT * CALL STFDGR(N,X,FCN,F,RNF,WRK1) KER = 0 DO 10 I = 1,N GS = MAX(ABS(F),ONE)/MAX(ABS(X(I)),ONE/TYPX(I)) IF (ABS(G(I)-WRK1(I)).GT.MAX(ABS(G(I)),GS)*ANALTL) KER = 1 10 CONTINUE IF (KER.EQ.0) GO TO 20 WRITE (IPR,9000) WRITE (IPR,9010) (I,G(I),WRK1(I),I=1,N) MSG = -8 20 CONTINUE RETURN 9000 FORMAT (' STCHKG PROBABLE ERROR IN CODING OF ANALYTIC', + ' GRADIENT FUNCTION.'/' STCHKG COMP',12X,'ANALYTIC', + 12X,'ESTIMATE') 9010 FORMAT (' STCHKG ',I5,3X,E20.13,3X,E20.13) END SUBROUTINE STCHKH(N,X,NPAIRS,NZ,IRN,LIRN,ICN,LICN,FCN,UGR,SCALE, + TYPX,GRDFLG,F,G,RNF,ANALTL,FCALL,CHECKH,LISTP, + NGRP,IPNTR,JPNTR,IWA,LIWA,FHESD,XD,ETA,HESS,IPR, + MSG) * * PURPOSE * ------- * * THIS ROUTINE CHECKS THE USER'S ANALYTIC HESSIAN AGAINST A * FINITE-DIFFERENCE HESSIAN. * * PARAMETERS: * ---------- * * N --> NUMBER OF VARIABLES IN THE PROBLEM * X --> CURRENT ITERATE * NPAIRS --> A POSITIVE INTEGER INPUT VARIABLE SET TO THE NUMBER * OF (IRN,ICN) PAIRS USED TO DESCRIBE THE SPARSITY * PATTERN OF THE HESSIAN MATRIX * NZ <-- ACTUAL NUMBER OF NONZEROS STORED IN THE LOWER * HALF OF THE HESSIAN MATRIX * IRN <-->ROW INDEX OF EACH NONZERO STORED IN THE LOWER * OR UPPER HALF OF THE HESSIAN MATRIX ON ENTRY. ON EXIT, * IT HOLDS THE ROW INDEX OF EACH NONZERO STORED IN THE * LOWER HALF OF THE HESSIAN MATRIX * ICN <-->COLUMN INDEX OF THE NONZEROS STORED IN THE LOWER * OR UPPER HALF OF THE HESSIAN MATRIX ON ENTRY. ON EXIT, * IT HOLDS THE COLUMN INDEX OF EACH NONZERO STORED IN THE * LOWER HALF OF THE HESSIAN MATRIX * FCN --> THE NAME OF A USER SUPPLIED SUBROUTINE THAT EVALUATES * THE FUNCTION F AT AN ARBITRARY VECTOR X * UGR --> THE NAME OF A USER SUPPLIED SUBROUTINE THAT RETURNS IN * THE VALUE OF THE GRADIENT * SCALE --> LOGICAL VARIABLE INDICATING WHETHER OR NOT SCALING * OF THE VARIABLES IS PERFORMED * TYPX --> TYPICAL SIZE OF THE COMPONENTS OF X * GRDFLG --> INTEGER FLAG DESIGNATING WHETHER OR NOT ANALYTIC * GRADIENT HAS BEEN SUPPLIED BY THE USER * F --> FUNCTION VALUE AT CURRENT ITERATE * G --> GRADIENT VALUE AT CURRENT ITERATE * RNF --> RELATIVE NOISE IN FUNCTION FCN * ANALTL --> TOLERANCE FOR COMPARISON OF ESTIMATED AND ANALYTICAL * GRADIENTS * FCALL --> A LOGICAL FLAG. IF THE HESSIAN MATRIX IS EVALUATED FOR * THE FIRST TIME, THEN FCALL IS SET TO TRUE; OTHERWISE * IT IS SET TO FALSE * CHECKH --> A LOGICAL FLAG. IF IT IS SET TO TRUE THEN THE USER'S * ANALYTIC HESSIAN IS CHECKED AGAINST THE FINITE * DIFFERENCE APPROXIMATION ONE * LISTP,NGRP,IPNTR,JPNTR,IWA --> WORSPACE FOR THE STFDHS SUBROUTINE * (SEE SUBROUTINE STFDHS FOR MORE DETAIL) * LIWA --> LENGTH OF ARRAY IWA * FHESD,XD,ETA --> WORKSPACE FOR THE STFDHS SUBROUTINE * HESS <--> ANALYTIC HESSIAN ON INPUT. ON OUTPUT, IT ALSO HOLDS * A FINITE-DIFFERENCE APPROXIMATION OF THE HESSIAN * (FROM NZ+1 TO 2*NZ) * IPR --> DEVICE TO WHICH TO SEND OUTPUT * MSG <-- MESSAGE OR ERROR CODE. IF PROBABLE CODING ERROR OF * HESSIAN IS DETECTED THEN MSG = -9 * INTEGER N,NPAIRS,NZ,GRDFLG,LIWA,IPR,MSG,LIRN,LICN INTEGER IRN(LIRN),ICN(LICN),LISTP(N),NGRP(N),IPNTR(N+1) INTEGER JPNTR(N+1),IWA(LIWA) REAL X(N),TYPX(N),F,G(N),RNF,ANALTL,FHESD(N) REAL XD(N),ETA(N),HESS(LICN) LOGICAL SCALE,FCALL,CHECKH * * LOCAL VARIABLES * INTEGER IST,IEND,KER,I,IR,J,JP,K,MAXGRP,MINGRP REAL HS,ONE EXTERNAL FCN,UGR INTRINSIC ABS,MAX,MIN DATA ONE/1.0E+0/ * * CHECK USER'S INPUT DATA. * * IF NPAIRS IS NOT POSITIVE THEN SET MSG TO -2 AND ABORT. * IF (NPAIRS.LE.0) THEN WRITE (IPR,9000) NPAIRS MSG = -4 RETURN END IF * * IF THE K-TH ELEMENT OF IRN OR THE K-TH ELEMENT OF ICN IS * NOT AN INTEGER BETWEEN 1 AND N THEN SET MSG TO -4 AND ABORT. * DO 10 K = 1,NPAIRS IF (IRN(K).LT.1 .OR. IRN(K).GT.N .OR. ICN(K).LT.1 .OR. + ICN(K).GT.N) THEN WRITE (IPR,9010) K,IRN(K),K,ICN(K) MSG = -5 RETURN END IF 10 CONTINUE * * GENERATE THE SPARSITY PATTERN FOR THE LOWER * TRIANGULAR PART OF HESSIAN. * DO 20 K = 1,NPAIRS I = IRN(K) J = ICN(K) IRN(K) = MAX(I,J) ICN(K) = MIN(I,J) 20 CONTINUE * * SORT THE DATA STRUCTURE BY COLUMNS * CALL STSORT(N,NPAIRS,HESS,IRN,ICN,JPNTR,IWA) * * COMPRESS THE DATA AND DETERMINE THE NUMBER OF NONZERO * ELEMENTS IN THE LOWER TRIANGULAR PART OF HESSIAN. * DO 30 I = 1,N IWA(I) = 0 30 CONTINUE NZ = 0 DO 50 J = 1,N K = NZ DO 40 JP = JPNTR(J),JPNTR(J+1) - 1 IR = IRN(JP) IF (IWA(IR).NE.J) THEN NZ = NZ + 1 IRN(NZ) = IR ICN(NZ) = J IWA(IR) = J END IF 40 CONTINUE JPNTR(J) = K + 1 50 CONTINUE JPNTR(N+1) = NZ + 1 IF (NZ.NE.NPAIRS) THEN MSG = -7 WRITE (IPR,9020) NPAIRS - NZ RETURN END IF * IF (.NOT.CHECKH) RETURN * * COMPUTE FINITE-DIFFERENCE APPROXIMATION OF THE HESSIAN * CALL STFDHS(N,NZ,X,IRN,ICN,G,RNF,GRDFLG,FCN,UGR,SCALE,TYPX,FCALL, + LISTP,NGRP,IPNTR,JPNTR,IWA,LIWA,FHESD,XD,ETA,MAXGRP, + MINGRP,HESS(NZ+1)) * * COMPARE USER'S ANALYTIC HESSIAN TO THE FINITE-DIFFERENCE * APPROXIMATION * KER = 0 DO 70 J = 1,N HS = MAX(ABS(G(J)),ONE)/MAX(ABS(X(J)),ONE/TYPX(J)) IST = JPNTR(J) IEND = JPNTR(J+1) DO 60 I = IST,IEND - 1 IF (ABS(HESS(I)-HESS(NZ+I)).GT. + MAX(ABS(HESS(I)),HS)*ANALTL) KER = 1 60 CONTINUE 70 CONTINUE * IF (KER.EQ.0) GO TO 90 WRITE (IPR,9030) DO 80 I = 1,NZ WRITE (IPR,9040) IRN(I),ICN(I),HESS(I),HESS(NZ+I) 80 CONTINUE MSG = -9 90 CONTINUE RETURN * 9000 FORMAT (' STCHKH ILLEGAL NUMBER OF NONZEROS NZ=',I10) 9010 FORMAT (' STCHKH ILLEGAL ROW OR COLUMN INDEX IN ANALYTIC', + ' HESSIAN'/,' STCHKH IRN( ',I6,' )=',I6,2X,' ICN( ',I6, + ' )=',I6) 9020 FORMAT (' STCHKH ',I5,' REDUNDANT ENTRIES IN SPARSITY', + ' PATTERN ENCOUNTERED') 9030 FORMAT (' STCHKH PROBABLE ERROR IN CODING OF ANALYTIC', + ' HESSIAN FUNCTION'/' STCHKH IRN ICN',14X,'ANALYTIC', + 14X,'(ESTIMATE)') 9040 FORMAT (' STCHKH ',2I5,2X,E20.13,2X,'(',E20.13,')') END SUBROUTINE STCHKI(N,NPAIRS,NZ,X,IRN,LIRN,ICN,LICN,LIWRK,LWRK, + SCALE,TYPX,FSCALE,GRADTL,STEPTL,ILIM,NDIGIT,EPS, + METHOD,GRDFLG,HSNFLG,STEPMX,IWA,LIWA,JPNTR,MSG, + IPR) * * PURPOSE: * ------- * * THIS ROUTINE CHECKS THE INPUT FOR REASONABLENESS. * * PARAMETERS: * ---------- * N --> DIMENSION OF PROBLEM * NPAIRS --> A POSITIVE INTEGER INPUT VARIABLE SET TO THE NUMBER * OF (IRN,ICN) PAIRS USED TO DESCRIBE THE SPARSITY * PATTERN OF THE HESSIAN MATRIX * NZ <-- NUMBER OF NONZEROS IN THE LOWER TRIANGULAR PART * OF THE HESSIAN MATRIX * X --> ON ENTRY, ESTIMATE TO ROOT OF FCN * IRN <--> AN INTEGER ARRAY OF LENGTH NPAIRS. ON INPUT IRN MUST * CONTAIN THE ROW INDICES OF THE NONZERO ELEMENTS IN * THE LOWER TRIANGULAR PART OF THE HESSIAN. ON OUTPUT * IRN IS PERMUTED SO THAT THE CORRESPONDING COLUMN * INDICES ARE IN NON-DECREASING ORDER. THE COLUMN * INDICES CAN BE RECOVERED FROM THE ARRAY JPNTR * LIRN --> LENGTH OF ARRAY IRN * ICN <--> AN INTEGER ARRAY OF LENGTH NPAIRS. ON INPUT ICN MUST * CONTAIN THE COLUMN INDICES OF THE NONZERO ELEMENTS * IN THE LOWER TRIANGULAR PART OF THE HESSIAN. ON * OUTPUT ICN IS PERMUTED SO THAT THE CORRESPONDING * ROW INDICES ARE IN NON-DECREASING ORDER * LICN --> LENGTH OF ARRAY ICN * LIWRK --> LENGTH OF WORKSPACE ARRAY IWRK * LWRK --> LENGTH OF WORKSPACE ARRAY WRK * SCALE <-- LOGICAL VARIABLE INDICATING WHETHER OR NOT SCALING * OF THE X VARIABLES IS PERFORMED * TYPX <--> TYPICAL SIZE OF EACH COMPONENT OF X * FSCALE <--> ESTIMATE OF SCALE OF OBJECTIVE FUNCTION FCN * GRADTL <--> TOLERANCE AT WHICH GRADIENT CONSIDERED CLOSE * ENOUGH TO ZERO TO TERMINATE ALGORITHM * STEPTL <--> TOLERANCE AT WHICH STEP LENGTH CONSIDERED CLOSE * ENOUGH TO ZERO TO TERMINATE ALGORITHM * ILIM <--> MAXIMUM NUMBER OF ALLOWABLE ITERATIONS * NDIGIT <--> NUMBER OF GOOD DIGITS IN OPTIMIZATION FUNCTION FCN * EPS --> MACHINE PRECISION * METHOD <--> METHOD TO BE USED (NEWTON OR TENSOR) * GRDFLG <--> =1 IF ANALYTIC GRADIENT SUPPLIED * HSNFLG <--> =1 IF ANALYTIC HESSIAN SUPPLIED * STEPMX <--> MAXIMUM STEP SIZE * IWA --> WORKSPACE ARRAY * LIWA <-- ACTUAL LENGTH OF ARRAY IWA ON EXIT * JPNTR <-- AN INTEGER OUTPUT ARRAY OF LENGTH N + 1 WHICH * SPECIFIES THE LOCATIONS OF THE ROW INDICES IN IRN. * THE ROW INDICES FOR COLUMN J ARE * IRN(K), K = JPNTR(J),...,JPNTR(J+1)-1. NOTE THAT * JPNTR(N+1)-1 IS THEN THE NUMBER OF NONZERO ELEMENTS * IN THE LOWER TRIANGULAR PART OF THE MATRIX HESSIAN * MSG <--> AN INTEGER VARIABLE THAT THE USER MAY SET ON INPUT TO * INHIBIT CERTAIN AUTOMATIC CHECKS OR OVERRIDE CERTAIN * DEFAULT CHARACTERISTICS OF THE PACKAGE. ON INPUT: * MSG = 0 : NO OUTPUT WILL BE PRODUCED * MSG = 1 : PRINT THE INPUT STATE, THE FINAL RESULTS, * AND THE STOPPING CONDITIONS * MSG = 2 : PRINT THE INTERMEDIATE RESULTS, THAT IS, THE * INPUT STATE, THE VALUES OF THE OBJECTIVE * FUNCTION AND THE SCALED GRADIENT AT EACH * ITERATION, AND THE FINAL RESULTS INCLUDING * THE STOPPING CONDITIONS AND THE NUMBER OF * FUNCTION, GRADIENT, AND HESSIAN * * ON OUTPUT, IF THE PROGRAM HAS TERMINATED BECAUSE OF * ERRONEOUS INPUT, MSG CONTAINS AN ERROR CODE INDICATING * THE REASON: * * MSG = -1 : ILLEGAL DIMENSION N; N <= 0. THE PROGRAM ABORTS * MSG = -2 : ILLEGAL LENGTH OF LIRN OR LICN; LIRN <= 0 OR * LICN <= 0. THE PROGRAM ABORTS * MSG = -3 : ILLEGAL LENGTH OF LIWRK OR LWRK; LIWRK < 2*LIRN * +12*N+2 OR LWRK < 7*N. THE PROGRAM ABORTS * MSG = -4 : ILLEGAL NUMBER OF NONZEROS NZ; NZ <= 0. THE * PROGRAM ABORTS * MSG = -5 : THE K-TH ELEMENT OF IRN OR THE K-TH ELEMENT OF * ICN IS NOT AN INTEGER BETWEEN 1 AND N; (IRN(K) * < 1 OR IRN(K) > N) OR (ICN(K) < 1 OR ICN(K) > N). * THE PROGRAM ABORTS * MSG = -6 : THE K-TH DIAGONAL ELEMENT IS NOT IN THE SPARSITY * PATTERN. THIS IS CHECKED ONLY IF HSNFLG = 0. * THE PROGRAM ABORTS * MSG = -7 : REDUNDANT ENTRIES IN SPARSITY PATTERN WAS * ENCOUNTERED. WHEN HSNFLG = 1 OR HSNFLG = 2, THE * PROGRAM ABORTS. WHEN HSNFLG = 0, THE PROGRAM * ELIMINATES THE REDUNDANT ENTRIES AND CONTINUES * THE EXECUTION * MSG = -8 : PROBABLE CODING ERROR IN THE USER'S ANALYTIC * GRADIENT ROUTINE UGR. THE PROGRAM ABORTS. (THIS * CHECK CAN BE OVERRIDDEN BY SETTING GRDFLG = 2.) * (SEE SUBROUTINE STCHKG) * MSG = -9 : PROBABLE CODING ERROR IN THE USER'S ANALYTIC * HESSIAN ROUTINE USH. THE PROGRAM ABORTS. (THIS * CHECK CAN BE OVERRIDDEN BY SETTING HSNFLG = 2.) * (SEE SUBROUTINE STCHKH) * * IPR --> DEVICE TO WHICH TO SEND OUTPUT * INTEGER N,NPAIRS,NZ,LIRN,LICN,LIWRK,LWRK,ILIM,NDIGIT, + METHOD INTEGER GRDFLG,HSNFLG,LIWA,MSG,IPR INTEGER IRN(LIRN),ICN(LICN),IWA(N),JPNTR(N+1) REAL X(N),TYPX(N),FSCALE,GRADTL,STEPTL,STEPMX LOGICAL SCALE * * LOCAL VARIABLES * INTEGER I,IR,J,JP,K REAL STPSIZ,TEMP,EPS,ZERO,ONE,THREE,TEN,THOUS INTRINSIC LOG10,MAX,MIN,SQRT DATA ZERO,ONE,THREE,TEN,THOUS/0.0E+0,1.0E+0,3.0E+0, + 10.0E+0,1000.0E+0/ * * CHECK THAT PARAMETERS ONLY TAKE ON ACCEPTABLE VALUES. * IF NOT, SET THEM TO DEFAULT VALUES. * * CHECK DIMENSION OF THE PROBLEM * IF (N.LE.0) THEN WRITE (IPR,9000) N MSG = -1 RETURN END IF * IF (LIRN.LE.0 .OR. LICN.LE.0) THEN WRITE (IPR,9010) LIRN,LICN MSG = -2 RETURN END IF * * IF LIWRK IS LESS THAN 2*LIRN+12*N+2 OR LWRK LESS THAN 7*N * SET MSG = -3 AND ABORT. * IF (LIWRK.LT.2*LIRN+12*N+2 .OR. LWRK.LT.7*N) THEN WRITE (IPR,9020) LIWRK,LWRK MSG = -3 RETURN END IF * * SET THE LENGTH OF LIWA * LIWA = 2*LIRN + 8*N * IF (HSNFLG.EQ.0) THEN * * IF NPAIRS IS NOT POSITIVE THEN SET MSG TO -4 AND ABORT. * IF (NPAIRS.LE.0) THEN WRITE (IPR,9030) NPAIRS MSG = -4 RETURN END IF * * IF THE K-TH ELEMENT OF IRN OR THE K-TH ELEMENT OF ICN IS * NOT AN INTEGER BETWEEN 1 AND N THEN SET MSG TO -5 AND ABORT. * DO 10 K = 1,NPAIRS IF (IRN(K).LT.1 .OR. IRN(K).GT.N .OR. ICN(K).LT.1 .OR. + ICN(K).GT.N) THEN WRITE (IPR,9040) K,IRN(K),K,ICN(K) MSG = -5 RETURN END IF 10 CONTINUE * * GENERATE THE SPARSITY PATTERN FOR THE LOWER * TRIANGULAR PART OF HESSIAN. * DO 20 K = 1,NPAIRS I = IRN(K) J = ICN(K) IRN(K) = MAX(I,J) ICN(K) = MIN(I,J) 20 CONTINUE * * IF HSNFLG = 0 AND THE K-TH DIAGONAL ELEMENT IS NOT IN THE * SPARSITY PATTERN THEN ABORT. * I = 0 DO 30 K = 1,N IWA(K) = 0 30 CONTINUE DO 40 K = 1,NPAIRS IF (IRN(K).EQ.ICN(K)) IWA(IRN(K)) = 1 40 CONTINUE DO 50 K = 1,N IF (IWA(K).NE.1) THEN WRITE (IPR,9050) K MSG = -6 RETURN END IF 50 CONTINUE * * SORT THE DATA STRUCTURE BY COLUMNS. * CALL SRTDAT(N,NPAIRS,IRN,ICN,JPNTR,IWA) * * COMPRESS THE DATA AND DETERMINE THE NUMBER OF NONZERO * ELEMENTS IN THE LOWER TRIANGULAR PART OF HESSIAN. * DO 60 I = 1,N IWA(I) = 0 60 CONTINUE NZ = 0 DO 80 J = 1,N K = NZ DO 70 JP = JPNTR(J),JPNTR(J+1) - 1 IR = IRN(JP) IF (IWA(IR).NE.J) THEN NZ = NZ + 1 IRN(NZ) = IR ICN(NZ) = J IWA(IR) = J END IF 70 CONTINUE JPNTR(J) = K + 1 80 CONTINUE JPNTR(N+1) = NZ + 1 * END IF * * CHECK METHOD USED * IF (METHOD.NE.0) METHOD = 1 * * CHECK GRADIENT FLAG * IF (GRDFLG.NE.2 .AND. GRDFLG.NE.1) GRDFLG = 0 * * CHECK HESSIAN FLAG * IF (HSNFLG.NE.2 .AND. HSNFLG.NE.1) HSNFLG = 0 * * CHECK MSG * IF (MSG.NE.2 .AND. MSG.NE.0) MSG = 1 * * CHECK SCALE MATRIX * DO 90 I = 1,N IF (TYPX(I).EQ.ZERO) TYPX(I) = ONE IF (TYPX(I).LT.ZERO) TYPX(I) = -TYPX(I) 90 CONTINUE * * SET THE LOGICAL VARIABLE SCALE * SCALE = .FALSE. DO 100 I = 1,N IF (TYPX(I).NE.ONE) THEN SCALE = .TRUE. GO TO 110 END IF 100 CONTINUE 110 CONTINUE * * CHECK MAXIMUM STEP SIZE * IF (STEPMX.GT.ZERO) GO TO 130 STPSIZ = ZERO DO 120 I = 1,N STPSIZ = STPSIZ + X(I)*X(I)/TYPX(I)*TYPX(I) 120 CONTINUE STPSIZ = SQRT(STPSIZ) STEPMX = MAX(THOUS*STPSIZ,THOUS) 130 CONTINUE * * CHECK FUNCTION SCALE * IF (FSCALE.EQ.ZERO) FSCALE = ONE IF (FSCALE.LT.ZERO) FSCALE = -FSCALE * * CHECK GRADIENT TOLERANCE * IF (GRADTL.LT.ZERO) GRADTL = EPS** (ONE/THREE) * * CHECK STEP TOLERANCE * IF (STEPTL.LT.ZERO) THEN TEMP = EPS** (ONE/THREE) STEPTL = TEMP*TEMP END IF * * CHECK ITERATION LIMIT * IF (ILIM.LE.0) ILIM = 150 * * CHECK NUMBER OF DIGITS OF ACCURACY IN FUNCTION FCN * IF (NDIGIT.LE.0) NDIGIT = -LOG10(EPS) IF (TEN** (-NDIGIT).LE.EPS) NDIGIT = -LOG10(EPS) RETURN * * ERROR EXITS AND WARNINGS * 9000 FORMAT (' STCHKI ILLEGAL DIMENSION N=',I10) 9010 FORMAT (' STCHKI ILLEGAL LENGTH OF LIRN',I10,' OR LICN=',I10) 9020 FORMAT (' STCHKI ILLEGAL LENGTH OF INTEGER WORKSPACE'/, + ' STCHKI IWRK=',I10,' OR REAL WORKSPACE WRK=',I10) 9030 FORMAT (' STCHKI ILLEGAL NUMBER OF NONZEROS NZ=',I10) 9040 FORMAT (' STCHKI ILLEGAL ROW OR COLUMN INDEX'/, + ' STCHKI IRN( ',I6,' )=',I6,2X,' ICN( ',I6,' )=',I5) 9050 FORMAT (' STCHKI DIAGONAL ELEMENT',I6,' IS NOT IN THE', + ' SPARSITY PATTERN'/, + ' STCHKI THE FINITE-DIFFERENCE HESSIAN APPROXIMATION', + ' REQUIRE'/, + ' STCHKI THAT DIAGONAL ELEMENTS BE IN THE SPARSITY', + ' PATTERN') END SUBROUTINE STCHKS(N,XPLS,FPLS,GPLS,X,ITNCNT,ICSCMX,ITRMCD,GRADTL, + STEPTL,FSCALE,ILIM,IRETCD,MXTAKE,RGX,IPR,MSG) * * PURPOSE: * ------- * * THIS ROUTINE CHECKS THE STOPPING CRITERIA AND TERMINATES THE * OPTIMIZATION ALGORITHM IF ANY OF THE FOLLOWING IS SATISFIED: * * 1. THE SCALED GRADIENT IS LESS THAN GRADTL * 2. THE LENGTH OF THE CURRENT STEP IS LESS THAN STEPTL * 3. THE CURRENT GLOBAL STEP FAILED TO LOCATE A POINT LOWER THAN XPLS * 4. THE ITERATION LIMIT HAS BEEN EXCEEDED * 5. FIVE CONSECUTIVE STEPS OF LENGTH STEPMX HAVE BEEN TAKEN * * PARAMETERS: * ---------- * * N --> DIMENSION OF PROBLEM * XPLS --> CURRENT ITERATE * FPLS --> FUNCTION VALUE AT XPLS * GPLS --> GRADIENT AT XPLS * X --> OLD ITERATE * ITNCNT --> CURRENT ITERATION NUMBER * ICSCMX <--> NUMBER CONSECUTIVE STEPS .GE. STEPMX * (RETAIN VALUE BETWEEN SUCCESSIVE CALLS) * ITRMCD <-- TERMINATION CODE * GRADTL --> TOLERANCE AT WHICH RELATIVE GRADIENT CONSIDERED * CLOSE ENOUGH TO ZERO TO TERMINATE ALGORITHM * STEPTL --> RELATIVE STEP SIZE AT WHICH SUCCESSIVE ITERATES * CONSIDERED CLOSE ENOUGH TO TERMINATE ALGORITHM * FSCALE --> ESTIMATE OF SCALE OF OBJECTIVE FUNCTION * ILIM --> MAXIMUM NUMBER OF ALLOWABLE ITERATIONS * IRETCD --> RETURN CODE FROM SUBROUTINE STLSCH * MXTAKE --> BOOLEAN FLAG INDICATING STEP OF MAXIMUM LENGTH * RGX <-- SCALED GRADIENT * IPR --> USED DEVICE TO WHICH TO SEND OUTPUT * MSG --> CONTROL OUTPUT * MSG = 0 MEANS NO INFORMATION IS OUTPUT * MSG >= 1 MEANS INFORMATION ABOUT STOPPING CONDITIONS * IS OUTPUT * INTEGER N,ITNCNT,ICSCMX,ITRMCD,ILIM,IRETCD,IPR,MSG REAL XPLS(N),FPLS,GPLS(N),X(N),GRADTL,STEPTL REAL FSCALE,RGX * * LOCAL VARIABLES * INTEGER I REAL D,RELGRD,RSX,RELSTP,ZERO,ONE LOGICAL MXTAKE INTRINSIC ABS,MAX DATA ZERO,ONE/0.0E+0,1.0E+0/ * ITRMCD = 0 * * LAST GLOBAL STEP FAILED TO LOCATE A POINT LOWER THAN X * IF (IRETCD.EQ.1) THEN ITRMCD = 3 IF (MSG.GE.1) WRITE (IPR,9020) RETURN END IF * * COMPUTE SCALED GRADIENT AND CHECK WHETHER IT IS WITHIN TOLERANCE * D = MAX(ABS(FPLS),FSCALE) * D=ONE RGX = ZERO DO 10 I = 1,N RELGRD = ABS(GPLS(I))*MAX(ABS(XPLS(I)),ONE)/D RGX = MAX(RGX,RELGRD) 10 CONTINUE IF (RGX.LE.GRADTL) THEN ITRMCD = 1 IF (MSG.GE.1) WRITE (IPR,9000) RETURN END IF * IF (ITNCNT.EQ.0) RETURN * * FIND DIRECTION IN WHICH RELATIVE STEPSIZE MAXIMUM * CHECK WHETHER WITHIN TOLERANCE * RSX = 0.0 DO 20 I = 1,N RELSTP = ABS(XPLS(I)-X(I))/MAX(ABS(XPLS(I)),ONE) RSX = MAX(RSX,RELSTP) 20 CONTINUE IF (RSX.LE.STEPTL) THEN ITRMCD = 2 IF (MSG.GE.1) WRITE (IPR,9010) RETURN END IF * * CHECK ITERATION LIMIT * IF (ITNCNT.GE.ILIM) THEN ITRMCD = 4 IF (MSG.GE.1) WRITE (IPR,9030) RETURN END IF * * CHECK NUMBER OF CONSECUTIVE STEPS \ STEPMX * IF (MXTAKE) THEN ICSCMX = ICSCMX + 1 IF (ICSCMX.GE.5) THEN ITRMCD = 5 IF (MSG.GE.1) WRITE (IPR,9040) RETURN END IF ELSE ICSCMX = 0 RETURN END IF * 9000 FORMAT (' STCHKS RELATIVE GRADIENT CLOSE TO ZERO'/ + ' STCHKS CURRENT ITERATE IS PROBABLY SOLUTION') 9010 FORMAT (' STCHKS SUCCESSIVE ITERATES WITHIN TOLERANCE'/ + ' STCHKS CURRENT ITERATE IS PROBABLY SOLUTION') 9020 FORMAT (' STCHKS LAST GLOBAL STEP FAILED TO LOCATE A POINT', + ' LOWER THAN X',/ + ' STCHKS EITHER X IS AN APPROXIMATE LOCAL MINIMUM', + ' OF THE FUNCTION',/ + ' STCHKS THE FUNCTION IS TOO NON-LINEAR FOR THIS', + ' ALGORITHM',/' STCHKS OR STEPTL IS TOO LARGE') 9030 FORMAT (' STCHKS ITERATION LIMIT EXCEEDED', + /' STCHKS ALGORITHM FAILED') 9040 FORMAT (' STCHKS MAXIMUM STEP SIZE EXCEEDED 5', + ' CONSECUTIVE TIMES',/ + ' STCHKS EITHER THE FUNCTION IS UNBOUNDED BELOW', + /' STCHKS BECOMES ASYMPTOTIC TO A FINITE VALUE', + ' FROM ABOVE IN SOME DIRECTION', + /' STCHKS OR STEPMX IS TOO SMALL') END SUBROUTINE STCZ3P(A1,A2,A3,XP,NZEROS) * * PURPOSE: * -------- * * THIS ROUTINE COMPUTES THE ROOTS OF A THIRD DEGREE * POLYNOMIAL OF THE FORM : X**3 + A1*X**2 + A2*X + A3. * * PARAMETERS: * ---------- * * A1, A2, A3 ---> COEFFICIENTS OF THE CUBIC POLYNOMIAL * XP <--- ZEROS OF THE CUBIC POLYNOMIAL * NZEROS <--- NUMBER OF ZEROS FOUND * * INTEGER NZEROS REAL A1,A2,A3,XP(3) * * LOCAL VARIABLES * REAL CST1,CST2,Q,R,D,S,T,THETA,PI,CONST REAL ZERO,HALF,ONE,TWO,THREE,FOUR,NINE REAL TSEVEN,FFOUR INTRINSIC ACOS,COS,SQRT PARAMETER (PI=3.141592741012573E+0) DATA ZERO,HALF,ONE,TWO,THREE,FOUR,NINE,TSEVEN, + FFOUR/0.0E+0,5.0E-1,1.0E+0,2.0E+0,3.0E+0,4.0E+0, + 9.0E+0,27.0E+0,54.0E+0/ * NZEROS = 0 CST1 = ONE/THREE CST2 = CST1*A1 Q = (THREE*A2-A1**TWO)/NINE R = (NINE*A1*A2-TSEVEN*A3-TWO*A1**THREE)/FFOUR D = Q**THREE + R**TWO IF (D.GT.ZERO) THEN S = R + SQRT(D) IF (S.LT.ZERO) THEN S = - ((-S)**CST1) ELSE S = S**CST1 END IF T = R - SQRT(D) IF (T.LT.ZERO) THEN T = - ((-T)**CST1) ELSE T = T**CST1 END IF * * COMPUTE THE ONLY REAL ZERO * NZEROS = 1 XP(1) = S + T - CST2 ELSE IF (D.EQ.ZERO) THEN IF (R.LT.ZERO) THEN S = - ((-R)**CST1) ELSE S = R**CST1 END IF T = S * * COMPUTE THE TWO REAL ZEROS * NZEROS = 2 XP(1) = S + T - CST2 XP(2) = -HALF* (S+T) ELSE THETA = ACOS(R/SQRT(-Q**THREE)) CONST = TWO*SQRT(-Q) THETA = CST1*THETA * * COMPUTE THE THREE REAL ZEROS * NZEROS = 3 XP(1) = CONST*COS(THETA) - CST2 XP(2) = CONST*COS(THETA+TWO*PI*CST1) - CST2 XP(3) = CONST*COS(THETA+FOUR*PI*CST1) - CST2 END IF RETURN END SUBROUTINE STDFLT(N,TYPX,FSCALE,GRADTL,STEPTL,ILIM,STEPMX,IPR, + METHOD,GRDFLG,HSNFLG,NDIGIT,INFORM,MSG) * * PURPOSE: *-------- * * THIS ROUTINE SETS THE DEFAULT VALUES OF THE PACKAGE. * * N --> A POSITIVE INTEGER VARIABLE SPECIFYING THE NUMBER OF * VARIABLES IN THE PROBLEM * TYPX <-- AN ARRAY OF LENGTH N IN WHICH THE TYPICAL SIZE OF THE * COMPONENTS OF X ARE SPECIFIED * FSCALE <-- A POSITIVE REAL NUMBER ESTIMATING THE MAGNITUDE * OF F(X) NEAR THE MINIMIZER * GRADTL <-- POSITIVE SCALAR GIVING THE TOLERANCE AT WHICH THE * SCALED GRADIENT OF F(X) IS CONSIDERED CLOSE ENOUGH TO * ZERO TO TERMINATE THE ALGORITHM * STEPTL <-- A POSITIVE SCALAR PROVIDING THE MINIMUM ALLOWABLE * RELATIVE STEP LENGTH * ILIM <-- POSITIVE INTEGER SPECIFYING THE MAXIMUM ITERATIONS TO BE * PERFORMED BEFORE THE PROGRAM IS TERMINATED * STEPMX <-- A POSITIVE SCALAR PROVIDING THE MAXIMUM ALLOWABLE SCALED * STEP LENGTH * IPR <-- THE UNIT ON WHICH THE ROUTINE OUTPUTS INFORMATION * METHOD <-- AN INTEGER FLAG DESIGNATING WHICH METHOD TO USE * METHOD = 0 : USE NEWTON'S METHOD * METHOD = 1 : USE THE TENSOR METHOD * GRDFLG <-- INTEGER FLAG DESIGNATING WHETHER OR NOT ANALYTIC HESSIAN * HAS BEEN SUPPLIED BY THE USER * GRDFLG = 0 : NO ANALYTIC GRADIENT SUPPLIED * GRDFLG = 1 : ANALYTIC GRADIENT SUPPLIED (WILL BE CHECKED * AGAINST FINITE DIFFERENCE GRADIENT) * GRDFLG = 2 : ANALYTIC GRADIENT SUPPLIED (WILL NOT BE * CHECKED AGAINST FINITE DIFFERENCE GRADIENT) * HSNFLG <-- INTEGER FLAG DESIGNATING WHETHER OR NOT ANALYTIC HESSIAN * HAS BEEN SUPPLIED BY THE USER * HSNFLG = 0 : NO ANALYTIC HESSIAN SUPPLIED * HSNFLG = 1 : ANALYTIC HESSIAN SUPPLIED (WILL BE CHECKED * AGAINST FINITE DIFFERENCE HESSIAN) * HSNFLG = 2 : ANALYTIC HESSIAN SUPPLIED (WILL NOT BE * CHECKED AGAINST FINITE DIFFERENCE HESSIAN) * NDIGIT <-- INTEGER ESTIMATING THE NUMBER OF ACCURATE DIGITS ON THE * OBJECTIVE FUNCTION F(X) * INFORM <-- AN INTEGER VARIABLE. IF IT IS SET TO 1, THE USER MUST * OBTAIN HESS TIMES VECTOR AND RE-ENTER WITH INFORM * UNCHANGED. THE RESULT OF HESS TIMES VECTOR MUST BE * STORED IN VECTOR. THE DEFAULT VALUE OF INFORM IS 0, * MEANING THAT HESS TIMES VECTOR IS COMPUTED BY THE * PACKAGE * MSG <-- AN INTEGER VARIABLE THAT THE USER MAY SET ON INPUT TO * INHIBIT CERTAIN AUTOMATIC CHECKS OR OVERRIDE CERTAIN * DEFAULT CHARACTERISTICS OF THE PACKAGE * INTEGER N,ILIM,IPR,METHOD,GRDFLG,HSNFLG,NDIGIT,INFORM,MSG REAL TYPX(N),FSCALE,GRADTL,STEPTL,STEPMX * * LOCAL VARIABLES * INTEGER I REAL EPS,TEMP,ZERO,ONE,THREE REAL R1MACH INTRINSIC LOG10 DATA ZERO,ONE,THREE/0.0E+0,1.0E+0,3.0E+0/ * EPS = R1MACH(4) METHOD = 1 FSCALE = ONE GRDFLG = 0 HSNFLG = 0 DO 10 I = 1,N TYPX(I) = ONE 10 CONTINUE TEMP = EPS** (ONE/THREE) GRADTL = TEMP STEPTL = TEMP*TEMP NDIGIT = -LOG10(EPS) * * SET ACTUAL DFAULT VALUE OF STEPMX IN STCHKI * STEPMX = ZERO ILIM = 500 IPR = 6 MSG = 1 INFORM = 0 RETURN END SUBROUTINE STDRUO(N,X,NPAIRS,IRN,LIRN,ICN,LICN,FCN,UGR,USH,TYPX, + FSCALE,GRADTL,STEPTL,ILIM,STEPMX,IPR,METHOD, + GRDFLG,HSNFLG,NDIGIT,MSG,XPLS,FPLS,GPLS,HESS,G, + S,D,DN,E,XD,BV,LISTP,NGRP,IPNTR,JPNTR,IWA,LIWRK, + LWRK,TERMCD,VECTOR,INFORM) * * PURPOSE: * ------- * * THIS ROUTINE IS THE DRIVER FOR SOLVING LARGE, SPARSE UNCONSTRAINED * OPTIMIZATION PROBLEMS USING TENSOR METHODS. * * PARAMETERS: * ---------- * * N --> A POSITIVE INTEGER VARIABLE SPECIFYING THE NUMBER OF * VARIABLES IN THE PROBLEM * X --> AN ARRAY OF LENGTH N THAT CONTAINS AN INITIAL * ESTIMATE OF THE MINIMIZER * NPAIRS --> AN INTEGER VARIABLE THAT MUST BE SET BY THE USER TO * THE NUMBER OF NONZEROS STORED IN THE LOWER OR UPPER * HALF OF THE HESSIAN MATRIX * IRN --> AN INTEGER ARRAY OF LENGTH LIRN. ON ENTRY, IT MUST * HOLD THE ROW INDEX OF EACH NONZERO STORED IN THE * LOWER OR UPPER HALF OF THE HESSIAN MATRIX * LIRN --> AN INTEGER VARIABLE THAT MUST BE SET BY THE USER TO * THE LENGTH OF ARRAY IRN. LIRN NEED NOT BE AS LARGE * AS LICN; NORMALLY IT NEED NOT BE VERY MUCH GREATER * THAN NPAIRS * ICN --> AN INTEGER ARRAY OF LENGTH LICN. ON ENTRY, IT MUST * HOLD THE COLUMN INDEX OF THE NONZEROS STORED IN THE * LOWER OR UPPER HALF OF THE HESSIAN MATRIX * LICN --> AN INTEGER VARIABLE THAT MUST BE SET BY THE USER TO THE * LENGTH OF THE HESSIAN ARRAY HESS AND ICN. LICN SHOULD * ORDINARILY BE 2 TO 4 TIMES AS LARGE AS NPAIRS * FCN --> THE NAME OF A USER SUPPLIED SUBROUTINE THAT EVALUATES * THE FUNCTION F AT AN ARBITRARY VECTOR X. THE SUBROUTINE * MUST BE DECLARED EXTERNAL IN THE USER'S CALLING PROGRAM * AND MUST CONFORM TO * CALL FCN(N, X, F) * UGR --> THE NAME OF A USER SUPPLIED SUBROUTINE THAT RETURNS IN * THE VALUE OF THE GRADIENT. UGR MUST BE DECLARED EXTERNAL * IN THE USER'S CALLING PROGRAM AND MUST CONFORM TO THE * USAGE * CALL UGR(N, X, G) * IF NO ANALYTIC GRADIENT IS SUPPLIED (GRDFLG = 0), THE * USER MUST USE THE DUMMY NAME STDUGR * USH --> THE NAME OF A USER SUPPLIED SUBROUTINE THAT RETURNS * IN HESS THE VALUE OF THE HESSIAN AT THE CURRENT POINT X. * USH MUST BE DECLARED EXTERNAL IN THE USER'S CALLING * PROGRAM AND MUST CONFORM TO THE USAGE * CALL USH(N,X,NPAIRS,LICN,HESS,IRN,ICN) * ONLY THE LOWER OR UPPER TRIANGULAR PART AND THE DIAGONAL * OF HESS SHOULD BE GIVEN, WITH THEIR CORRESPONDING ROW * AND COLUMN INDICES. IF NO ANALYTIC GRADIENT IS SUPPLIED * (HSNFLG = 0), THE USER MUST USE THE DUMMY NAME STDUSH * TYPX --> AN ARRAY OF LENGTH N IN WHICH THE TYPICAL SIZE OF THE * COMPONENTS OF X ARE SPECIFIED * FSCALE --> A POSITIVE REAL NUMBER ESTIMATING THE MAGNITUDE * OF F(X) NEAR THE MINIMIZER * GRADTL --> POSITIVE SCALAR GIVING THE TOLERANCE AT WHICH THE * SCALED GRADIENT OF F(X) IS CONSIDERED CLOSE ENOUGH TO * ZERO TO TERMINATE THE ALGORITHM * STEPTL --> A POSITIVE SCALAR PROVIDING THE MINIMUM ALLOWABLE * RELATIVE STEP LENGTH * ILIM --> POSITIVE INTEGER SPECIFYING THE MAXIMUM ITERATIONS * TO BE PERFORMED BEFORE THE PROGRAM IS TERMINATED * STEPMX --> A POSITIVE SCALAR PROVIDING THE MAXIMUM ALLOWABLE * SCALED STEP LENGTH * IPR --> THE UNIT ON WHICH THE ROUTINE OUTPUTS INFORMATION * METHOD --> AN INTEGER FLAG DESIGNATING WHICH METHOD TO USE * METHOD = 0 : USE NEWTON'S METHOD * METHOD = 1 : USE THE TENSOR METHOD * GRDFLG --> INTEGER FLAG DESIGNATING WHETHER OR NOT ANALYTIC * HESSIAN HAS BEEN SUPPLIED BY THE USER * GRDFLG = 0 : NO ANALYTIC GRADIENT SUPPLIED * GRDFLG = 1 : ANALYTIC GRADIENT SUPPLIED (WILL BE CHECKED * AGAINST FINITE DIFFERENCE GRADIENT) * GRDFLG = 2 : ANALYTIC GRADIENT SUPPLIED (WILL NOT BE * CHECKED AGAINST FINITE DIFFERENCE GRADIENT) * HSNFLG --> INTEGER FLAG DESIGNATING WHETHER OR NOT ANALYTIC * HESSIAN HAS BEEN SUPPLIED BY THE USER * HSNFLG = 0 : NO ANALYTIC HESSIAN SUPPLIED * HSNFLG = 1 : ANALYTIC HESSIAN SUPPLIED (WILL BE CHECKED * AGAINST FINITE DIFFERENCE HESSIAN) * HSNFLG = 2 : ANALYTIC HESSIAN SUPPLIED (WILL NOT BE * CHECKED AGAINST FINITE DIFFERENCE HESSIAN) * NDIGIT --> INTEGER ESTIMATING THE NUMBER OF ACCURATE DIGITS IN THE * OBJECTIVE FUNCTION F(X) * MSG <--> AN INTEGER VARIABLE THAT THE USER MAY SET ON INPUT TO * INHIBIT CERTAIN AUTOMATIC CHECKS OR OVERRIDE CERTAIN * DEFAULT CHARACTERISTICS OF THE PACKAGE * XPLS <-- AN ARRAY OF LENGTH N CONTAINING THE BEST APPROXIMATION * TO THE MINIMIZER UPON RETURN. (IF THE ALGORITHM HAS NOT * CONVERGED, THE LAST ITERATE IS RETURNED) * FPLS <-- A SCALAR VARIABLE THAT CONTAINS THE FUNCTION VALUE AT * THE FINAL ITERATE XPLS * GPLS <-- AN ARRAY OF LENGTH N CONTAINING THE GRADIENT VALUE * AT XPLS * HESS <-- AN ARRAY THAT IS USED TO STORE THE HESSIAN MATRIX AT * EACH ITERATION. IT NEEDS TO BE AT LEAST OF DIMENSION * LICN. ON EXIT, HESS CONTAINS THE HESSIAN VALUE AT THE * MINIMIZER * G,S,D,DN,E,XD,BV,LISTP,NGRP,IPNTR,JPNTR,IWA --> WORKSPACE * LIWRK --> AN INTEGER VARIABLE. IT MUST BE SET BY THE USER TO THE * LENGTH OF ARRAY IWRK AND IS NOT ALTERED BY THE PACKAGE * (SEE SUBROUTINE STUMCD (STUMCS)) * LWRK --> AN INTEGER VARIABLE. IT MUST BE SET BY THE USER TO THE * LENGTH OF ARRAY WRK AND IS NOT ALTERED BY THE PACKAGE * (SEE SUBROUTINE STUMCD (STUMCS)) * TERMCD <-- AN INTEGER THAT SPECIFIES THE REASON WHY THE ALGORITHM * HAS TERMINATED * VECTOR<--> AN ARRAY OF LENGTH N. IT NEED NOT BE SET BY THE USER * ON ENTRY. IF INFORM IS SET TO 1, A RE-ENTRY MUST BE * MADE WITH VECTOR SET TO HESS TIMES VECTOR (SEE INFORM) * INFORM<--> AN INTEGER VARIABLE. IF IT IS SET TO 1, THE USER MUST * OBTAIN HESS TIMES VECTOR AND RE-ENTER WITH INFORM * UNCHANGED. THE RESULT OF HESS TIMES VECTOR MUST BE * STORED IN VECTOR. THE DEFAULT VALUE OF INFORM IS 0, * MEANING THAT HESS TIMES VECTOR IS COMPUTED BY THE * PACKAGE * * BLAS SUBROUTINES: DCOPY,DDOT,DNRM2 * INTEGER N,NPAIRS,LIRN,LICN,ILIM,IPR,METHOD,GRDFLG,HSNFLG INTEGER NDIGIT,MSG,LIWRK,LWRK,TERMCD,INFORM INTEGER IRN(LIRN),ICN(LICN),LISTP(N),NGRP(N),IPNTR(N+1) INTEGER JPNTR(N+1),IWA(*) REAL X(N),TYPX(N),FSCALE,GRADTL,STEPTL,STEPMX REAL XPLS(N),FPLS,GPLS(N),HESS(LICN),G(N) REAL S(N),D(N),DN(N),E(N),XD(N),BV(N),VECTOR(N) LOGICAL SCALE * * LOCAL VARIABLES * INTEGER ITNNO,ICSCMX,I,MAXGRP,MINGRP,IFCNT,IGCNT,IHCNT INTEGER IFLAG,IRETCD,NZ,LIWA REAL RNF,ANALTL,F,GNORM,FP,SUM1,SUM2 REAL ALMBDA,OCENT,ONE,TWO,THREE,TEN REAL EPS,GAMMA,ALPHA,BETA,FN,RGX REAL SDOT,SNRM2,R1MACH LOGICAL MXTAKE,DSCENT,FCALL,CHECKH COMMON /ALMB/ALMBDA COMMON /COUNT/IFCNT,IGCNT,IHCNT EXTERNAL FCN,UGR,USH EXTERNAL SCOPY,SNRM2,SDOT DATA OCENT,ONE,TWO,THREE,TEN/1.0E-2,1.0E+0,2.0E+0, + 3.0E+0,10.0E+0/ * * IF INFORM = 2 THEN USER MUST SUPPLY VECTOR = HESS * S * IF (INFORM.EQ.2) GO TO 160 * * COMPUTE MACHINE EPSILON * EPS = R1MACH(4) * * CHECK USER'S INPUT DATA * CALL STCHKI(N,NPAIRS,NZ,X,IRN,LIRN,ICN,LICN,LIWRK,LWRK,SCALE,TYPX, + FSCALE,GRADTL,STEPTL,ILIM,NDIGIT,EPS,METHOD,GRDFLG, + HSNFLG,STEPMX,IWA,LIWA,JPNTR,MSG,IPR) IF (MSG.LT.0) RETURN * * PRINT OUT USED PARAMETERS * IF (MSG.GE.1) THEN WRITE (IPR,9000) GRDFLG WRITE (IPR,9010) HSNFLG WRITE (IPR,9020) METHOD WRITE (IPR,9030) ILIM WRITE (IPR,9040) EPS WRITE (IPR,9050) STEPTL WRITE (IPR,9060) GRADTL WRITE (IPR,9070) STEPMX END IF * * INITIALIZATION * ITNNO = 0 ICSCMX = 0 TERMCD = 0 IFCNT = 0 IGCNT = 0 IHCNT = 0 * * SCALE X * IF (SCALE) THEN DO 10 I = 1,N X(I) = X(I)/TYPX(I) 10 CONTINUE END IF * * INITIAL ITERATION * RNF = MAX(TEN** (-NDIGIT),EPS) ANALTL = MAX(OCENT,SQRT(RNF)) * * UNSCALE X AND COMPUTE F AND G * IF (SCALE) THEN DO 20 I = 1,N VECTOR(I) = X(I)*TYPX(I) 20 CONTINUE ELSE DO 30 I = 1,N VECTOR(I) = X(I) 30 CONTINUE END IF IFCNT = IFCNT + 1 CALL FCN(N,VECTOR,F) IGCNT = IGCNT + 1 IF (GRDFLG.EQ.1) THEN CALL UGR(N,VECTOR,G) CALL STCHKG(N,VECTOR,FCN,F,G,TYPX,RNF,ANALTL,XD,MSG,IPR) IF (MSG.LT.0) RETURN ELSE IF (GRDFLG.EQ.2) THEN CALL UGR(N,VECTOR,G) ELSE CALL STFDGR(N,VECTOR,FCN,F,RNF,G) END IF * * SCALE G * IF (SCALE) THEN DO 40 I = 1,N G(I) = G(I)*TYPX(I) 40 CONTINUE END IF * GNORM = SNRM2(N,G,1) * * PRINT OUT INITIAL ITERATION * IF (MSG.GE.1) THEN CALL STRSLT(N,VECTOR,F,G,RGX,ITNNO,TERMCD,IPR) END IF * * TEST WHETHER INITIAL GUESS SATISFIES THE STOPPING CRITERIA * IF (GNORM.LE.GRADTL) THEN TERMCD = 1 RETURN END IF * * ITERATION 1 * ITNNO = ITNNO + 1 * * COMPUTE HESSIAN * IHCNT = IHCNT + 1 IF (HSNFLG.EQ.1) THEN CALL USH(N,VECTOR,NPAIRS,LICN,HESS,IRN,ICN) FCALL = .TRUE. CHECKH = .TRUE. CALL STCHKH(N,VECTOR,NPAIRS,NZ,IRN,LIRN,ICN,LICN,FCN,UGR, + SCALE,TYPX,GRDFLG,F,G,RNF,ANALTL,FCALL,CHECKH, + LISTP,NGRP,IPNTR,JPNTR,IWA,LIWA,D,XD,E,HESS,IPR, + MSG) IF (MSG.LT.0) RETURN FCALL = .FALSE. ELSE IF (HSNFLG.EQ.2) THEN CALL USH(N,VECTOR,NPAIRS,LICN,HESS,IRN,ICN) CHECKH = .FALSE. CALL STCHKH(N,VECTOR,NPAIRS,NZ,IRN,LIRN,ICN,LICN,FCN,UGR, + SCALE,TYPX,GRDFLG,F,G,RNF,ANALTL,FCALL,CHECKH, + LISTP,NGRP,IPNTR,JPNTR,IWA,LIWA,D,XD,E,HESS,IPR, + MSG) IF (MSG.LT.0) RETURN ELSE FCALL = .TRUE. CALL STFDHS(N,NZ,VECTOR,IRN,ICN,G,RNF,GRDFLG,FCN,UGR,SCALE, + TYPX,FCALL,LISTP,NGRP,IPNTR,JPNTR,IWA,LIWA,D,XD,E, + MAXGRP,MINGRP,HESS) FCALL = .FALSE. END IF * * SCALE HESS * IF (SCALE) THEN DO 50 I = 1,NZ HESS(I) = HESS(I)*TYPX(IRN(I))*TYPX(ICN(I)) 50 CONTINUE END IF * * SOLVE FOR NEWTON STEP D * DO 60 I = 1,N D(I) = -G(I) 60 CONTINUE * * SPARSE CHOLESKY DECOMPOSITION FOR HESS (HESS = L D LT) * CALL STMA27(N,NZ,IRN,ICN,HESS,LICN,IWA,LIWA,D,XD,1,.TRUE.,EPS, + IFLAG) * * COOMPUTE NEWTON STEP USING A LINE SEARCH * CALL STLSCH(N,X,F,G,D,XPLS,FPLS,MXTAKE,IRETCD,STEPMX,STEPTL,SCALE, + TYPX,FCN,VECTOR) * * UPDATE GRADIENT AT XPLS * * UNSCALE XPLS AND COMPUTE GPLS * IF (SCALE) THEN DO 70 I = 1,N VECTOR(I) = XPLS(I)*TYPX(I) 70 CONTINUE ELSE DO 80 I = 1,N VECTOR(I) = XPLS(I) 80 CONTINUE END IF IGCNT = IGCNT + 1 IF (GRDFLG.EQ.0) THEN CALL STFDGR(N,VECTOR,FCN,FPLS,RNF,GPLS) ELSE CALL UGR(N,VECTOR,GPLS) END IF * * SCALE GPLS * IF (SCALE) THEN DO 90 I = 1,N GPLS(I) = GPLS(I)*TYPX(I) 90 CONTINUE END IF * * CHECK STOPPING CONDITIONS * CALL STCHKS(N,XPLS,FPLS,GPLS,X,ITNNO,ICSCMX,TERMCD,GRADTL,STEPTL, + FSCALE,ILIM,IRETCD,MXTAKE,RGX,IPR,MSG) * * IF TERMCD > 0 THEN STOPPING CONDITIONS SATISFIED * IF (TERMCD.GT.0) GO TO 300 * * UPDATE X, F AND S FOR TENSOR MODEL * FP = F F = FPLS DO 100 I = 1,N S(I) = X(I) - XPLS(I) X(I) = XPLS(I) 100 CONTINUE * * IF MSG >= 2 THEN PRINT OUT EACH ITERATION * IF (MSG.GE.2) THEN CALL STRSLT(N,XPLS,FPLS,GPLS,RGX,ITNNO,TERMCD,IPR) END IF * * ITERATION > 1 * * UNSCALE X AND COMPUTE HESS * 110 CONTINUE * IF (SCALE) THEN DO 120 I = 1,N VECTOR(I) = X(I)*TYPX(I) 120 CONTINUE ELSE DO 130 I = 1,N VECTOR(I) = X(I) 130 CONTINUE END IF IHCNT = IHCNT + 1 IF (HSNFLG.EQ.0) THEN CALL STFDHS(N,NZ,VECTOR,IRN,ICN,GPLS,RNF,GRDFLG,FCN,UGR,SCALE, + TYPX,FCALL,LISTP,NGRP,IPNTR,JPNTR,IWA,LIWA,D,XD,E, + MAXGRP,MINGRP,HESS) ELSE CALL USH(N,VECTOR,NZ,LICN,HESS,IRN,ICN) END IF * * SCALE HESS * IF (SCALE) THEN DO 140 I = 1,NZ HESS(I) = HESS(I)*TYPX(IRN(I))*TYPX(ICN(I)) 140 CONTINUE END IF * * IF METHOD = 0 THEN USE NEWTON STEP ONLY * IF (METHOD.EQ.0) THEN * * COMPUTE NEWTON STEP * DO 150 I = 1,N D(I) = -GPLS(I) 150 CONTINUE * * CHOLESKY DECOMPOSITION FOR HESS * CALL STMA27(N,NZ,IRN,ICN,HESS,LICN,IWA,LIWA,D,XD,1,.TRUE.,EPS, + IFLAG) GO TO 190 * END IF * IF (INFORM.EQ.1) THEN INFORM = 2 CALL SCOPY(N,S(1),1,VECTOR(1),1) RETURN ELSE CALL STHMUV(N,NZ,IRN,ICN,HESS,S,VECTOR) GO TO 170 END IF * 160 INFORM = 1 170 CONTINUE * * FORM TENSOR MODEL * CALL STFTSM(N,F,FP,GPLS,G,S,ALPHA,BETA,VECTOR,D) * SUM1 = SDOT(N,S(1),1,S(1),1) SUM2 = SDOT(N,S(1),1,D(1),1) * DO 180 I = 1,N BV(I) = (THREE*SUM1*D(I)-TWO*S(I)*SUM2)/ (THREE*SUM1**3) 180 CONTINUE GAMMA = BETA/SUM1**4 * * SOLVE TENSOR MODEL AND COMPUTE TENSOR STEP * CALL STMSLV(N,D,S,GPLS,E,VECTOR,DN,BV,GAMMA,NZ,IRN,ICN,HESS,LICN, + IWA,LIWA,XD,EPS,DSCENT,IFLAG) * IF (.NOT.DSCENT) THEN CALL SCOPY(N,DN(1),1,D(1),1) END IF * 190 ITNNO = ITNNO + 1 CALL SCOPY(N,GPLS(1),1,G(1),1) * * COMPUTE TENSOR (OR NEWTON) STEP USING A LINE SEARCH * CALL STLSCH(N,X,F,G,D,XPLS,FPLS,MXTAKE,IRETCD,STEPMX,STEPTL,SCALE, + TYPX,FCN,VECTOR) * IF (METHOD.EQ.0) GO TO 260 IF (ALMBDA.EQ.ONE) GO TO 240 * IF (DSCENT) THEN * * FULL TENSOR STEP IS A DESCENT DIRECTION BUT DOES NOT * PROVIDE ENOUGH DECREASE IN THE OBJECTIVE FUNCTION. * CALCULATE NEWTON STEP ONLY IF HESSIAN IS INDEFINITE; * OTHERWISE NEWTON STEP WAS ALREADY COMPUTED AS PART * OF THE TENSOR STEP COMPUTATION * IF (IFLAG.EQ.3) THEN DO 200 I = 1,N DN(I) = -E(I) 200 CONTINUE GO TO 230 END IF CALL SCOPY(N,G(1),1,DN(1),1) CALL STMA27(N,NZ,IRN,ICN,HESS,LICN,IWA,LIWA,DN,XD,3,.TRUE., + EPS,IFLAG) IF (IFLAG.EQ.30) THEN DO 210 I = 1,N DN(I) = -E(I) 210 CONTINUE ELSE DO 220 I = 1,N DN(I) = -DN(I) 220 CONTINUE END IF 230 CONTINUE * * CALCULATE A STEP IN THE NEWTON DIRECTION * CALL STLSCH(N,X,F,G,DN,XD,FN,MXTAKE,IRETCD,STEPMX,STEPTL, + SCALE,TYPX,FCN,VECTOR) * * COMPARE TENSOR STEP TO NEWTON STEP * IF NEWTON STEP IS BETTER, SET NEXT ITERATE TO NEW NEWTON POINT * IF (FN.LT.FPLS) THEN FPLS = FN CALL SCOPY(N,DN(1),1,D(1),1) CALL SCOPY(N,XD(1),1,XPLS(1),1) END IF END IF 240 CONTINUE DO 250 I = 1,N D(I) = XPLS(I) - X(I) 250 CONTINUE * 260 CONTINUE * * UNSCALE XPLS, AND COMPUTE FPLS AND GPLS * IF (SCALE) THEN DO 270 I = 1,N VECTOR(I) = XPLS(I)*TYPX(I) 270 CONTINUE ELSE DO 280 I = 1,N VECTOR(I) = XPLS(I) 280 CONTINUE END IF IGCNT = IGCNT + 1 IF (GRDFLG.EQ.0) THEN CALL STFDGR(N,VECTOR,FCN,FPLS,RNF,GPLS) ELSE CALL UGR(N,VECTOR,GPLS) END IF * * SCALE GPLS * IF (SCALE) THEN DO 290 I = 1,N GPLS(I) = GPLS(I)*TYPX(I) 290 CONTINUE END IF * * CHECK STOPPING CONDITIONS * CALL STCHKS(N,XPLS,FPLS,GPLS,X,ITNNO,ICSCMX,TERMCD,GRADTL,STEPTL, + FSCALE,ILIM,IRETCD,MXTAKE,RGX,IPR,MSG) * * IF TERMCD = 0 THEN EXECUTION CONTINUES * IF (TERMCD.EQ.0) GO TO 330 * 300 CONTINUE * * TRANSFORM XPLS BACK TO ORIGINAL SPACE * IF (SCALE) THEN DO 310 I = 1,N XPLS(I) = XPLS(I)*TYPX(I) 310 CONTINUE END IF * * IF MSG >= 1 THEN PRINT OUT FINAL ITERATION * IF (MSG.GE.1) THEN CALL STRSLT(N,XPLS,FPLS,GPLS,RGX,ITNNO,TERMCD,IPR) END IF * * UPDATE THE HESSIAN * IHCNT = IHCNT + 1 IF (HSNFLG.EQ.0) THEN CALL STFDHS(N,NZ,VECTOR,IRN,ICN,GPLS,RNF,GRDFLG,FCN,UGR,SCALE, + TYPX,FCALL,LISTP,NGRP,IPNTR,JPNTR,IWA,LIWA,D,XD,E, + MAXGRP,MINGRP,HESS) ELSE CALL USH(N,XPLS,NZ,LICN,HESS,IRN,ICN) END IF * * SCALE HESS * IF (SCALE) THEN DO 320 I = 1,NZ HESS(I) = HESS(I)*TYPX(IRN(I))*TYPX(ICN(I)) 320 CONTINUE END IF RETURN * * UPDATE INFORMATION AT THE CURRENT POINT * 330 CONTINUE CALL SCOPY(N,XPLS(1),1,X(1),1) DO 340 I = 1,N S(I) = -D(I) 340 CONTINUE * * IF ITERATION LIMIT EXCEEDED THEN RETURN * IF (ITNNO.GT.ILIM) GO TO 300 * * IF MSG >= 2 THEN PRINT OUT EACH ITERATION * IF (MSG.GE.2) THEN CALL STRSLT(N,XPLS,FPLS,GPLS,RGX,ITNNO,TERMCD,IPR) END IF * * UPDATE F * FP = F F = FPLS * * PERFORM NEXT ITERATION * GO TO 110 * * END OF ITERATION > 1 * * 9000 FORMAT (' STDRUO GRADIENT FLAG = ',I1) 9010 FORMAT (' STDRUO HESSIAN FLAG = ',I1) 9020 FORMAT (' STDRUO METHOD = ',I1) 9030 FORMAT (' STDRUO ITERATION LIMIT =',I5) 9040 FORMAT (' STDRUO MACHINE EPSILON = ',E20.13) 9050 FORMAT (' STDRUO STEP TOLERANCE = ',E20.13) 9060 FORMAT (' STDRUO GRADIENT TOLERANCE = ',E20.13) 9070 FORMAT (' STDRUO MAXIMUM STEP SIZE = ',E20.13) END SUBROUTINE STDUGR(N,X,G) * * PURPOSE: * ------- * * THIS IS A DUMMY ROUTINE TO PREVENT UNSATISFIED EXTERNAL DIAGNOSTIC * WHEN SPECIFIC ANALYTIC GRADIENT IS NOT SUPPLIED. * INTEGER N REAL X(N),G(N) RETURN END SUBROUTINE STDUSH(N,X,NZ,LICN,HESS,IRN,ICN) * * PURPOSE: * ------- * * THIS IS A DUMMY ROUTINE TO PREVENT UNSATISFIED EXTERNAL DIAGNOSTIC * WHEN SPECIFIC ANALYTIC HESSIAN IS NOT SUPPLIED. * INTEGER N,NZ,LICN INTEGER IRN(NZ),ICN(LICN) REAL X(N),HESS(LICN) RETURN END SUBROUTINE STFDGR(N,XPLS,FCN,FPLS,RNOISE,FDG) * * PURPOSE * ------- * * THIS ROUTINE FINDS A FORWARD FINITE-DIFFERENCE APPROXIMATION "FDG" * TO THE FIRST DERIVATIVE OF THE FUNCTION DEFINED BY THE SUBPROGRAM * "FCN" EVALUATED AT THE NEW ITERATE "XPLS". * * PARAMETERS: * ---------- * N --> DIMENSION OF PROBLEM * XPLS --> NEW ITERATE * FCN --> NAME OF SUBROUTINE THAT EVALUATES THE * OPTIMIZATION FUNCTION * FPLS --> FUNCTION VALUE AT NEW ITERATE * RNOISE --> RELATIVE NOISE IN FCN * FDG <-- FINITE-DIFFERENCE APPROXIMATION OF GRADIENT AT XPLS * INTEGER N REAL XPLS(N),FPLS,RNOISE,FDG(N) * * LOCAL VARIABLES * INTEGER J REAL XTMPJ,STEPSZ,FHAT,ONE EXTERNAL FCN INTRINSIC ABS,MAX,SQRT DATA ONE/1.0E+0/ * * FIND DERIVATIVE OF FCN WITH RESPECT TO XPLS(J), J=1,N * DO 10 J = 1,N XTMPJ = XPLS(J) STEPSZ = SQRT(RNOISE)*MAX(ABS(XPLS(J)),ONE) XPLS(J) = XTMPJ + STEPSZ CALL FCN(N,XPLS,FHAT) XPLS(J) = XTMPJ FDG(J) = (FHAT-FPLS)/STEPSZ 10 CONTINUE RETURN END SUBROUTINE STFDHS(N,NZ,X,IRN,ICN,G,RNF,GRDFLG,FCN,UGR,SCALE,TYPX, + FCALL,LISTP,NGRP,IPNTR,JPNTR,IWA,LIWA,FHESD,XD, + ETA,MAXGRP,MINGRP,HESS) * * PURPOSE: * ------- * * THIS ROUTINE FINDS A FORWARD FINITE-DIFFERENCE APPROXIMATION "HESS" * TO THE SECOND DERIVATIVE (HESSIAN) OF THE FUNCTION DEFINED BY THE * SUBPROGRAM "FCN" EVALUATED AT THE NEW ITERATE "XPLS". * * PARAMETERS: * ---------- * * N --> DIMENSION OF PROBLEM * NZ --> NUMBER OF NONZEROS IN THE LOWER TRIANGULAR PART * OF THE HESSIAN MATRIX * X --> CURRENT ITERATE * IRN --> ROW INDEX OF THE NONZERO ELEMENTS IN THE LOWER * TRIANGULAR PART OF THE HESSIAN * ICN --> COLUMN INDEX OF THE NONZERO ELEMENTS IN THE LOWER * TRIANGULAR PART OF THE HESSIAN * G --> GRADIENT VALUE AT X * RNF --> RELATIVE NOISE IN OPTIMIZATION FUNCTION FCN * GRDFLG --> =1 IF ANALYTIC GRADIENT SUPPLIED * FCN --> NAME OF A USER SUPPLIED SUBROUTINE THAT EVALUATES * THE FUNCTION F AT AN ARBITRARY VECTOR X * UGR --> NAME OF A USER SUPPLIED SUBROUTINE THAT RETURNS IN * THE VALUE OF THE GRADIENT AT AN ARBITRARY VECTOR X * SCALE --> LOGICAL VARIABLE INDICATING WHETHER OR NOT SCALING * OF THE VARIABLES IS PERFORMED * TYPX --> AN ARRAY OF LENGTH N IN WHICH THE TYPICAL SIZE OF THE * COMPONENTS OF X ARE SPECIFIED * FCALL --> LOGICAL FLAG. IF THE HESSIAN MATRIX IS EVALUATED FOR * THE FIRST TIME, THEN FCALL IS SET TO TRUE; OTHERWISE * IT IS SET TO FALSE * LISTP,NGRP,IPNTR,JPNTR,IWA --> WORKSPACE * (SEE SUBROUTINE DSSM FOR DETAILS) * LIWA --> LENGTH OF IWA ARRAY * FHESD,XD,ETA --> WORKSPACE * MAXGRP,MINGRP<-- SEE SUBROUTINE DSSM FOR DETAILS * HESS <-- FINITE-DIFFERENCE HESSIAN ON EXIT * * INTEGER N,NZ,GRDFLG,LIWA,MAXGRP,MINGRP INTEGER IRN(NZ),ICN(NZ),LISTP(N),NGRP(N),IPNTR(N+1) INTEGER JPNTR(N+1),IWA(LIWA) REAL X(N),G(N),TYPX(N),RNF,FHESD(N) REAL XD(N),ETA(N),HESS(NZ) LOGICAL SCALE * * LOCAL VARIABLES * INTEGER JP,I,INFO,J,METHOD,NUMGRP REAL FD,ONE LOGICAL FCALL EXTERNAL FCN,UGR INTRINSIC ABS,MAX,SQRT DATA ONE/1.0E+0/ * METHOD = 1 * * CALL DSM ONLY IF IT IS THE FIRST TIME * IF (FCALL) THEN CALL DSSM(N,NZ,IRN,ICN,METHOD,LISTP,NGRP,MAXGRP,MINGRP,INFO, + IPNTR,JPNTR,IWA,LIWA) ELSE CALL SETR(N,N,IRN,JPNTR,ICN,IPNTR,IWA) END IF * * UNSCALE G * IF (SCALE) THEN DO 10 I = 1,N G(I) = G(I)/TYPX(I) 10 CONTINUE END IF * * APPROXIMATE THE HESSIAN MATRIX * DO 40 NUMGRP = 1,MAXGRP DO 20 J = 1,N ETA(J) = SQRT(RNF)*MAX(ABS(X(J)),ONE) XD(J) = X(J) IF (NGRP(J).EQ.NUMGRP) XD(J) = X(J) + ETA(J) 20 CONTINUE IF (GRDFLG.EQ.0) THEN CALL FCN(N,XD,FD) CALL STFDGR(N,XD,FCN,FD,RNF,FHESD) ELSE CALL UGR(N,XD,FHESD) END IF DO 30 I = 1,N FHESD(I) = FHESD(I) - G(I) 30 CONTINUE CALL FDHS(N,IRN,JPNTR,ICN,IPNTR,LISTP,NGRP,MAXGRP,NUMGRP,ETA, + FHESD,HESS,IWA) 40 CONTINUE DO 60 J = 1,N DO 50 JP = JPNTR(J),JPNTR(J+1) - 1 ICN(JP) = J 50 CONTINUE 60 CONTINUE * * SCALE G * IF (SCALE) THEN DO 70 I = 1,N G(I) = G(I)*TYPX(I) 70 CONTINUE END IF * END SUBROUTINE STFTSM(N,F,FP,G,GP,S,ALPHA,BETA,SH,A) * * PURPOSE: * ------- * * THIS ROUTINE FORMS THE TENSOR MODEL. * * PARAMETERS: * ---------- * * N --> DIMENSION OF PROBLEM * F --> CURRENT FUNCTION VALUE * FP --> PREVIOUS FUNCTION VALUE * G --> CURRENT GRADIENT * GP --> PREVIOUS GRADIENT * S --> STEP TO PREVIOUS POINT * ALPHA <-- SCALAR TO FORM 3RD ORDER TERM OF TENSOR MODEL * BETA <-- SCALAR TO FORM 4TH ORDER TERM OF TENSOR MODEL * SH <-- HESSIAN MATRIX TIMES S (SEE SUBROUTINE STHMUV) * A <-- A=2*(GP-G-SH-S*BETA/(6*(S-TRANS*S))) * * BLAS SUBROUTINES: DDOT * INTEGER N REAL F,FP,G(N),GP(N),S(N),ALPHA,BETA,SH(N),A(N) * * LOCAL VARIABLES * INTEGER I REAL GS,GPS,SHS,B1,B2,STS REAL HALF,TWO,SIX,TFOUR,STWO REAL SDOT EXTERNAL SDOT DATA HALF,TWO,SIX,TFOUR,STWO/5.0E-1,2.0E+0,6.0E+0, + 24.0E+0,72.0E+0/ * GS = SDOT(N,G(1),1,S(1),1) GPS = SDOT(N,GP(1),1,S(1),1) SHS = SDOT(N,SH(1),1,S(1),1) B1 = GPS - GS - SHS B2 = FP - F - GS - HALF*SHS ALPHA = TFOUR*B2 - SIX*B1 BETA = TFOUR*B1 - STWO*B2 * * COMPUTE A * STS = SDOT(N,S(1),1,S(1),1) DO 10 I = 1,N A(I) = TWO* (GP(I)-G(I)-SH(I)-S(I)*BETA/ (SIX*STS)) 10 CONTINUE RETURN END SUBROUTINE STHMUV(N,NZ,IRN,ICN,HESS,S,SH) * * PURPOSE: * ------- * * THIS ROUTINE COMPUTES SH = HESS * S. * * PARAMETERS: * ---------- * N --> A POSITIVE INTEGER VARIABLE SPECIFYING THE NUMBER OF * VARIABLES IN THE PROBLEM * NZ --> NUMBER OF NONZEROS STORED IN THE LOWER HALF OF * THE HESSIAN MATRIX * IRN --> AN INTEGER ARRAY OF LENGTH LIRN. ON ENTRY, IT MUST * HOLD THE ROW INDEX OF EACH NONZERO STORED IN THE LOWER * HALF OF THE HESSIAN MATRIX * ICN --> AN INTEGER ARRAY OF LENGTH LICN. ON ENTRY, IT MUST * HOLD THE COLUMN INDEX OF THE NONZEROS STORED IN LOWER * HALF OF THE HESSIAN MATRIX * HESS --> HESSIAN MATRIX AT CURRENT ITERATE * S --> STEP TO PREVIOUS POINT * SH <-- HESS * S ON EXIT * * INTEGER N,NZ INTEGER IRN(NZ),ICN(NZ) REAL HESS(NZ),S(N),SH(N) * * LOCAL VARIABLES * INTEGER I,II,JJ REAL ZERO DATA ZERO/0.0E0/ * DO 10 I = 1,N SH(I) = ZERO 10 CONTINUE DO 20 I = 1,NZ II = IRN(I) JJ = ICN(I) SH(II) = SH(II) + HESS(I)*S(JJ) IF (II.NE.JJ) THEN SH(JJ) = SH(JJ) + HESS(I)*S(II) END IF 20 CONTINUE RETURN END SUBROUTINE STLSCH(N,X,F,G,P,XPLS,FPLS,MXTAKE,IRETCD,STEPMX,STEPTL, + SCALE,TYPX,FCN,W2) * * PURPOSE: * ------- * * THIS ROUTINE FIND A NEXT ITERATE BY A LINE SEARCH. * THIS IS THE ALPHA CONDITION ONLY LINE SEARCH. * * PARAMETERS: * ---------- * N --> DIMENSION OF PROBLEM * X --> OLD ITERATE * F --> FUNCTION VALUE AT OLD ITERATE * G --> GRADIENT AT OLD ITERATE * P --> NEWTON OR TENSOR DIRECTION * XPLS <-- NEW ITERATE * FPLS <-- FUNCTION VALUE AT NEW ITERATE * MXTAKE <-- BOOLEAN FLAG INDICATING STEP OF MAXIMUM LENGTH USED * IRETCD <-- RETURN CODE * STEPMX --> MAXIMUM ALLOWABLE STEP SIZE * STEPTL --> RELATIVE STEP SIZE AT WHICH SUCCESSIVE ITERATES * CONSIDERED CLOSE ENOUGH TO TERMINATE ALGORITHM * SCALE --> LOGICAL VARIABLE INDICATING WHETHER OR NOT SCALING * OF THE VARIABLES IS PERFORMED * TYPX --> DIAGONAL SCALING MATRIX FOR X * FCN --> NAME OF SUBROUTINE TO EVALUATE OPTIMIZATION FUNCTION * W2 --> WORKING SPACE * * BLAS SUBROUTINES: DDOT,DSCAL * INTEGER N,IRETCD REAL X(N),F,G(N),P(N),XPLS(N),FPLS,STEPMX REAL STEPTL,TYPX(N),W2(N) LOGICAL SCALE * * LOCAL VARIABLES * INTEGER I,K INTEGER IFCNT,IGCNT,IHCNT REAL TMP,SLN,SCL,SLP,RLN REAL TEMP,TEMP1,TEMP2,ALMBMN,ALMBDA REAL TLMBDA,T1,T2,T3,PLMBDA REAL A,B,DISC,PFPLS,ZERO,TENTH REAL HALF,Z99,ONE,TWO,THREE,ALPHA REAL SDOT LOGICAL MXTAKE COMMON /COUNT/IFCNT,IGCNT,IHCNT COMMON /ALMB/ALMBDA EXTERNAL FCN,SSCAL,SDOT INTRINSIC ABS,MAX,SQRT DATA ZERO,TENTH,HALF,Z99,ONE,TWO,THREE,ALPHA/0.0E+0, + 1.0E-1,5.0E-1,0.99E+0,1.0E+0,2.0E+0,3.0E+0, + 1.0E-4/ * MXTAKE = .FALSE. IRETCD = 2 *$ WRITE(IPR,954) *$ WRITE(IPR,955) (P(I),I=1,N) TMP = ZERO DO 10 I = 1,N TMP = TMP + P(I)*P(I) 10 CONTINUE SLN = SQRT(TMP) IF (SLN.LE.STEPMX) GO TO 20 * * STEP LONGER THAN MAXIMUM ALLOWED * SCL = STEPMX/SLN CALL SSCAL(N,SCL,P(1),1) SLN = STEPMX *$ WRITE(IPR,954) *$ WRITE(IPR,955) (P(I),I=1,N) 20 CONTINUE SLP = SDOT(N,G(1),1,P(1),1) RLN = ZERO DO 30 I = 1,N TEMP = ONE TEMP1 = ABS(X(I)) TEMP2 = MAX(TEMP1,TEMP) TEMP1 = ABS(P(I)) RLN = MAX(RLN,TEMP1/TEMP2) 30 CONTINUE ALMBMN = STEPTL/RLN ALMBDA = ONE *$ WRITE(IPR,952) SLN,SLP,RMNLMB,STEPMX,STEPTL * * LOOP * CHECK IF NEW ITERATE SATISFACTORY. GENERATE NEW LAMBDA IF NECESSARY. * 40 CONTINUE IF (IRETCD.LT.2) THEN RETURN END IF DO 50 I = 1,N XPLS(I) = X(I) + ALMBDA*P(I) 50 CONTINUE IFCNT = IFCNT + 1 IF (SCALE) THEN DO 60 K = 1,N W2(K) = XPLS(K)*TYPX(K) 60 CONTINUE CALL FCN(N,W2,FPLS) ELSE CALL FCN(N,XPLS,FPLS) END IF *$ WRITE(IPR,956) ALMBDA *$ WRITE(IPR,951) *$ WRITE(IPR,955) (XPLS(I),I=1,N) *$ WRITE(IPR,953) FPLS IF (FPLS.GT.F+SLP*ALPHA*ALMBDA) GO TO 70 * IF(FPLS.LE. F+SLP*1.E-4*ALMBDA) * THEN * * SOLUTION FOUND * IRETCD = 0 IF (ALMBDA.EQ.TENTH .AND. SLN.GT.Z99*STEPMX) MXTAKE = .TRUE. GO TO 40 * * SOLUTION NOT (YET) FOUND * * ELSE 70 IF (ALMBDA.GE.ALMBMN) GO TO 80 * IF(ALMBDA .LT. ALMBMN) * THEN * * NO SATISFACTORY XPLS FOUND SUFFICIENTLY DISTINCT FROM X * IRETCD = 1 GO TO 40 * ELSE * * CALCULATE NEW LAMBDA * 80 IF (ALMBDA.NE.ONE) GO TO 90 * IF(ALMBDA.EQ.1.0) * THEN * * FIRST BACKTRACK: QUADRATIC FIT * TLMBDA = -SLP/ (TWO* (FPLS-F-SLP)) GO TO 120 * ELSE * * ALL SUBSEQUENT BACKTRACKS: CUBIC FIT * 90 T1 = FPLS - F - ALMBDA*SLP T2 = PFPLS - F - PLMBDA*SLP T3 = ONE/ (ALMBDA-PLMBDA) A = T3* (T1/ (ALMBDA*ALMBDA)-T2/ (PLMBDA*PLMBDA)) B = T3* (T2*ALMBDA/ (PLMBDA*PLMBDA)-T1*PLMBDA/ (ALMBDA*ALMBDA)) DISC = B*B - THREE*A*SLP IF (DISC.LE.B*B) GO TO 100 * IF(DISC.GT. B*B) * THEN * * ONLY ONE POSITIVE CRITICAL POINT, MUST BE MINIMUM * TLMBDA = (-B+SIGN(ONE,A)*SQRT(DISC))/ (THREE*A) GO TO 110 * ELSE * * BOTH CRITICAL POINTS POSITIVE, FIRST IS MINIMUM * 100 TLMBDA = (-B-SIGN(ONE,A)*SQRT(DISC))/ (THREE*A) * ENDIF 110 IF (TLMBDA.GT.HALF*ALMBDA) TLMBDA = HALF*ALMBDA * ENDIF 120 PLMBDA = ALMBDA PFPLS = FPLS IF (TLMBDA.GE.ALMBDA*TENTH) GO TO 130 * IF(TLMBDA.LT.ALMBDA/TENTH) * THEN ALMBDA = ALMBDA*TENTH GO TO 140 * ELSE 130 ALMBDA = TLMBDA * ENDIF * ENDIF * ENDIF 140 GO TO 40 *$951 FORMAT(' STLSCH NEW ITERATE (XPLS)') *$952 FORMAT(' STLSCH SLN =',E20.13/ *$ * ' STLSCH SLP =',E20.13/ *$ * ' STLSCH ALMBMN=',E20.13/ *$ * ' STLSCH STEPMX=',E20.13/ *$ * ' STLSCH STEPTL=',E20.13) *$953 FORMAT(' STLSCH F(XPLS)=',E20.13) *$954 FORMAT(' STLSCH COMPUTED STEP (P)') *$955 FORMAT(' STLSCH ',5(E20.13,3X)) *$956 FORMAT(' STLSCH ALMBDA=',E20.13) END SUBROUTINE STMA27(N,NZ,IRN,ICN,A,MAXA,IW,MAXIW,RHS,W,ISLV,NEWTON, + EPS,IFLAG) * * PURPOSE: * -------- * * THIS ROUTINE SOLVES A SPARSE SYMMETRIC SYSTEM OF LINEAR EQUATIONS: * A * X = RHS. * * PARAMETERS: * ---------- * * N --> DIMENSION OF MATRIX A * NZ --> NUMBER OF NONZEROS IN THE LOWER HALF OF MATRIX A * IRN --> ROW INDEX OF A. IRN IS UNALTERED BY MA27 * ICN --> COLUMN INDEX OF A. ICN IS UNALTERED BY MA27 * A <-> A SYMMETRIC MATRIX OF DIMENSION N. ON EXIT, IT HOLDS * THE FACTORS L AND D OF THE L D L-TRANS FACTORIZATION * MAXA --> MAXIMUM LENGTH OF MATRIX A * IW <-- WORKSPACE OF DIMENSION MAXIW. IT IS USED TO * HOLD INTEGER INFORMATION ON THE FACTORS L AND D * RHS <-> RIGHT HAND SIDE VECTOR OF THE EQUATIONS BEING SOLVED. * ON EXIT, IT HOLDS THE SOLUTION VECTOR * W --> WORKSPACE OF DIMENSION N * ISLV --> FLAG WITH THE FOLLOWING MEANINGS: * ISLV = 0 : ANALAYSE AND FACTORIZE MATRIX A * ISLV = 1 : ANALAYSE, FACTORIZE, AND SOLVE SYSTEMS OF * EQUATIONS * ISLV = 2 : PERFORM A SOLVE ONLY (A HAS ALREADY BEEN * FACTORIZED PREVIOUSLY) * ISLV = 3 : CHANGE THE NEGATIVE SIGNS OF THE EIGENVALUES * TO PLUS IF ANY * NEWTON --> LOGICAL PARAMETER. IT IS SET TO TRUE IF NEWTON'S METHOD * IS USED AND TO FALSE IF THE TENSOR METHOD IS USED * EPS --> MACHINE PRECISION * IFLAG <-- A VALUE OF ZERO INDICATES THAT THE SUBROUTINE HAS * PERFORMED SUCCESSFULLY * INTEGER N,NZ,MAXA,MAXIW,ISLV,IFLAG INTEGER IRN(NZ),ICN(MAXA),IW(MAXIW) REAL A(MAXA),RHS(N),W(N),EPS LOGICAL NEWTON * * LOCAL VARIABLES * INTEGER NEG1,NEG2,LDIAG,LP,MP,ISTIW,LIW,NSTEPS,MAXFRT REAL U,TENTH COMMON /MA27DD/U,LP,MP,LDIAG DATA TENTH/0.1E+0/ * IF (ISLV.EQ.2) GO TO 20 IF (ISLV.EQ.3) GO TO 10 U = TENTH LDIAG = 0 * * SET UP SUBDIVISION OF ARRAYS * ISTIW = 5*N + 1 LIW = MAXIW - 5*N IFLAG = 0 * * ANALYSE SPARSITY PATTERN * CALL MA27A(N,NZ,IRN,ICN,IW(ISTIW),LIW,IW,IW(3*N+1),NSTEPS,IFLAG) * * FACTORIZE MATRIX * CALL MA27B(N,NZ,IRN,ICN,A,MAXA,IW(ISTIW),LIW,IW,NSTEPS,MAXFRT, + IW(3*N+1),IFLAG) * * CHANGE THE NEGATIVE SIGNS OF THE EIGENVALUES TO PLUS * 10 CONTINUE IF (NEWTON .OR. IFLAG.EQ.3) THEN CALL DSYPRC(MAXA,LIW,A,IW(ISTIW),EPS,NEG1,NEG2) END IF * IF (ISLV.EQ.3 .AND. (NEG1.EQ.0.AND.NEG2.EQ.0)) THEN IFLAG = 30 GO TO 30 END IF * 20 CONTINUE IF (ISLV.EQ.0) GO TO 30 * * SOLVE THE EQUATIONS * CALL MA27C(N,A,MAXA,IW(ISTIW),LIW,W,MAXFRT,RHS,IW(3*N+1),NSTEPS) 30 RETURN END SUBROUTINE STMSLV(N,DT,S,G,W1,W2,W3,BV,GAMMA,NZ,IRN,ICN,HESS,LICN, + IWA,LIWA,WK,EPS,DSCENT,IFLAG) * * PURPOSE: * ------- * * THIS ROUTINE COMPUTES THE TENSOR STEP. IT ALSO COMPUTES THE * NEWTON STEP IF THE TENSOR STEP IS NOT DESCENT. * * PARAMETERS: * ---------- * * N --> DIMENSION OF PROBLEM * DT <-- TENSOR STEP ON EXIT * S --> STEP TO PREVIOUS POINT * G --> GRADIENT AT CURRENT ITERATE * W1 --> WORKSPACE ARRAY OF DIMENSION N. IT IS USED TO HOLD * THE SOLUTION OF HESS W1 = G * W2 --> WORKSPACE ARRAY OF DIMENSION N. IT IS USED TO HOLD * THE SOLUTION OF HESS W2 = S * W3 <-- NEWTON STEP (IF REQUIRED) * BV --> (3*(S-TRANS*S)*D-2*(S-TRANS*D)*S)/(3*(S-TRANS*S)**3) * (SEE D IN SUBROUTINE STFTSM) * GAMMA --> BETA/(S-TRANS*S)**4 (SEE BETA IN SUBROUTINE STFTSM) * NZ --> NUMBER OF NONZEROS IN THE LOWER TRIANGULAR PART * OF THE HESSIAN MATRIX * IRN --> ROW INDEX OF THE NONZEROS STORED IN THE LOWER * HALF OF THE HESSIAN MATRIX * ICN --> COLUMN INDEX OF THE NONZEROS STORED IN THE LOWER * HALF OF THE HESSIAN MATRIX * HESS --> HESSIAN MATRIX * LICN --> LENGTH OF ARRAY HESS * IWA --> WORKSPACE * LIWA --> LENGTH OF ARRAY IWA * WK --> WORKSPACE OF DIMENSION N * EPS --> MACHINE PRECISION * DSCENT <-- LOGICAL VARIABLE INDICATING WHETHER OR NOT THE TENSOR * STEP IS DESCENT * IFLAG <-- OUTPUT PARAMETER FROM THE MA27 PACKAGE. A VALUE OF * ZERO INDICATES THAT THE FACTORIZATION WAS PERFORMED * SUCCESSFULLY * * BLAS SUBROUTINES: DCOPY,DDOT * INTEGER N,NZ,LICN,LIWA,IFLAG INTEGER IRN(NZ),ICN(LICN),IWA(6*N) REAL DT(N),S(N),G(N),W1(N),W2(N),W3(N) REAL BV(N),GAMMA,HESS(LICN),WK(N),EPS LOGICAL DSCENT * * LOCAL VARIABLES * INTEGER NZEROS,I REAL BETA(3),THETA(3),UU,WW,VV,YY,ZZ REAL AA,A1,A2,A3,AMINB,AMINT,GD REAL SDOT,ZERO,HALF,ONE,OHALF,TWO,SIX DATA ZERO,HALF,ONE,OHALF,TWO,SIX/0.0E0,0.5E+0,1.0E+0, + 1.5E+0,2.0E+0,6.0E+0/ * DSCENT = .TRUE. CALL STMA27(N,NZ,IRN,ICN,HESS,LICN,IWA,LIWA,W1,WK,0,.FALSE.,EPS, + IFLAG) * * SOLVE HESS W1 = G FOR W1 * CALL SCOPY(N,G(1),1,W1(1),1) CALL STMA27(N,NZ,IRN,ICN,HESS,LICN,IWA,LIWA,W1,WK,2,.FALSE.,EPS, + IFLAG) * * SOLVE HESS W2 = S FOR W2 * CALL SCOPY(N,S(1),1,W2(1),1) CALL STMA27(N,NZ,IRN,ICN,HESS,LICN,IWA,LIWA,W2,WK,2,.FALSE.,EPS, + IFLAG) * * SOLVE HESS W3 = B FOR W3 * CALL SCOPY(N,BV(1),1,W3(1),1) CALL STMA27(N,NZ,IRN,ICN,HESS,LICN,IWA,LIWA,W3,WK,2,.FALSE.,EPS, + IFLAG) * * COMPUTE COEFFICIENTS OF THE SYSTEM OF TWO EQUATIONS * OF THE 3RD ORDER IN 2 UNKNOWS * UU = SDOT(N,S(1),1,W1(1),1) WW = SDOT(N,S(1),1,W2(1),1) VV = SDOT(N,S(1),1,W3(1),1) YY = SDOT(N,BV(1),1,W1(1),1) ZZ = SDOT(N,BV(1),1,W3(1),1) * * CALCULATE COEFFICIENTS OF THIRD DEGREE POLYNOMIAL * AA = HALF*WW*ZZ - (ONE/SIX)*GAMMA*WW - HALF*VV**2 A1 = -OHALF*VV/AA A2 = (YY*WW-UU*VV-ONE)/AA A3 = -UU/AA * * COMPUTE ZEROS OF THIRD DEGREE POLYNOMIAL * CALL STCZ3P(A1,A2,A3,BETA,NZEROS) * DO 10 I = 1,NZEROS THETA(I) = - (UU+BETA(I)+HALF*VV*BETA(I)**2+ + (ONE/SIX)*GAMMA*WW*BETA(I)**3)/ (WW*BETA(I)) 10 CONTINUE * * COMPUTE SMALLEST BETA IN ABSOLUTE VALUE * AMINB = BETA(1) AMINT = THETA(1) IF (NZEROS.EQ.2) THEN IF (AMINB.GT.ABS(BETA(2))) THEN AMINB = BETA(2) AMINT = THETA(2) END IF ELSE IF (NZEROS.EQ.3) THEN IF (AMINB.GT.ABS(BETA(2))) THEN AMINB = BETA(2) AMINT = THETA(2) END IF IF (AMINB.GT.ABS(BETA(3))) THEN AMINB = BETA(3) AMINT = THETA(3) END IF END IF * * COMPUTE TENSOR STEP * DO 20 I = 1,N DT(I) = -W1(I) - (ONE/TWO)*W3(I)*AMINB**2 - + W2(I)*AMINB*AMINT - (ONE/SIX)*W2(I)*GAMMA*AMINB**3 20 CONTINUE * * IF TENSOR STEP IS NOT A DESCENT DIRECTION THEN COMPUTE NEWTON STEP * GD = SDOT(N,G(1),1,DT(1),1) * IF (GD.GT.ZERO) THEN * * COMPUTE THE NEWTON STEP * DSCENT = .FALSE. IF (IFLAG.EQ.3) THEN DO 30 I = 1,N W3(I) = -W1(I) 30 CONTINUE GO TO 60 END IF CALL SCOPY(N,G(1),1,W3(1),1) CALL STMA27(N,NZ,IRN,ICN,HESS,LICN,IWA,LIWA,W3,WK,3,.TRUE., + EPS,IFLAG) IF (IFLAG.EQ.30) THEN DO 40 I = 1,N W3(I) = -W1(I) 40 CONTINUE ELSE DO 50 I = 1,N W3(I) = -W3(I) 50 CONTINUE END IF END IF * 60 CONTINUE RETURN END SUBROUTINE STRSLT(N,XPLS,FVAL,GPLS,RGX,ITN,TERMCD,IPR) * * PURPOSE: * ------- * * THIS ROUTINE PRINTS INFORMATION. * * PARAMETERS: * ---------- * * N ---> DIMENSION OF PROBLEM * XPLS ---> ITERATE TO BE PRINTED OUT * FVAL ---> FUNCTION VALUE AT XPLS * GPLS ---> GRADIENT AT XPLS * RGX ---> MAXIMUM RELATIVE GRADIENT * ITN ---> ITERATION NUMBER * IPR ---> DEVICE TO WHICH TO SEND OUTPUT * * INTEGER N,ITN,TERMCD,IPR REAL XPLS(N),FVAL,GPLS(N),RGX * * LOCAL VARIABLES * INTEGER I,IFCNT,IGCNT,IHCNT REAL D,RELGRD,ZERO,ONE COMMON /COUNT/IFCNT,IGCNT,IHCNT DATA ZERO,ONE/0.0E0,1.0E0/ * * FIND DIRECTION IN WHICH RELATIVE GRADIENT MAXIMUM * CHECK WHETHER WITHIN TOLERANCE * IF (ITN.EQ.0) THEN D = ONE RGX = ZERO DO 10 I = 1,N RELGRD = ABS(GPLS(I))*MAX(ABS(XPLS(I)),ONE)/D RGX = MAX(RGX,RELGRD) 10 CONTINUE END IF WRITE (IPR,9000) WRITE (IPR,9010) WRITE (IPR,9020) ITN WRITE (IPR,9030) WRITE (IPR,9040) FVAL WRITE (IPR,9050) WRITE (IPR,9060) RGX WRITE (IPR,9010) WRITE (IPR,9000) IF (TERMCD.GT.0) THEN WRITE (IPR,9070) IFCNT,IGCNT,IHCNT END IF * * RETURN 9000 FORMAT (' ') 9010 FORMAT ('---------------------------------------------') 9020 FORMAT (' STRSLT ITERATION K =',I5) 9030 FORMAT (' STRSLT FUNCTION AT X(K)') 9040 FORMAT (' STRSLT ',E20.13) 9050 FORMAT (' STRSLT SCALED GRADIENT AT X(K)') 9060 FORMAT (' STRSLT ',E20.13) 9070 FORMAT (' STRSLT NUMBER OF FUNCTION EVALUATIONS',I5/, + ' STRSLT NUMBER OF GRADIENT EVALUATIONS',I5/, + ' STRSLT NUMBER OF HESSIAN EVALUATIONS',I5/) END SUBROUTINE STSORT(N,NNZ,A,INDROW,INDCOL,JPNTR,IWA) INTEGER N,NNZ INTEGER INDROW(NNZ),INDCOL(NNZ),JPNTR(N+1),IWA(N) REAL A(NNZ) * ********** * * SUBROUTINE STSORT * * GIVEN THE NONZERO ELEMENTS OF AN M BY N MATRIX A IN * ARBITRARY ORDER AS SPECIFIED BY THEIR ROW AND COLUMN * INDICES, THIS SUBROUTINE PERMUTES THESE ELEMENTS SO * THAT THEIR COLUMN INDICES ARE IN NON-DECREASING ORDER. * IT ALSO PERMUTES THE CORRESPONDING A ELEMENTS. * * ON INPUT IT IS ASSUMED THAT THE ELEMENTS ARE SPECIFIED IN * * INDROW(K),INDCOL(K), A(K), K = 1,...,NNZ. * * ON OUTPUT THE ELEMENTS ARE PERMUTED SO THAT INDCOL IS * IN NON-DECREASING ORDER. IN ADDITION, THE ARRAY JPNTR * IS SET SO THAT THE ROW INDICES FOR COLUMN J ARE * * INDROW(K), K = JPNTR(J),...,JPNTR(J+1)-1. * * THE SUBROUTINE STATEMENT IS * * SUBROUTINE STSORT(N,NNZ,A,INDROW,INDCOL,JPNTR,IWA) * * WHERE * * N IS A POSITIVE INTEGER INPUT VARIABLE SET TO THE NUMBER * OF COLUMNS OF A. * * NNZ IS A POSITIVE INTEGER INPUT VARIABLE SET TO THE NUMBER * OF NONZERO ELEMENTS OF A. * * A IS AN ARRAY OF LENGTH NNZ. ON INPUT A CONTAINS THE * THE NONZEROS ELEMENTS. ON OUTPUT A IS PERMUTED SO THAT * THE CORRESPONDING COLUMN INDICES OF INDCOL ARE IN * NON-DECREASING ORDER. * * INDROW IS AN INTEGER ARRAY OF LENGTH NNZ. ON INPUT INDROW * MUST CONTAIN THE ROW INDICES OF THE NONZERO ELEMENTS OF A. * ON OUTPUT INDROW IS PERMUTED SO THAT THE CORRESPONDING * COLUMN INDICES OF INDCOL ARE IN NON-DECREASING ORDER. * * INDCOL IS AN INTEGER ARRAY OF LENGTH NNZ. ON INPUT INDCOL * MUST CONTAIN THE COLUMN INDICES OF THE NONZERO ELEMENTS * OF A. ON OUTPUT INDCOL IS PERMUTED SO THAT THESE INDICES * ARE IN NON-DECREASING ORDER. * * JPNTR IS AN INTEGER OUTPUT ARRAY OF LENGTH N + 1 WHICH * SPECIFIES THE LOCATIONS OF THE ROW INDICES IN THE OUTPUT * INDROW. THE ROW INDICES FOR COLUMN J ARE * * INDROW(K), K = JPNTR(J),...,JPNTR(J+1)-1. * * NOTE THAT JPNTR(1) IS SET TO 1 AND THAT JPNTR(N+1)-1 * IS THEN NNZ. * * IWA IS AN INTEGER WORK ARRAY OF LENGTH N. * * SUBPROGRAMS CALLED * * FORTRAN-SUPPLIED ... MAX * * ARGONNE NATIONAL LABORATORY. MINPACK PROJECT. JULY 1983. * THOMAS F. COLEMAN, BURTON S. GARBOW, JORGE J. MORE' * MODIFIED BY ALI BOUARICHA, OCTOBER 1994. * * ********** INTEGER I,J,K,L REAL S * * STORE IN ARRAY IWA THE COUNTS OF NONZEROES IN THE COLUMNS. * DO 10 J = 1,N IWA(J) = 0 10 CONTINUE DO 20 K = 1,NNZ IWA(INDCOL(K)) = IWA(INDCOL(K)) + 1 20 CONTINUE * * SET POINTERS TO THE START OF THE COLUMNS IN INDROW. * JPNTR(1) = 1 DO 30 J = 1,N JPNTR(J+1) = JPNTR(J) + IWA(J) IWA(J) = JPNTR(J) 30 CONTINUE K = 1 * * BEGIN IN-PLACE SORT. * 40 CONTINUE J = INDCOL(K) IF (K.GE.JPNTR(J)) THEN * * CURRENT ELEMENT IS IN POSITION. NOW EXAMINE THE * NEXT ELEMENT OR THE FIRST UN-SORTED ELEMENT IN * THE J-TH GROUP. * K = MAX(K+1,IWA(J)) ELSE * * CURRENT ELEMENT IS NOT IN POSITION. PLACE ELEMENT * IN POSITION AND MAKE THE DISPLACED ELEMENT THE * CURRENT ELEMENT. * L = IWA(J) IWA(J) = IWA(J) + 1 I = INDROW(K) S = A(K) INDROW(K) = INDROW(L) INDCOL(K) = INDCOL(L) A(K) = A(L) INDROW(L) = I INDCOL(L) = J A(L) = S END IF IF (K.LE.NNZ) GO TO 40 RETURN * * LAST CARD OF SUBROUTINE SRTDAT. * END SUBROUTINE STUMCS(N,X,NPAIRS,IRN,LIRN,ICN,LICN,FCN,UGR,USH,TYPX, + FSCALE,GRADTL,STEPTL,ILIM,STEPMX,IPR,METHOD, + GRDFLG,HSNFLG,NDIGIT,MSG,XPLS,FPLS,GPLS,HESS, + WRK,LWRK,IWRK,LIWRK,TERMCD,VECTOR,INFORM) * * PURPOSE: * ------- * * THIS ROUTINE PROVIDES A COMPLETE INTERFACE TO STENMIN. THE USER * HAS FULL CONTROL OVER THE OPTIONS. * * PARAMETERS: * ---------- * * N --> A POSITIVE INTEGER VARIABLE SPECIFYING THE NUMBER * OF VARIABLES IN THE PROBLEM * X --> AN ARRAY OF LENGTH N THAT CONTAINS AN INITIAL * ESTIMATE OF THE MINIMIZER * NPAIRS --> AN INTEGER VARIABLE THAT MUST BE SET BY THE USER TO * THE NUMBER OF NONZEROS STORED IN THE LOWER OR UPPER * HALF OF THE HESSIAN MATRIX * IRN --> AN INTEGER ARRAY OF LENGTH LIRN. ON ENTRY, IT MUST * HOLD THE ROW INDEX OF EACH NONZERO STORED IN THE * LOWER OR UPPER HALF OF THE HESSIAN MATRIX * LIRN --> AN INTEGER VARIABLE THAT MUST BE SET BY THE USER TO * THE LENGTH OF ARRAY IRN. LIRN NEED NOT BE AS LARGE * AS LICN; NORMALLY IT NEED NOT BE VERY MUCH GREATER * THAN NPAIRS * ICN --> AN INTEGER ARRAY OF LENGTH LICN. ON ENTRY, IT MUST * HOLD THE COLUMN INDEX OF THE NONZEROS STORED IN THE * LOWER OR UPPER HALF OF THE HESSIAN MATRIX * LICN --> AN INTEGER VARIABLE THAT MUST BE SET BY THE USER TO * THE LENGTH OF THE HESSIAN ARRAY HESS AND ICN. LICN * SHOULD ORDINARILY BE 2 TO 4 TIMES AS LARGE AS NPAIRS * FCN --> THE NAME OF A USER SUPPLIED SUBROUTINE THAT EVALUATES * THE FUNCTION F AT AN ARBITRARY VECTOR X. THE SUBROUTINE * MUST BE DECLARED EXTERNAL IN THE USER'S CALLING PROGRAM * AND MUST CONFORM TO * CALL FCN(N, X, F) * UGR --> THE NAME OF A USER SUPPLIED SUBROUTINE THAT RETURNS IN * THE VALUE OF THE GRADIENT. UGR MUST BE DECLARED EXTERNAL * IN THE USER'S CALLING PROGRAM AND MUST CONFORM TO THE * USAGE * CALL UGR(N, X, G) * IF NO ANALYTIC GRADIENT IS SUPPLIED (GRDFLG = 0), THE * USER MUST USE THE DUMMY NAME STDUGR * USH --> THE NAME OF A USER SUPPLIED SUBROUTINE THAT RETURNS * IN HESS THE VALUE OF THE HESSIAN AT THE CURRENT POINT X. * USH MUST BE DECLARED EXTERNAL IN THE USER'S CALLING * PROGRAM AND MUST CONFORM TO THE USAGE * CALL USH(N,X,NPAIRS,LICN,HESS,IRN,ICN) * ONLY THE LOWER OR UPPER TRIANGULAR PART AND THE DIAGONAL * OF HESS SHOULD BE GIVEN, WITH THEIR CORRESPONDING ROW * AND COLUMN INDICES. IF NO ANALYTIC GRADIENT IS SUPPLIED * (HSNFLG = 0), THE USER MUST USE THE DUMMY NAME STDUSH * TYPX --> AN ARRAY OF LENGTH N IN WHICH THE TYPICAL SIZE OF THE * COMPONENTS OF X ARE SPECIFIED * FSCALE --> A POSITIVE REAL NUMBER ESTIMATING THE MAGNITUDE * OF F(X) NEAR THE MINIMIZER * GRADTL --> POSITIVE SCALAR GIVING THE TOLERANCE AT WHICH THE * SCALED GRADIENT OF F(X) IS CONSIDERED CLOSE ENOUGH TO * ZERO TO TERMINATE THE ALGORITHM * STEPTL --> A POSITIVE SCALAR PROVIDING THE MINIMUM ALLOWABLE * RELATIVE STEP LENGTH * ILIM --> POSITIVE INTEGER SPECIFYING THE MAXIMUM ITERATIONS TO BE * PERFORMED BEFORE THE PROGRAM IS TERMINATED * STEPMX --> A POSITIVE SCALAR PROVIDING THE MAXIMUM ALLOWABLE SCALED * STEP LENGTH * IPR --> THE UNIT ON WHICH THE ROUTINE OUTPUTS INFORMATION * METHOD --> AN INTEGER FLAG DESIGNATING WHICH METHOD TO USE * METHOD = 0 : USE NEWTON'S METHOD * METHOD = 1 : USE THE TENSOR METHOD * GRDFLG --> INTEGER FLAG DESIGNATING WHETHER OR NOT ANALYTIC HESSIAN * HAS BEEN SUPPLIED BY THE USER * GRDFLG = 0 : NO ANALYTIC GRADIENT SUPPLIED * GRDFLG = 1 : ANALYTIC GRADIENT SUPPLIED (WILL BE CHECKED * AGAINST FINITE DIFFERENCE GRADIENT) * GRDFLG = 2 : ANALYTIC GRADIENT SUPPLIED (WILL NOT BE * CHECKED AGAINST FINITE DIFFERENCE GRADIENT) * HSNFLG --> INTEGER FLAG DESIGNATING WHETHER OR NOT ANALYTIC HESSIAN * HAS BEEN SUPPLIED BY THE USER * HSNFLG = 0 : NO ANALYTIC HESSIAN SUPPLIED * HSNFLG = 1 : ANALYTIC HESSIAN SUPPLIED (WILL BE CHECKED * AGAINST FINITE DIFFERENCE HESSIAN) * HSNFLG = 2 : ANALYTIC HESSIAN SUPPLIED (WILL NOT BE * CHECKED AGAINST FINITE DIFFERENCE HESSIAN) * NDIGIT --> INTEGER ESTIMATING THE NUMBER OF ACCURATE DIGITS IN THE * OBJECTIVE FUNCTION F(X) * MSG <--> AN INTEGER VARIABLE THAT THE USER MAY SET ON INPUT TO * INHIBIT CERTAIN AUTOMATIC CHECKS OR OVERRIDE CERTAIN * DEFAULT CHARACTERISTICS OF THE PACKAGE * XPLS <-- AN ARRAY OF LENGTH N CONTAINING THE BEST APPROXIMATION * TO THE MINIMIZER UPON RETURN. (IF THE ALGORITHM HAS NOT * CONVERGED, THE LAST ITERATE IS RETURNED) * FPLS <-- A SCALAR VARIABLE THAT CONTAINS THE FUNCTION VALUE AT * THE FINAL ITERATE XPLS * GPLS <-- AN ARRAY OF LENGTH N CONTAINING THE GRADIENT VALUE * AT XPLS * HESS <-- AN ARRAY THAT IS USED TO STORE THE HESSIAN MATRIX AT * EACH ITERATION. IT NEEDS TO BE AT LEAST OF DIMENSION * LICN. ON EXIT, HESS CONTAINS THE HESSIAN VALUE AT THE * MINIMIZER * WRK --> AN ARRAY OF LENGTH LWRK. THIS IS USED AS WORKSPACE * BY THE PACKAGE. ITS LENGTH MUST BE AT LEAST 7*N * LWRK --> AN INTEGER VARIABLE. IT MUST BE SET BY THE USER TO THE * LENGTH OF ARRAY WRK AND IS NOT ALTERED BY THE PACKAGE * IWRK --> AN INTEGER ARRAY OF LENGTH LIWRK. THIS IS USED AS * WORKSPACE BY THE PACKAGE. ITS LENGTH MUST BE AT LEAST * 2*NPAIRS+12*N+2 * LIWRK --> AN INTEGER VARIABLE. IT MUST BE SET BY THE USER TO THE * LENGTH OF ARRAY IWRK AND IS NOT ALTERED BY THE PACKAGE * TERMCD <-- AN INTEGER THAT SPECIFIES THE REASON WHY THE ALGORITHM * HAS TERMINATED * VECTOR<--> AN ARRAY OF LENGTH N. IT NEED NOT BE SET BY THE USER * ON ENTRY. IF INFORM IS SET TO 1, A RE-ENTRY MUST BE * MADE WITH VECTOR SET TO HESS TIMES VECTOR (SEE INFORM) * INFORM<--> AN INTEGER VARIABLE. IF IT IS SET TO 1, THE USER MUST * OBTAIN HESS TIMES VECTOR AND RE-ENTER WITH INFORM * UNCHANGED. THE RESULT OF HESS TIMES VECTOR MUST BE * STORED IN VECTOR. THE DEFAULT VALUE OF INFORM IS 0, * MEANING THAT HESS TIMES VECTOR IS COMPUTED BY THE * PACKAGE * INTEGER N,NPAIRS,LIRN,LICN,ILIM,IPR,METHOD INTEGER GRDFLG,HSNFLG,NDIGIT,MSG,LWRK,LIWRK,TERMCD INTEGER INFORM INTEGER IRN(LIRN),ICN(LICN),IWRK(LIWRK) REAL X(N),TYPX(N),FSCALE,GRADTL,STEPTL,STEPMX REAL XPLS(N),FPLS,GPLS(N),HESS(LICN),WRK(LWRK) REAL VECTOR(N) * EXTERNAL FCN,UGR,USH * * LOCAL VARIABLES * * NONE * * * EQUIVALENCE FOR DOUBLE PRECISION/REAL ARRAYS * * EQUIVALENCE WRK(1) = G(N) * WRK(N+1) = S(N) * WRK(2*N+1) = D(N) * WRK(3*N+1) = DN(N) * WRK(4*N+1) = E(N) * WRK(5*N+1) = XD(N) * WRK(6*N+1) = BV(N) * * EQUIVALENCE FOR INTEGER ARRAYS * * EQUIVALENCE IWRK(1) = LISTP(N) * IWRK(N+1) = NGRP(N) * IWRK(2*N+1) = IPNTR(N+1) * IWRK(3*N+2) = JPNTR(N+1) * IWRK(4*N+3) = IWA(2*NZ+8*N) * CALL STDRUO(N,X,NPAIRS,IRN,LIRN,ICN,LICN,FCN,UGR,USH,TYPX,FSCALE, + GRADTL,STEPTL,ILIM,STEPMX,IPR,METHOD,GRDFLG,HSNFLG, + NDIGIT,MSG,XPLS,FPLS,GPLS,HESS,WRK(1),WRK(N+1), + WRK(2*N+1),WRK(3*N+1),WRK(4*N+1),WRK(5*N+1), + WRK(6*N+1),IWRK(1),IWRK(N+1),IWRK(2*N+1),IWRK(3*N+2), + IWRK(4*N+3),LIWRK,LWRK,TERMCD,VECTOR,INFORM) RETURN END SUBROUTINE STUMSS(N,X,NPAIRS,IRN,LIRN,ICN,LICN,FCN,TYPX,MSG,XPLS, + FPLS,GPLS,HESS,WRK,LWRK,IWRK,LIWRK,TERMCD) * * PURPOSE: * ------- * * THIS ROUTINE PROVIDES A SIMPLE INTERFACE TO STENMIN. THE USER * HAS NO CONTROL OVER THE OPTIONS. * * PARAMETERS: * ---------- * * N --> A POSITIVE INTEGER VARIABLE SPECIFYING THE NUMBER OF * VARIABLES IN THE PROBLEM * X --> AN ARRAY OF LENGTH N THAT CONTAINS AN INITIAL * ESTIMATE OF THE MINIMIZER * NPAIRS --> AN INTEGER VARIABLE THAT MUST BE SET BY THE USER TO * THE NUMBER OF NONZEROS STORED IN THE LOWER OR UPPER * HALF OF THE HESSIAN MATRIX * IRN --> AN INTEGER ARRAY OF LENGTH LIRN. ON ENTRY, IT MUST * HOLD THE ROW INDEX OF EACH NONZERO STORED IN THE LOWER * OR UPPER HALF OF THE HESSIAN MATRIX * LIRN --> AN INTEGER VARIABLE THAT MUST BE SET BY THE USER TO * THE LENGTH OF ARRAY IRN. LIRN NEED NOT BE AS LARGE * AS LICN; NORMALLY IT NEED NOT BE VERY MUCH GREATER * THAN NPAIRS * ICN --> AN INTEGER ARRAY OF LENGTH LICN. ON ENTRY, IT MUST * HOLD THE COLUMN INDEX OF THE NONZEROS STORED IN THE * LOWER OR UPPER HALF OF THE HESSIAN MATRIX. * LICN --> AN INTEGER VARIABLE THAT MUST BE SET BY THE USER TO THE * LENGTH OF THE HESSIAN ARRAY HESS AND ICN. LICN SHOULD * ORDINARILY BE 2 TO 4 TIMES AS LARGE AS NPAIRS * FCN --> THE NAME OF A USER SUPPLIED SUBROUTINE THAT EVALUATES * THE FUNCTION F AT AN ARBITRARY VECTOR X. THE SUBROUTINE * MUST BE DECLARED EXTERNAL IN THE USER'S CALLING PROGRAM * AND MUST CONFORM TO * CALL FCN(N, X, F) * TYPX --> AN ARRAY OF LENGTH N IN WHICH THE TYPICAL SIZE OF THE * COMPONENTS OF X ARE SPECIFIED. IT NEED NOT BE SET ON * ENTRY * MSG <--> AN INTEGER VARIABLE THAT THE USER MAY SET ON INPUT TO * INHIBIT CERTAIN AUTOMATIC CHECKS OR OVERRIDE CERTAIN * DEFAULT CHARACTERISTICS OF THE PACKAGE * XPLS <-- AN ARRAY OF LENGTH N CONTAINING THE BEST APPROXIMATION * TO THE MINIMIZER UPON RETURN. (IF THE ALGORITHM HAS NOT * CONVERGED, THE LAST ITERATE IS RETURNED) * FPLS <-- A SCALAR VARIABLE THAT CONTAINS THE FUNCTION VALUE AT * THE FINAL ITERATE XPLS * GPLS <-- AN ARRAY OF LENGTH N CONTAINING THE GRADIENT VALUE * AT XPLS * HESS <-- AN ARRAY THAT IS USED TO STORE THE HESSIAN MATRIX AT * EACH ITERATION. IT NEEDS TO BE AT LEAST OF DIMENSION * LICN. ON EXIT, HESS CONTAINS THE HESSIAN VALUE AT THE * MINIMIZER * WRK --> AN ARRAY OF LENGTH LWRK. THIS IS USED AS WORKSPACE * BY THE PACKAGE. ITS LENGTH MUST BE AT LEAST 8*N * LWRK --> AN INTEGER VARIABLE. IT MUST BE SET BY THE USER TO THE * LENGTH OF ARRAY WRK AND IS NOT ALTERED BY THE PACKAGE * IWRK --> AN INTEGER ARRAY OF LENGTH LIWRK. THIS IS USED AS * WORKSPACE BY THE PACKAGE. ITS LENGTH MUST BE AT LEAST * 2*NPAIRS+12*N+2 * LIWRK --> AN INTEGER VARIABLE. IT MUST BE SET BY THE USER TO THE * LENGTH OF ARRAY IWRK AND IS NOT ALTERED BY THE PACKAGE * TERMCD <-- AN INTEGER THAT SPECIFIES THE REASON WHY THE ALGORITHM * HAS TERMINATED * * INTEGER N,NPAIRS,LIRN,LICN INTEGER MSG,LWRK,LIWRK,TERMCD INTEGER IRN(LIRN),ICN(LICN),IWRK(LIWRK) REAL X(N),TYPX(N),XPLS(N),FPLS,GPLS(N) REAL HESS(LICN),WRK(LWRK) * * LOCAL VARIABLES * INTEGER ILIM,IPR,METHOD,GRDFLG,HSNFLG,NDIGIT,INFORM REAL FSCALE,GRADTL,STEPTL,STEPMX * EXTERNAL FCN,STDUGR,STDUSH * * EQUIVALENCE FOR DOUBLE PRECISION/REAL ARRAYS * * EQUIVALENCE WRK(1) = G(N) * WRK(N+1) = S(N) * WRK(2*N+1) = D(N) * WRK(3*N+1) = DN(N) * WRK(4*N+1) = E(N) * WRK(5*N+1) = XD(N) * WRK(6*N+1) = BV(N) * WRK(7*N+1) = VECTOR(N) * * EQUIVALENCE FOR INTEGER ARRAYS * * EQUIVALENCE IWRK(1) = LISTP(N) * IWRK(N+1) = NGRP(N) * IWRK(2*N+1) = IPNTR(N+1) * IWRK(3*N+2) = JPNTR(N+1) * IWRK(4*N+3) = IWA(2*NZ+8*N) * CALL STDFLT(N,TYPX,FSCALE,GRADTL,STEPTL,ILIM,STEPMX,IPR,METHOD, + GRDFLG,HSNFLG,NDIGIT,INFORM,MSG) * CALL STDRUO(N,X,NPAIRS,IRN,LIRN,ICN,LICN,FCN,STDUGR,STDUSH,TYPX, + FSCALE,GRADTL,STEPTL,ILIM,STEPMX,IPR,METHOD,GRDFLG, + HSNFLG,NDIGIT,MSG,XPLS,FPLS,GPLS,HESS,WRK(1),WRK(N+1), + WRK(2*N+1),WRK(3*N+1),WRK(4*N+1),WRK(5*N+1), + WRK(6*N+1),IWRK(1),IWRK(N+1),IWRK(2*N+1),IWRK(3*N+2), + IWRK(4*N+3),LIWRK,LWRK,TERMCD,WRK(7*N+1),INFORM) RETURN END SUBROUTINE DSYPRC(LA,LIW,A,IW,EPSMCH,NEG1,NEG2) INTEGER LA,LIW,NEG1,NEG2 INTEGER IW(LIW) REAL A(LA),EPSMCH * * THE GILL-MURRAY-PONCELEON-SAUNDERS CODE FOR MODIFYING THE NEGATIVE * EIGEN-COMPONENTS OBTAINED WHEN FACTORIZING A SYMMETRIC INDEFINITE * MATRIX USING THE HARWELL CODE MA27. (SEE SOL 90-8, P.19-21) * * NICK GOULD, 20TH JULY 1990. * INTEGER ALEN,APOS,IBLK,NBLK,IPOS,NROWS,NCOLS,J,K REAL ZERO,ONE,TWO,ALPHA,BETA,GAMMA,TAU,T,C,S,E1,E2 LOGICAL SINGLE INTRINSIC ABS,SQRT * * SET DATA. * PARAMETER (ZERO=0.0E+0,ONE=1.0E+0,TWO=2.0E+0) * * NEG1 AND NEG2 ARE THE NUMBER OF NEGATIVE EIGENVALUES WHICH ARISE * FROM NEGATIVE 1x1 AND 2x2 BLOCK PIVOTS. * NEG1 = 0 NEG2 = 0 NBLK = ABS(IW(1)) IPOS = 2 APOS = 1 * * LOOP OVER ALL THE BLOCK PIVOTS. * DO 20 IBLK = 1,NBLK NCOLS = IW(IPOS) IF (NCOLS.LT.0) THEN NROWS = 1 NCOLS = -NCOLS ELSE IPOS = IPOS + 1 NROWS = IW(IPOS) END IF * * PROCESS THE DIAGONALS IN THIS BLOCK. * ALEN = NCOLS SINGLE = .TRUE. DO 10 K = IPOS + 1,IPOS + NROWS IF (SINGLE) THEN ALPHA = A(APOS) J = IW(K) SINGLE = J .GT. 0 IF (SINGLE) THEN * * NEGATIVE 1x1 BLOCK. * IF (ALPHA.LT.ZERO) THEN NEG1 = NEG1 + 1 A(APOS) = -ALPHA ELSE IF (ALPHA.GT.ONE/EPSMCH) THEN NEG1 = NEG1 + 1 A(APOS) = ONE/EPSMCH END IF END IF ELSE BETA = A(APOS+1) GAMMA = A(APOS+ALEN) * * 2x2 BLOCK: ( ALPHA BETA ) = ( C S ) ( E1 ) ( C S ) * ( BETA GAMMA ) ( S -C ) ( E2 ) ( S -C ) * IF (ALPHA*GAMMA.LT.BETA**2) THEN TAU = (GAMMA-ALPHA)/ (TWO*BETA) T = -ONE/ (ABS(TAU)+SQRT(ONE+TAU**2)) IF (TAU.LT.ZERO) T = -T C = ONE/ (ONE+T**2) S = T*C E1 = ALPHA + BETA*T E2 = GAMMA - BETA*T * * CHANGE E1 AND E2 TO THEIR ABSOLUTE VALUES AND THEN MULTIPLY THE * THREE 2 * 2 MATRICES TO GET THE MODIFIED ALPHA, BETA AND GAMMA. * IF (E1.LT.ZERO) THEN NEG2 = NEG2 + 1 E1 = -E1 END IF IF (E2.LT.ZERO) THEN NEG2 = NEG2 + 1 E2 = -E2 END IF A(APOS) = C**2*E1 + S**2*E2 A(APOS+1) = C*S* (E1-E2) A(APOS+ALEN) = S**2*E1 + C**2*E2 END IF END IF ELSE SINGLE = .TRUE. END IF APOS = APOS + ALEN ALEN = ALEN - 1 10 CONTINUE IPOS = IPOS + NCOLS + 1 20 CONTINUE RETURN END SHAR_EOF fi # end of overwriting check if test -f 'r1mach.f' then echo shar: will not over-write existing file "'r1mach.f'" else cat << \SHAR_EOF > 'r1mach.f' REAL FUNCTION R1MACH(I) INTEGER I C C SINGLE-PRECISION MACHINE CONSTANTS C C R1MACH(1) = B**(EMIN-1), THE SMALLEST POSITIVE MAGNITUDE. C C R1MACH(2) = B**EMAX*(1 - B**(-T)), THE LARGEST MAGNITUDE. C C R1MACH(3) = B**(-T), THE SMALLEST RELATIVE SPACING. C C R1MACH(4) = B**(1-T), THE LARGEST RELATIVE SPACING. C C R1MACH(5) = LOG10(B) C C TO ALTER THIS FUNCTION FOR A PARTICULAR ENVIRONMENT, C THE DESIRED SET OF DATA STATEMENTS SHOULD BE ACTIVATED BY C REMOVING THE C FROM COLUMN 1. C C FOR IEEE-ARITHMETIC MACHINES (BINARY STANDARD), THE FIRST C SET OF CONSTANTS BELOW SHOULD BE APPROPRIATE. C C WHERE POSSIBLE, DECIMAL, OCTAL OR HEXADECIMAL CONSTANTS ARE USED C TO SPECIFY THE CONSTANTS EXACTLY. SOMETIMES THIS REQUIRES USING C EQUIVALENT INTEGER ARRAYS. IF YOUR COMPILER USES HALF-WORD C INTEGERS BY DEFAULT (SOMETIMES CALLED INTEGER*2), YOU MAY NEED TO C CHANGE INTEGER TO INTEGER*4 OR OTHERWISE INSTRUCT YOUR COMPILER C TO USE FULL-WORD INTEGERS IN THE NEXT 5 DECLARATIONS. C C COMMENTS JUST BEFORE THE END STATEMENT (LINES STARTING WITH *) C GIVE C SOURCE FOR R1MACH. C INTEGER SMALL(2) INTEGER LARGE(2) INTEGER RIGHT(2) INTEGER DIVER(2) INTEGER LOG10(2) INTEGER CRAY1, SC COMMON /D8MACH/ CRAY1 C/6S C/7S SAVE SMALL, LARGE, RIGHT, DIVER, LOG10, SC C/ REAL RMACH(5) C EQUIVALENCE (RMACH(1),SMALL(1)) EQUIVALENCE (RMACH(2),LARGE(1)) EQUIVALENCE (RMACH(3),RIGHT(1)) EQUIVALENCE (RMACH(4),DIVER(1)) EQUIVALENCE (RMACH(5),LOG10(1)) C C MACHINE CONSTANTS FOR IEEE ARITHMETIC MACHINES, SUCH AS THE AT&T C 3B SERIES, MOTOROLA 68000 BASED MACHINES (E.G. SUN 3 AND AT&T C PC 7300), AND 8087 BASED MICROS (E.G. IBM PC AND AT&T 6300). C DATA SMALL(1) / 8388608 / DATA LARGE(1) / 2139095039 / DATA RIGHT(1) / 864026624 / DATA DIVER(1) / 872415232 / DATA LOG10(1) / 1050288283 /, SC/987/ C MACHINE CONSTANTS FOR AMDAHL MACHINES. C C DATA SMALL(1) / 1048576 / C DATA LARGE(1) / 2147483647 / C DATA RIGHT(1) / 990904320 / C DATA DIVER(1) / 1007681536 / C DATA LOG10(1) / 1091781651 /, SC/987/ C C MACHINE CONSTANTS FOR THE BURROUGHS 1700 SYSTEM. C C DATA RMACH(1) / Z400800000 / C DATA RMACH(2) / Z5FFFFFFFF / C DATA RMACH(3) / Z4E9800000 / C DATA RMACH(4) / Z4EA800000 / C DATA RMACH(5) / Z500E730E8 /, SC/987/ C C MACHINE CONSTANTS FOR THE BURROUGHS 5700/6700/7700 SYSTEMS. C C DATA RMACH(1) / O1771000000000000 / C DATA RMACH(2) / O0777777777777777 / C DATA RMACH(3) / O1311000000000000 / C DATA RMACH(4) / O1301000000000000 / C DATA RMACH(5) / O1157163034761675 /, SC/987/ C C MACHINE CONSTANTS FOR FTN4 ON THE CDC 6000/7000 SERIES. C C DATA RMACH(1) / 00564000000000000000B / C DATA RMACH(2) / 37767777777777777776B / C DATA RMACH(3) / 16414000000000000000B / C DATA RMACH(4) / 16424000000000000000B / C DATA RMACH(5) / 17164642023241175720B /, SC/987/ C C MACHINE CONSTANTS FOR FTN5 ON THE CDC 6000/7000 SERIES. C C DATA RMACH(1) / O"00564000000000000000" / C DATA RMACH(2) / O"37767777777777777776" / C DATA RMACH(3) / O"16414000000000000000" / C DATA RMACH(4) / O"16424000000000000000" / C DATA RMACH(5) / O"17164642023241175720" /, SC/987/ C C MACHINE CONSTANTS FOR CONVEX C-1. C C DATA RMACH(1) / '00800000'X / C DATA RMACH(2) / '7FFFFFFF'X / C DATA RMACH(3) / '34800000'X / C DATA RMACH(4) / '35000000'X / C DATA RMACH(5) / '3F9A209B'X /, SC/987/ C C MACHINE CONSTANTS FOR THE CRAY 1, XMP, 2, AND 3. C C DATA RMACH(1) / 200034000000000000000B / C DATA RMACH(2) / 577767777777777777776B / C DATA RMACH(3) / 377224000000000000000B / C DATA RMACH(4) / 377234000000000000000B / C DATA RMACH(5) / 377774642023241175720B /, SC/987/ C C MACHINE CONSTANTS FOR THE DATA GENERAL ECLIPSE S/200. C C NOTE - IT MAY BE APPROPRIATE TO INCLUDE THE FOLLOWING LINE - C STATIC RMACH(5) C C DATA SMALL/20K,0/,LARGE/77777K,177777K/ C DATA RIGHT/35420K,0/,DIVER/36020K,0/ C DATA LOG10/40423K,42023K/, SC/987/ C C MACHINE CONSTANTS FOR THE HARRIS SLASH 6 AND SLASH 7. C C DATA SMALL(1),SMALL(2) / '20000000, '00000201 / C DATA LARGE(1),LARGE(2) / '37777777, '00000177 / C DATA RIGHT(1),RIGHT(2) / '20000000, '00000352 / C DATA DIVER(1),DIVER(2) / '20000000, '00000353 / C DATA LOG10(1),LOG10(2) / '23210115, '00000377 /, SC/987/ C C MACHINE CONSTANTS FOR THE HONEYWELL DPS 8/70 SERIES. C C DATA RMACH(1) / O402400000000 / C DATA RMACH(2) / O376777777777 / C DATA RMACH(3) / O714400000000 / C DATA RMACH(4) / O716400000000 / C DATA RMACH(5) / O776464202324 /, SC/987/ C C MACHINE CONSTANTS FOR THE IBM 360/370 SERIES, C THE XEROX SIGMA 5/7/9 AND THE SEL SYSTEMS 85/86. C C DATA RMACH(1) / Z00100000 / C DATA RMACH(2) / Z7FFFFFFF / C DATA RMACH(3) / Z3B100000 / C DATA RMACH(4) / Z3C100000 / C DATA RMACH(5) / Z41134413 /, SC/987/ C C MACHINE CONSTANTS FOR THE INTERDATA 8/32 C WITH THE UNIX SYSTEM FORTRAN 77 COMPILER. C C FOR THE INTERDATA FORTRAN VII COMPILER REPLACE C THE Z'S SPECIFYING HEX CONSTANTS WITH Y'S. C C DATA RMACH(1) / Z'00100000' / C DATA RMACH(2) / Z'7EFFFFFF' / C DATA RMACH(3) / Z'3B100000' / C DATA RMACH(4) / Z'3C100000' / C DATA RMACH(5) / Z'41134413' /, SC/987/ C C MACHINE CONSTANTS FOR THE PDP-10 (KA OR KI PROCESSOR). C C DATA RMACH(1) / "000400000000 / C DATA RMACH(2) / "377777777777 / C DATA RMACH(3) / "146400000000 / C DATA RMACH(4) / "147400000000 / C DATA RMACH(5) / "177464202324 /, SC/987/ C C MACHINE CONSTANTS FOR PDP-11 FORTRANS SUPPORTING C 32-BIT INTEGERS (EXPRESSED IN INTEGER AND OCTAL). C C DATA SMALL(1) / 8388608 / C DATA LARGE(1) / 2147483647 / C DATA RIGHT(1) / 880803840 / C DATA DIVER(1) / 889192448 / C DATA LOG10(1) / 1067065499 /, SC/987/ C C DATA RMACH(1) / O00040000000 / C DATA RMACH(2) / O17777777777 / C DATA RMACH(3) / O06440000000 / C DATA RMACH(4) / O06500000000 / C DATA RMACH(5) / O07746420233 /, SC/987/ C C MACHINE CONSTANTS FOR PDP-11 FORTRANS SUPPORTING C 16-BIT INTEGERS (EXPRESSED IN INTEGER AND OCTAL). C C DATA SMALL(1),SMALL(2) / 128, 0 / C DATA LARGE(1),LARGE(2) / 32767, -1 / C DATA RIGHT(1),RIGHT(2) / 13440, 0 / C DATA DIVER(1),DIVER(2) / 13568, 0 / C DATA LOG10(1),LOG10(2) / 16282, 8347 /, SC/987/ C C DATA SMALL(1),SMALL(2) / O000200, O000000 / C DATA LARGE(1),LARGE(2) / O077777, O177777 / C DATA RIGHT(1),RIGHT(2) / O032200, O000000 / C DATA DIVER(1),DIVER(2) / O032400, O000000 / C DATA LOG10(1),LOG10(2) / O037632, O020233 /, SC/987/ C C MACHINE CONSTANTS FOR THE SEQUENT BALANCE 8000. C C DATA SMALL(1) / $00800000 / C DATA LARGE(1) / $7F7FFFFF / C DATA RIGHT(1) / $33800000 / C DATA DIVER(1) / $34000000 / C DATA LOG10(1) / $3E9A209B /, SC/987/ C C MACHINE CONSTANTS FOR THE UNIVAC 1100 SERIES. C C DATA RMACH(1) / O000400000000 / C DATA RMACH(2) / O377777777777 / C DATA RMACH(3) / O146400000000 / C DATA RMACH(4) / O147400000000 / C DATA RMACH(5) / O177464202324 /, SC/987/ C C MACHINE CONSTANTS FOR THE VAX UNIX F77 COMPILER. C C DATA SMALL(1) / 128 / C DATA LARGE(1) / -32769 / C DATA RIGHT(1) / 13440 / C DATA DIVER(1) / 13568 / C DATA LOG10(1) / 547045274 /, SC/987/ C C MACHINE CONSTANTS FOR THE VAX-11 WITH C FORTRAN IV-PLUS COMPILER. C C DATA RMACH(1) / Z00000080 / C DATA RMACH(2) / ZFFFF7FFF / C DATA RMACH(3) / Z00003480 / C DATA RMACH(4) / Z00003500 / C DATA RMACH(5) / Z209B3F9A /, SC/987/ C C MACHINE CONSTANTS FOR VAX/VMS VERSION 2.2. C C DATA RMACH(1) / '80'X / C DATA RMACH(2) / 'FFFF7FFF'X / C DATA RMACH(3) / '3480'X / C DATA RMACH(4) / '3500'X / C DATA RMACH(5) / '209B3F9A'X /, SC/987/ C C *** ISSUE STOP 777 IF ALL DATA STATEMENTS ARE COMMENTED... IF (SC .NE. 987) THEN * *** CHECK FOR AUTODOUBLE *** SMALL(2) = 0 RMACH(1) = 1E13 IF (SMALL(2) .NE. 0) THEN * *** AUTODOUBLED *** IF ( SMALL(1) .EQ. 1117925532 * .AND. SMALL(2) .EQ. -448790528) THEN * *** IEEE BIG ENDIAN *** SMALL(1) = 1048576 SMALL(2) = 0 LARGE(1) = 2146435071 LARGE(2) = -1 RIGHT(1) = 1017118720 RIGHT(2) = 0 DIVER(1) = 1018167296 DIVER(2) = 0 LOG10(1) = 1070810131 LOG10(2) = 1352628735 ELSE IF ( SMALL(2) .EQ. 1117925532 * .AND. SMALL(1) .EQ. -448790528) THEN * *** IEEE LITTLE ENDIAN *** SMALL(2) = 1048576 SMALL(1) = 0 LARGE(2) = 2146435071 LARGE(1) = -1 RIGHT(2) = 1017118720 RIGHT(1) = 0 DIVER(2) = 1018167296 DIVER(1) = 0 LOG10(2) = 1070810131 LOG10(1) = 1352628735 ELSE IF ( SMALL(1) .EQ. -2065213935 * .AND. SMALL(2) .EQ. 10752) THEN * *** VAX WITH D_FLOATING *** SMALL(1) = 128 SMALL(2) = 0 LARGE(1) = -32769 LARGE(2) = -1 RIGHT(1) = 9344 RIGHT(2) = 0 DIVER(1) = 9472 DIVER(2) = 0 LOG10(1) = 546979738 LOG10(2) = -805796613 ELSE IF ( SMALL(1) .EQ. 1267827943 * .AND. SMALL(2) .EQ. 704643072) THEN * *** IBM MAINFRAME *** SMALL(1) = 1048576 SMALL(2) = 0 LARGE(1) = 2147483647 LARGE(2) = -1 RIGHT(1) = 856686592 RIGHT(2) = 0 DIVER(1) = 873463808 DIVER(2) = 0 LOG10(1) = 1091781651 LOG10(2) = 1352628735 ELSE WRITE(*,9010) STOP 777 END IF ELSE RMACH(1) = 1234567. IF (SMALL(1) .EQ. 1234613304) THEN * *** IEEE *** SMALL(1) = 8388608 LARGE(1) = 2139095039 RIGHT(1) = 864026624 DIVER(1) = 872415232 LOG10(1) = 1050288283 ELSE IF (SMALL(1) .EQ. -1271379306) THEN * *** VAX *** SMALL(1) = 128 LARGE(1) = -32769 RIGHT(1) = 13440 DIVER(1) = 13568 LOG10(1) = 547045274 ELSE IF (SMALL(1) .EQ. 1175639687) THEN * *** IBM MAINFRAME *** SMALL(1) = 1048576 LARGE(1) = 2147483647 RIGHT(1) = 990904320 DIVER(1) = 1007681536 LOG10(1) = 1091781651 ELSE IF (SMALL(1) .EQ. 1251390520) THEN * *** CONVEX C-1 *** SMALL(1) = 8388608 LARGE(1) = 2147483647 RIGHT(1) = 880803840 DIVER(1) = 889192448 LOG10(1) = 1067065499 ELSE * CRAY1 = 4617762693716115456 CRAY1 = 4617762 CRAY1 = 1000000*CRAY1 + 693716 CRAY1 = 1000000*CRAY1 + 115456 IF (SMALL(1) .NE. CRAY1) THEN WRITE(*,9020) STOP 777 END IF * *** CRAY 1, XMP, 2, AND 3 *** * SMALL(1) = 2306828171632181248 SMALL(1) = 2306828 SMALL(1) = 1000000*SMALL(1) + 171632 SMALL(1) = 1000000*SMALL(1) + 181248 * LARGE(1) = 6917247552664371198 LARGE(1) = 6917247 LARGE(1) = 1000000*LARGE(1) + 552664 LARGE(1) = 1000000*LARGE(1) + 371198 * RIGHT(1) = 4598878906987053056 RIGHT(1) = 4598878 RIGHT(1) = 1000000*RIGHT(1) + 906987 RIGHT(1) = 1000000*RIGHT(1) + 053056 * DIVER(1) = 4599160381963763712 DIVER(1) = 4599160 DIVER(1) = 1000000*DIVER(1) + 381963 DIVER(1) = 1000000*DIVER(1) + 763712 * LOG10(1) = 4611574008272714704 LOG10(1) = 4611574 LOG10(1) = 1000000*LOG10(1) + 008272 LOG10(1) = 1000000*LOG10(1) + 714704 END IF END IF SC = 987 END IF C C *** ISSUE STOP 776 IF ALL DATA STATEMENTS ARE OBVIOUSLY WRONG... IF (RMACH(4) .GE. 1.0) STOP 776 *C/6S *C IF (I .LT. 1 .OR. I .GT. 5) *C 1 CALL SETERR(24HR1MACH - I OUT OF BOUNDS,24,1,2) *C/7S * IF (I .LT. 1 .OR. I .GT. 5) * 1 CALL SETERR('R1MACH - I OUT OF BOUNDS',24,1,2) *C/ C IF (I .LT. 1 .OR. I .GT. 5) THEN WRITE(*,*) 'R1MACH(I): I =',I,' is out of bounds.' STOP END IF R1MACH = RMACH(I) RETURN C/6S C9010 FORMAT(/42H Adjust autodoubled R1MACH by getting data/ C *42H appropriate for your machine from D1MACH.) C9020 FORMAT(/46H Adjust R1MACH by uncommenting data statements/ C *30H appropriate for your machine.) C/7S 9010 FORMAT(/' Adjust autodoubled R1MACH by getting data'/ *' appropriate for your machine from D1MACH.') 9020 FORMAT(/' Adjust R1MACH by uncommenting data statements'/ *' appropriate for your machine.') C/ C * /* C source for R1MACH -- remove the * in column 1 */ *#include *#include *#include * *float r1mach_(long *i) *{ * switch(*i){ * case 1: return FLT_MIN; * case 2: return FLT_MAX; * case 3: return FLT_EPSILON/FLT_RADIX; * case 4: return FLT_EPSILON; * case 5: return log10(FLT_RADIX); * } * * fprintf(stderr, "invalid argument: r1mach(%ld)\n", *i); * exit(1); * return 0; /* for compilers that complain of missing return values */ * } END SHAR_EOF fi # end of overwriting check if test -f 'colmor.f' then echo shar: will not over-write existing file "'colmor.f'" else cat << \SHAR_EOF > 'colmor.f' SUBROUTINE DSSM(N,NPAIRS,INDROW,INDCOL,METHOD,LISTP,NGRP, * MAXGRP,MINGRP,INFO,IPNTR,JPNTR,IWA,LIWA) INTEGER N,NPAIRS,METHOD,MAXGRP,MINGRP,INFO,LIWA INTEGER INDROW(NPAIRS),INDCOL(NPAIRS),LISTP(N),NGRP(N), * IPNTR(N+1),JPNTR(N+1),IWA(LIWA) C ********** C C SUBROUTINE DSSM C C GIVEN THE SPARSITY PATTERN OF A SYMMETRIC MATRIX A OF ORDER N, C THIS SUBROUTINE DETERMINES A SYMMETRIC PERMUTATION OF A AND A C PARTITION OF THE COLUMNS OF A CONSISTENT WITH THE DETERMINATION C OF A BY A LOWER TRIANGULAR SUBSTITUTION METHOD. C C THE SPARSITY PATTERN OF THE MATRIX A IS SPECIFIED BY THE C ARRAYS INDROW AND INDCOL. ON INPUT THE INDICES FOR THE C NON-ZERO ELEMENTS IN THE LOWER TRIANGULAR PART OF A ARE C C (INDROW(K),INDCOL(K)), K = 1,2,...,NPAIRS. C C THE (INDROW(K),INDCOL(K)) PAIRS MAY BE SPECIFIED IN ANY ORDER. C DUPLICATE INPUT PAIRS ARE PERMITTED, BUT THE SUBROUTINE C ELIMINATES THEM. THE SUBROUTINE REQUIRES THAT ALL THE DIAGONAL C ELEMENTS BE PART OF THE SPARSITY PATTERN AND REPLACES ANY PAIR C (INDROW(K),INDCOL(K)) WHERE INDROW(K) IS LESS THAN INDCOL(K) C BY THE PAIR (INDCOL(K),INDROW(K)). C C THE DIRECT METHOD (METHOD = 1) FIRST DETERMINES A PARTITION C OF THE COLUMNS OF A SUCH THAT TWO COLUMNS IN A GROUP HAVE A C NON-ZERO ELEMENT IN ROW K ONLY IF COLUMN K IS IN AN EARLIER C GROUP. USING THIS PARTITION, THE SUBROUTINE THEN COMPUTES A C SYMMETRIC PERMUTATION OF A CONSISTENT WITH THE DETERMINATION C OF A BY A LOWER TRIANGULAR SUBSTITUTION METHOD. C C THE INDIRECT METHOD FIRST COMPUTES A SYMMETRIC PERMUTATION OF A C WHICH MINIMIZES THE MAXIMUM NUMBER OF NON-ZERO ELEMENTS IN ANY C ROW OF L, WHERE L IS THE LOWER TRIANGULAR PART OF THE PERMUTED C MATRIX. THE SUBROUTINE THEN PARTITIONS THE COLUMNS OF L INTO C GROUPS SUCH THAT COLUMNS OF L IN A GROUP DO NOT HAVE A NON-ZERO C IN THE SAME ROW POSITION. C C THE SUBROUTINE STATEMENT IS C C SUBROUTINE DSSM(N,NPAIRS,INDROW,INDCOL,METHOD,LISTP,NGRP, C MAXGRP,MINGRP,INFO,IPNTR,JPNTR,IWA,LIWA) C C WHERE C C N IS A POSITIVE INTEGER INPUT VARIABLE SET TO THE ORDER OF A. C C NPAIRS IS A POSITIVE INTEGER INPUT VARIABLE SET TO THE NUMBER C OF (INDROW,INDCOL) PAIRS USED TO DESCRIBE THE SPARSITY C PATTERN OF A. C C INDROW IS AN INTEGER ARRAY OF LENGTH NPAIRS. ON INPUT INDROW C MUST CONTAIN THE ROW INDICES OF THE NON-ZERO ELEMENTS IN C THE LOWER TRIANGULAR PART OF A. ON OUTPUT INDROW IS C PERMUTED SO THAT THE CORRESPONDING COLUMN INDICES ARE IN C NON-DECREASING ORDER. THE COLUMN INDICES CAN BE RECOVERED C FROM THE ARRAY JPNTR. C C INDCOL IS AN INTEGER ARRAY OF LENGTH NPAIRS. ON INPUT INDCOL C MUST CONTAIN THE COLUMN INDICES OF THE NON-ZERO ELEMENTS C IN THE LOWER TRIANGULAR PART OF A. ON OUTPUT INDCOL IS C PERMUTED SO THAT THE CORRESPONDING ROW INDICES ARE IN C NON-DECREASING ORDER. THE ROW INDICES CAN BE RECOVERED C FROM THE ARRAY IPNTR. C C METHOD IS AN INTEGER INPUT VARIABLE. IF METHOD = 1, THE C DIRECT METHOD IS USED TO DETERMINE THE PARTITION AND C SYMMETRIC PERMUTATION. OTHERWISE, THE INDIRECT METHOD IS C USED TO DETERMINE THE SYMMETRIC PERMUTATION AND PARTITION. C C LISTP IS AN INTEGER OUTPUT ARRAY OF LENGTH N WHICH SPECIFIES C THE SYMMETRIC PERMUTATION OF THE MATRIX A. ELEMENT (I,J) C OF A IS THE (LISTP(I),LISTP(J)) ELEMENT OF THE PERMUTED C MATRIX. C C NGRP IS AN INTEGER OUTPUT ARRAY OF LENGTH N WHICH SPECIFIES C THE PARTITION OF THE COLUMNS OF A. COLUMN J BELONGS TO C GROUP NGRP(J). C C MAXGRP IS AN INTEGER OUTPUT VARIABLE WHICH SPECIFIES THE C NUMBER OF GROUPS IN THE PARTITION OF THE COLUMNS OF A. C C MINGRP IS AN INTEGER OUTPUT VARIABLE WHICH SPECIFIES A LOWER C BOUND FOR THE NUMBER OF GROUPS IN ANY PARTITION OF THE C COLUMNS OF A CONSISTENT WITH THE DETERMINATION OF A BY A C LOWER TRIANGULAR SUBSTITUTION METHOD. C C INFO IS AN INTEGER OUTPUT VARIABLE SET AS FOLLOWS. FOR C NORMAL TERMINATION INFO = 1. C C IPNTR IS AN INTEGER OUTPUT ARRAY OF LENGTH N + 1 WHICH C SPECIFIES THE LOCATIONS OF THE COLUMN INDICES IN INDCOL. C THE COLUMN INDICES FOR ROW I ARE C C INDCOL(K), K = IPNTR(I),...,IPNTR(I+1)-1. C C NOTE THAT IPNTR(N+1)-1 IS THEN THE NUMBER OF NON-ZERO C ELEMENTS IN THE LOWER TRIANGULAR PART OF THE MATRIX A. C C JPNTR IS AN INTEGER OUTPUT ARRAY OF LENGTH N + 1 WHICH C SPECIFIES THE LOCATIONS OF THE ROW INDICES IN INDROW. C THE ROW INDICES FOR COLUMN J ARE C C INDROW(K), K = JPNTR(J),...,JPNTR(J+1)-1. C C NOTE THAT JPNTR(N+1)-1 IS THEN THE NUMBER OF NON-ZERO C ELEMENTS IN THE LOWER TRIANGULAR PART OF THE MATRIX A. C C IWA IS AN INTEGER WORK ARRAY OF LENGTH LIWA. C C LIWA IS A POSITIVE INTEGER INPUT VARIABLE NOT LESS THAN 6*N. C C SUBPROGRAMS CALLED C C MINPACK-SUPPLIED ... DEGR,IDO,IDOG,NUMSRT,SDPT,SEQ,SETR, C SLO,SLOG,SRTDAT C C FORTRAN-SUPPLIED ... MAX,MIN C C ARGONNE NATIONAL LABORATORY. MINPACK PROJECT. DECEMBER 1984. C THOMAS F. COLEMAN, BURTON S. GARBOW, JORGE J. MORE' C C ********** INTEGER I,J,JP,MAXID,MAXVD,MAXCLQ,NNZ,NUMGRP C INFO = 1 C C EXTEND THE DATA STRUCTURE TO ROWS. C CALL SETR(N,N,INDROW,JPNTR,INDCOL,IPNTR,IWA) C C DETERMINE THE SMALLEST-LAST ORDERING OF THE VERTICES OF THE C ADJACENCY GRAPH OF A, AND FROM IT DETERMINE A LOWER BOUND C FOR THE NUMBER OF GROUPS. C CALL SLOG(N,INDROW,JPNTR,INDCOL,IPNTR,IWA(1),MAXCLQ, * MAXVD,IWA(N+1),IWA(2*N+1),IWA(3*N+1)) MINGRP = 1 + MAXVD C C USE THE SELECTED METHOD. C IF (METHOD .EQ. 1) THEN C C DIRECT METHOD. DETERMINE A PARTITION OF THE COLUMNS C OF A BY THE POWELL-TOINT METHOD. C CALL SDPT(N,INDROW,JPNTR,INDCOL,IPNTR,NGRP,MAXGRP, * IWA(N+1),IWA(2*N+1)) C C DEFINE A SYMMETRIC PERMUTATION OF A ACCORDING TO THE C ORDERING OF THE COLUMN GROUP NUMBERS IN THE PARTITION. C CALL NUMSRT(N,MAXGRP,NGRP,1,IWA(1),IWA(2*N+1),IWA(N+1)) DO 80 I = 1, N LISTP(IWA(I)) = I 80 CONTINUE ELSE C C INDIRECT METHOD. DETERMINE THE INCIDENCE DEGREE ORDERING C OF THE VERTICES OF THE ADJACENCY GRAPH OF A AND, TOGETHER C WITH THE SMALLEST-LAST ORDERING, DEFINE A SYMMETRIC C PERMUTATION OF A. C CALL IDOG(N,INDROW,JPNTR,INDCOL,IPNTR,LISTP,MAXCLQ, * MAXID,IWA(N+1),IWA(2*N+1),IWA(3*N+1)) IF (MAXID .GT. MAXVD) THEN DO 90 I = 1, N LISTP(I) = IWA(I) 90 CONTINUE END IF C C GENERATE THE SPARSITY PATTERN FOR THE LOWER C TRIANGULAR PART L OF THE PERMUTED MATRIX. C DO 110 J = 1, N DO 100 JP = JPNTR(J), JPNTR(J+1)-1 I = INDROW(JP) INDROW(JP) = MAX(LISTP(I),LISTP(J)) INDCOL(JP) = MIN(LISTP(I),LISTP(J)) 100 CONTINUE 110 CONTINUE C C SORT THE DATA STRUCTURE BY COLUMNS. C CALL SRTDAT(N,NNZ,INDROW,INDCOL,JPNTR,IWA) C C EXTEND THE DATA STRUCTURE TO ROWS. C CALL SETR(N,N,INDROW,JPNTR,INDCOL,IPNTR,IWA) C C DETERMINE THE DEGREE SEQUENCE FOR THE INTERSECTION C GRAPH OF THE COLUMNS OF L. C CALL DEGR(N,INDROW,JPNTR,INDCOL,IPNTR,IWA(5*N+1),IWA(N+1)) C C COLOR THE INTERSECTION GRAPH OF THE COLUMNS OF L C WITH THE SMALLEST-LAST (SL) ORDERING. C CALL SLO(N,INDROW,JPNTR,INDCOL,IPNTR,IWA(5*N+1),IWA(4*N+1), * MAXCLQ,IWA(1),IWA(N+1),IWA(2*N+1),IWA(3*N+1)) CALL SEQ(N,INDROW,JPNTR,INDCOL,IPNTR,IWA(4*N+1),IWA(1), * MAXGRP,IWA(N+1)) DO 120 J = 1, N NGRP(J) = IWA(LISTP(J)) 120 CONTINUE C C EXIT IF THE SMALLEST-LAST ORDERING IS OPTIMAL. C IF (MAXGRP .EQ. MAXCLQ) GO TO 140 C C COLOR THE INTERSECTION GRAPH OF THE COLUMNS OF L C WITH THE INCIDENCE DEGREE (ID) ORDERING. C CALL IDO(N,N,INDROW,JPNTR,INDCOL,IPNTR,IWA(5*N+1),IWA(4*N+1), * MAXCLQ,IWA(1),IWA(N+1),IWA(2*N+1),IWA(3*N+1)) CALL SEQ(N,INDROW,JPNTR,INDCOL,IPNTR,IWA(4*N+1),IWA(1), * NUMGRP,IWA(N+1)) C C RETAIN THE BETTER OF THE TWO ORDERINGS. C IF (NUMGRP .LT. MAXGRP) THEN MAXGRP = NUMGRP DO 130 J = 1, N NGRP(J) = IWA(LISTP(J)) 130 CONTINUE END IF 140 CONTINUE C C GENERATE THE SPARSITY PATTERN FOR THE LOWER C TRIANGULAR PART OF THE ORIGINAL MATRIX. C DO 150 J = 1, N IWA(LISTP(J)) = J 150 CONTINUE DO 170 J = 1, N DO 160 JP = JPNTR(J), JPNTR(J+1)-1 I = INDROW(JP) INDROW(JP) = MAX(IWA(I),IWA(J)) INDCOL(JP) = MIN(IWA(I),IWA(J)) 160 CONTINUE 170 CONTINUE C C SORT THE DATA STRUCTURE BY COLUMNS. C CALL SRTDAT(N,NNZ,INDROW,INDCOL,JPNTR,IWA) C C EXTEND THE DATA STRUCTURE TO ROWS. C CALL SETR(N,N,INDROW,JPNTR,INDCOL,IPNTR,IWA) END IF RETURN C C LAST CARD OF SUBROUTINE DSSM. C END SUBROUTINE IDOG(N,NGHBRP,NPNTRP,NGHBRS,NPNTRS,LISTP, * MAXCLQ,MAXID,IWA1,IWA2,IWA3) INTEGER N,MAXCLQ,MAXID INTEGER NGHBRP(*),NPNTRP(N+1),NGHBRS(*),NPNTRS(N+1),LISTP(N), * IWA1(0:N-1),IWA2(N),IWA3(N) C ********** C C SUBROUTINE IDOG C C GIVEN A LOOPLESS GRAPH G = (V,E), THIS SUBROUTINE DETERMINES C THE INCIDENCE DEGREE ORDERING OF THE VERTICES OF G. C C THE INCIDENCE DEGREE ORDERING IS DETERMINED RECURSIVELY BY C LETTING LIST(K), K = 1,...,N BE A VERTEX WITH MAXIMAL C INCIDENCE TO THE SUBGRAPH SPANNED BY THE ORDERED VERTICES. C AMONG ALL THE VERTICES OF MAXIMAL INCIDENCE, A VERTEX OF C MAXIMAL DEGREE IS CHOSEN. THIS SUBROUTINE DETERMINES THE C INVERSE OF THE INCIDENCE DEGREE ORDERING, THAT IS, AN ARRAY C LISTP SUCH THAT LISTP(LIST(K)) = K FOR K = 1,2,...,N. C C THE SUBROUTINE STATEMENT IS C C SUBROUTINE IDOG(N,NGHBRP,NPNTRP,NGHBRS,NPNTRS,LISTP, C MAXCLQ,MAXID,IWA1,IWA2,IWA3) C C WHERE C C N IS A POSITIVE INTEGER INPUT VARIABLE SET TO THE NUMBER C OF VERTICES OF G. C C NGHBRP IS AN INTEGER INPUT ARRAY WHICH CONTAINS THE C PREDECESSOR ADJACENCY LISTS FOR THE GRAPH G. C C NPNTRP IS AN INTEGER INPUT ARRAY OF LENGTH N + 1 WHICH C SPECIFIES THE LOCATIONS OF THE PREDECESSOR ADJACENCY C LISTS IN NGHBRP. THE VERTICES PRECEDING AND ADJACENT C TO VERTEX J ARE C C NGHBRP(K), K = NPNTRP(J),...,NPNTRP(J+1)-1. C C NOTE THAT NPNTRP(N+1)-1 IS THEN THE NUMBER OF VERTICES C PLUS EDGES OF THE GRAPH G. C C NGHBRS IS AN INTEGER INPUT ARRAY WHICH CONTAINS THE C SUCCESSOR ADJACENCY LISTS FOR THE GRAPH G. C C NPNTRS IS AN INTEGER INPUT ARRAY OF LENGTH N + 1 WHICH C SPECIFIES THE LOCATIONS OF THE SUCCESSOR ADJACENCY C LISTS IN NGHBRS. THE VERTICES SUCCEEDING AND ADJACENT C TO VERTEX J ARE C C NGHBRS(K), K = NPNTRS(J),...,NPNTRS(J+1)-1. C C NOTE THAT NPNTRS(N+1)-1 IS THEN THE NUMBER OF VERTICES C PLUS EDGES OF THE GRAPH G. C C LISTP IS AN INTEGER OUTPUT ARRAY OF LENGTH N WHICH SPECIFIES C THE INVERSE OF THE INCIDENCE DEGREE ORDERING OF THE C VERTICES. VERTEX J IS IN POSITION LISTP(J) OF THIS ORDERING. C C MAXCLQ IS AN INTEGER OUTPUT VARIABLE SET TO THE SIZE C OF THE LARGEST CLIQUE FOUND DURING THE ORDERING. C C MAXID IS AN INTEGER OUTPUT VARIABLE SET TO THE MAXIMUM C INCIDENCE DEGREE FOUND DURING THE ORDERING. C C IWA1,IWA2, AND IWA3 ARE INTEGER WORK ARRAYS OF LENGTH N. C C SUBPROGRAMS CALLED C C MINPACK-SUPPLIED ... NUMSRT C C FORTRAN-SUPPLIED ... MAX C C ARGONNE NATIONAL LABORATORY. MINPACK PROJECT. DECEMBER 1984. C THOMAS F. COLEMAN, BURTON S. GARBOW, JORGE J. MORE' C C ********** INTEGER I,J,K,MAXINC,MAXDEG,MAXLST,NCOMP,NUMDEG,NUMINC,NUMORD C C INITIALIZATION BLOCK. C DO 10 J = 1, N LISTP(J) = (NPNTRP(J+1) - NPNTRP(J) - 1) + * (NPNTRS(J+1) - NPNTRS(J) - 1) 10 CONTINUE MAXLST = (NPNTRP(N+1) + NPNTRS(N+1))/N C C SORT THE DEGREE SEQUENCE. C CALL NUMSRT(N,N-1,LISTP,1,IWA1,IWA2,IWA3) C C CREATE A DOUBLY-LINKED LIST TO ACCESS THE INCIDENCES OF THE C VERTICES. THE POINTERS FOR THE LINKED LIST ARE AS FOLLOWS. C C EACH UN-ORDERED VERTEX I IS IN A LIST (THE INCIDENCE LIST) C OF VERTICES WITH THE SAME INCIDENCE. C C IWA1(NUMINC) IS THE FIRST VERTEX IN THE NUMINC LIST C UNLESS IWA1(NUMINC) = 0. IN THIS CASE THERE ARE C NO VERTICES IN THE NUMINC LIST. C C IWA2(I) IS THE VERTEX BEFORE I IN THE INCIDENCE LIST C UNLESS IWA2(I) = 0. IN THIS CASE I IS THE FIRST C VERTEX IN THIS INCIDENCE LIST. C C IWA3(I) IS THE VERTEX AFTER I IN THE INCIDENCE LIST C UNLESS IWA3(I) = 0. IN THIS CASE I IS THE LAST C VERTEX IN THIS INCIDENCE LIST. C C IF I IS AN UN-ORDERED VERTEX, THEN -LISTP(I) IS THE C INCIDENCE OF I TO THE GRAPH INDUCED BY THE ORDERED C VERTICES. IF J IS AN ORDERED VERTEX, THEN LISTP(J) C IS THE INCIDENCE DEGREE ORDER OF VERTEX J. C MAXINC = 0 DO 20 J = 1, N I = IWA1(J-1) IWA1(J-1) = 0 IWA2(I) = 0 IWA3(I) = IWA1(0) IF (IWA1(0) .GT. 0) IWA2(IWA1(0)) = I IWA1(0) = I LISTP(J) = 0 20 CONTINUE MAXCLQ = 0 MAXID = 0 NUMORD = 1 C C BEGINNING OF ITERATION LOOP. C 30 CONTINUE C C CHOOSE A VERTEX J OF MAXIMAL DEGREE AMONG THE C VERTICES OF MAXIMAL INCIDENCE MAXINC. C 40 CONTINUE K = IWA1(MAXINC) IF (K .GT. 0) GO TO 50 MAXINC = MAXINC - 1 GO TO 40 50 CONTINUE MAXDEG = -1 DO 60 I = 1, MAXLST NUMDEG = (NPNTRP(K+1) - NPNTRP(K) - 1) + * (NPNTRS(K+1) - NPNTRS(K) - 1) IF (NUMDEG .GT. MAXDEG) THEN MAXDEG = NUMDEG J = K END IF K = IWA3(K) IF (K .LE. 0) GO TO 70 60 CONTINUE 70 CONTINUE LISTP(J) = NUMORD MAXID = MAX(MAXID,MAXINC) C C UPDATE THE SIZE OF THE LARGEST CLIQUE C FOUND DURING THE ORDERING. C IF (MAXINC .EQ. 0) NCOMP = 0 NCOMP = NCOMP + 1 IF (MAXINC+1 .EQ. NCOMP) MAXCLQ = MAX(MAXCLQ,NCOMP) C C TERMINATION TEST. C NUMORD = NUMORD + 1 IF (NUMORD .GT. N) GO TO 100 C C DELETE VERTEX J FROM THE MAXINC LIST. C IF (IWA2(J) .EQ. 0) THEN IWA1(MAXINC) = IWA3(J) ELSE IWA3(IWA2(J)) = IWA3(J) END IF IF (IWA3(J) .GT. 0) IWA2(IWA3(J)) = IWA2(J) C C DETERMINE ALL THE NEIGHBORS OF VERTEX J WHICH PRECEDE J C IN THE SUBGRAPH SPANNED BY THE UN-ORDERED VERTICES. C DO 80 K = NPNTRP(J), NPNTRP(J+1)-1 I = NGHBRP(K) C C UPDATE THE POINTERS TO THE CURRENT INCIDENCE LISTS. C NUMINC = -LISTP(I) IF (NUMINC .GE. 0) THEN LISTP(I) = LISTP(I) - 1 MAXINC = MAX(MAXINC,-LISTP(I)) C C DELETE VERTEX I FROM THE NUMINC LIST. C IF (IWA2(I) .EQ. 0) THEN IWA1(NUMINC) = IWA3(I) ELSE IWA3(IWA2(I)) = IWA3(I) END IF IF (IWA3(I) .GT. 0) IWA2(IWA3(I)) = IWA2(I) C C ADD VERTEX I TO THE NUMINC+1 LIST. C IWA2(I) = 0 IWA3(I) = IWA1(NUMINC+1) IF (IWA1(NUMINC+1) .GT. 0) IWA2(IWA1(NUMINC+1)) = I IWA1(NUMINC+1) = I END IF 80 CONTINUE C C DETERMINE ALL THE NEIGHBORS OF VERTEX J WHICH SUCCEED J C IN THE SUBGRAPH SPANNED BY THE UN-ORDERED VERTICES. C DO 90 K = NPNTRS(J), NPNTRS(J+1)-1 I = NGHBRS(K) C C UPDATE THE POINTERS TO THE CURRENT INCIDENCE LISTS. C NUMINC = -LISTP(I) IF (NUMINC .GE. 0) THEN LISTP(I) = LISTP(I) - 1 MAXINC = MAX(MAXINC,-LISTP(I)) C C DELETE VERTEX I FROM THE NUMINC LIST. C IF (IWA2(I) .EQ. 0) THEN IWA1(NUMINC) = IWA3(I) ELSE IWA3(IWA2(I)) = IWA3(I) END IF IF (IWA3(I) .GT. 0) IWA2(IWA3(I)) = IWA2(I) C C ADD VERTEX I TO THE NUMINC+1 LIST. C IWA2(I) = 0 IWA3(I) = IWA1(NUMINC+1) IF (IWA1(NUMINC+1) .GT. 0) IWA2(IWA1(NUMINC+1)) = I IWA1(NUMINC+1) = I END IF 90 CONTINUE C C END OF ITERATION LOOP. C GO TO 30 100 CONTINUE RETURN C C LAST CARD OF SUBROUTINE IDOG. C END SUBROUTINE SDPT(N,NGHBRP,NPNTRP,NGHBRS,NPNTRS,NGRP,MAXGRP, * IWA1,IWA2) INTEGER N,MAXGRP INTEGER NGHBRP(*),NPNTRP(N+1),NGHBRS(*),NPNTRS(N+1),NGRP(N), * IWA1(0:N-1),IWA2(N) C ********** C C SUBROUTINE SDPT C C GIVEN A LOOPLESS GRAPH G = (V,E), THIS SUBROUTINE DETERMINES C A SYMMETRIC COLORING OF G BY THE POWELL-TOINT DIRECT METHOD. C C THE POWELL-TOINT METHOD ASSIGNS THE K-TH COLOR BY EXAMINING C THE UN-COLORED VERTICES U(K) IN ORDER OF NON-INCREASING DEGREE C AND ASSIGNING COLOR K TO VERTEX V IF THERE ARE NO PATHS OF C LENGTH 1 OR 2 (IN THE GRAPH INDUCED BY U(K)) BETWEEN V AND C SOME K-COLORED VERTEX. C C THE SUBROUTINE STATEMENT IS C C SUBROUTINE SDPT(N,NGHBRP,NPNTRP,NGHBRS,NPNTRS,NGRP,MAXGRP, C IWA1,IWA2) C C WHERE C C N IS A POSITIVE INTEGER INPUT VARIABLE SET TO THE NUMBER C OF VERTICES OF G. C C NGHBRP IS AN INTEGER INPUT ARRAY WHICH CONTAINS THE C PREDECESSOR ADJACENCY LISTS FOR THE GRAPH G. C C NPNTRP IS AN INTEGER INPUT ARRAY OF LENGTH N + 1 WHICH C SPECIFIES THE LOCATIONS OF THE PREDECESSOR ADJACENCY C LISTS IN NGHBRP. THE VERTICES PRECEDING AND ADJACENT C TO VERTEX J ARE C C NGHBRP(K), K = NPNTRP(J),...,NPNTRP(J+1)-1. C C NOTE THAT NPNTRP(N+1)-1 IS THEN THE NUMBER OF VERTICES C PLUS EDGES OF THE GRAPH G. C C NGHBRS IS AN INTEGER INPUT ARRAY WHICH CONTAINS THE C SUCCESSOR ADJACENCY LISTS FOR THE GRAPH G. C C NPNTRS IS AN INTEGER INPUT ARRAY OF LENGTH N + 1 WHICH C SPECIFIES THE LOCATIONS OF THE SUCCESSOR ADJACENCY C LISTS IN NGHBRS. THE VERTICES SUCCEEDING AND ADJACENT C TO VERTEX J ARE C C NGHBRS(K), K = NPNTRS(J),...,NPNTRS(J+1)-1. C C NOTE THAT NPNTRS(N+1)-1 IS THEN THE NUMBER OF VERTICES C PLUS EDGES OF THE GRAPH G. C C NGRP IS AN INTEGER OUTPUT ARRAY OF LENGTH N WHICH SPECIFIES C THE SYMMETRIC COLORING OF G. VERTEX J IS COLORED WITH C COLOR NGRP(J). C C MAXGRP IS AN INTEGER OUTPUT VARIABLE WHICH SPECIFIES THE C NUMBER OF COLORS IN THE SYMMETRIC COLORING OF G. C C IWA1 AND IWA2 ARE INTEGER WORK ARRAYS OF LENGTH N. C C SUBPROGRAMS CALLED C C FORTRAN-SUPPLIED ... MAX C C ARGONNE NATIONAL LABORATORY. MINPACK PROJECT. DECEMBER 1984. C THOMAS F. COLEMAN, BURTON S. GARBOW, JORGE J. MORE' C C ********** INTEGER J,JP,K,KP,L,MAXDEG,NUMDEG,NUMV C C INITIALIZATION BLOCK. NUMV IS THE CURRENT NUMBER OF UN-COLORED C VERTICES, MAXDEG IS THE MAXIMUM INDUCED DEGREE OF THESE C VERTICES, AND MAXGRP IS THE CURRENT GROUP NUMBER (COLOR). C NUMV = N MAXDEG = 0 DO 10 J = 1, N NGRP(J) = (NPNTRP(J) - NPNTRP(J+1) + 1) + * (NPNTRS(J) - NPNTRS(J+1) + 1) MAXDEG = MAX(MAXDEG,-NGRP(J)) IWA2(J) = -J 10 CONTINUE MAXGRP = 0 C C BEGINNING OF ITERATION LOOP. C 20 CONTINUE C C SORT THE LIST OF UN-COLORED VERTICES SO THAT THEIR C INDUCED DEGREES ARE IN NON-DECREASING ORDER. C DO 30 NUMDEG = 0, MAXDEG IWA1(NUMDEG) = 0 30 CONTINUE DO 40 L = 1, NUMV NUMDEG = -NGRP(-IWA2(L)) IWA1(NUMDEG) = IWA1(NUMDEG) + 1 40 CONTINUE K = 1 DO 50 NUMDEG = MAXDEG, 0, -1 L = IWA1(NUMDEG) IWA1(NUMDEG) = K K = K + L 50 CONTINUE K = 1 60 CONTINUE J = IWA2(K) IF (J .GT. 0) THEN K = IWA1(-NGRP(J)) ELSE NUMDEG = -NGRP(-J) L = IWA1(NUMDEG) IWA2(K) = IWA2(L) IWA2(L) = -J IWA1(NUMDEG) = IWA1(NUMDEG) + 1 END IF IF (K .LE. NUMV) GO TO 60 MAXGRP = MAXGRP + 1 C C DETERMINE THE VERTICES IN GROUP MAXGRP. C DO 160 L = 1, NUMV J = IWA2(L) C C EXAMINE EACH VERTEX K PRECEDING VERTEX J AND ALL C THE NEIGHBORS OF VERTEX K TO DETERMINE IF VERTEX C J CAN BE CONSIDERED FOR GROUP MAXGRP. C DO 90 JP = NPNTRP(J), NPNTRP(J+1)-1 K = NGHBRP(JP) IF (NGRP(K) .EQ. MAXGRP) GO TO 150 IF (NGRP(K) .LE. 0) THEN DO 70 KP = NPNTRP(K), NPNTRP(K+1)-1 IF (NGRP(NGHBRP(KP)) .EQ. MAXGRP) GO TO 150 70 CONTINUE DO 80 KP = NPNTRS(K), NPNTRS(K+1)-1 IF (NGRP(NGHBRS(KP)) .EQ. MAXGRP) GO TO 150 80 CONTINUE END IF 90 CONTINUE C C EXAMINE EACH VERTEX K SUCCEEDING VERTEX J AND ALL C THE NEIGHBORS OF VERTEX K TO DETERMINE IF VERTEX C J CAN BE ADDED TO GROUP MAXGRP. C DO 120 JP = NPNTRS(J), NPNTRS(J+1)-1 K = NGHBRS(JP) IF (NGRP(K) .EQ. MAXGRP) GO TO 150 IF (NGRP(K) .LE. 0) THEN DO 100 KP = NPNTRP(K), NPNTRP(K+1)-1 IF (NGRP(NGHBRP(KP)) .EQ. MAXGRP) GO TO 150 100 CONTINUE DO 110 KP = NPNTRS(K), NPNTRS(K+1)-1 IF (NGRP(NGHBRS(KP)) .EQ. MAXGRP) GO TO 150 110 CONTINUE END IF 120 CONTINUE C C ADD VERTEX J TO GROUP MAXGRP AND REMOVE VERTEX J C FROM THE LIST OF UN-COLORED VERTICES. C NGRP(J) = MAXGRP IWA2(L) = 0 C C UPDATE THE DEGREES OF THE NEIGHBORS OF VERTEX J. C DO 130 JP = NPNTRP(J), NPNTRP(J+1)-1 K = NGHBRP(JP) IF (NGRP(K) .LT. 0) NGRP(K) = NGRP(K) + 1 130 CONTINUE DO 140 JP = NPNTRS(J), NPNTRS(J+1)-1 K = NGHBRS(JP) IF (NGRP(K) .LT. 0) NGRP(K) = NGRP(K) + 1 140 CONTINUE 150 CONTINUE 160 CONTINUE C C COMPRESS THE UPDATED LIST OF UN-COLORED VERTICES. C RESET NUMV AND RECOMPUTE MAXDEG. C K = 0 MAXDEG = 0 DO 170 L = 1, NUMV IF (IWA2(L) .NE. 0) THEN K = K + 1 IWA2(K) = -IWA2(L) MAXDEG = MAX(MAXDEG,-NGRP(IWA2(L))) END IF 170 CONTINUE NUMV = K C C END OF ITERATION LOOP. C IF (NUMV .GT. 0) GO TO 20 RETURN C C LAST CARD OF SUBROUTINE SDPT. C END SUBROUTINE SLOG(N,NGHBRP,NPNTRP,NGHBRS,NPNTRS,LISTP, * MAXCLQ,MAXVD,IWA1,IWA2,IWA3) INTEGER N,MAXCLQ,MAXVD INTEGER NGHBRP(*),NPNTRP(N+1),NGHBRS(*),NPNTRS(N+1),LISTP(N), * IWA1(0:N-1),IWA2(N),IWA3(N) C ********** C C SUBROUTINE SLOG C C GIVEN A LOOPLESS GRAPH G = (V,E), THIS SUBROUTINE DETERMINES C THE SMALLEST-LAST ORDERING OF THE VERTICES OF G. C C THE SMALLEST-LAST ORDERING IS DETERMINED RECURSIVELY BY C LETTING LIST(K), K = N,...,1 BE A VERTEX WITH LEAST DEGREE C IN THE SUBGRAPH SPANNED BY THE UN-ORDERED VERTICES. C THIS SUBROUTINE DETERMINES THE INVERSE OF THE SMALLEST-LAST C ORDERING, THAT IS, AN ARRAY LISTP SUCH THAT LISTP(LIST(K)) = K C FOR K = 1,2,...,N. C C THE SUBROUTINE STATEMENT IS C C SUBROUTINE SLOG(N,NGHBRP,NPNTRP,NGHBRS,NPNTRS,LISTP, C MAXCLQ,MAXVD,IWA1,IWA2,IWA3) C C WHERE C C N IS A POSITIVE INTEGER INPUT VARIABLE SET TO THE NUMBER C OF VERTICES OF G. C C NGHBRP IS AN INTEGER INPUT ARRAY WHICH CONTAINS THE C PREDECESSOR ADJACENCY LISTS FOR THE GRAPH G. C C NPNTRP IS AN INTEGER INPUT ARRAY OF LENGTH N + 1 WHICH C SPECIFIES THE LOCATIONS OF THE PREDECESSOR ADJACENCY C LISTS IN NGHBRP. THE VERTICES PRECEDING AND ADJACENT C TO VERTEX J ARE C C NGHBRP(K), K = NPNTRP(J),...,NPNTRP(J+1)-1. C C NOTE THAT NPNTRP(N+1)-1 IS THEN THE NUMBER OF VERTICES C PLUS EDGES OF THE GRAPH G. C C NGHBRS IS AN INTEGER INPUT ARRAY WHICH CONTAINS THE C SUCCESSOR ADJACENCY LISTS FOR THE GRAPH G. C C NPNTRS IS AN INTEGER INPUT ARRAY OF LENGTH N + 1 WHICH C SPECIFIES THE LOCATIONS OF THE SUCCESSOR ADJACENCY C LISTS IN NGHBRS. THE VERTICES SUCCEEDING AND ADJACENT C TO VERTEX J ARE C C NGHBRS(K), K = NPNTRS(J),...,NPNTRS(J+1)-1. C C NOTE THAT NPNTRS(N+1)-1 IS THEN THE NUMBER OF VERTICES C PLUS EDGES OF THE GRAPH G. C C LISTP IS AN INTEGER OUTPUT ARRAY OF LENGTH N WHICH SPECIFIES C THE INVERSE OF THE SMALLEST-LAST ORDERING OF THE VERTICES. C VERTEX J IS IN POSITION LISTP(J) OF THIS ORDERING. C C MAXCLQ IS AN INTEGER OUTPUT VARIABLE SET TO THE SIZE C OF THE LARGEST CLIQUE FOUND DURING THE ORDERING. C C MAXVD IS AN INTEGER OUTPUT VARIABLE SET TO THE MAXIMUM C VERTEX DEGREE FOUND DURING THE ORDERING. C C IWA1,IWA2, AND IWA3 ARE INTEGER WORK ARRAYS OF LENGTH N. C C SUBPROGRAMS CALLED C C FORTRAN-SUPPLIED ... MAX,MIN C C ARGONNE NATIONAL LABORATORY. MINPACK PROJECT. DECEMBER 1984. C THOMAS F. COLEMAN, BURTON S. GARBOW, JORGE J. MORE' C C ********** INTEGER I,J,K,MINDEG,NUMDEG,NUMORD C C INITIALIZATION BLOCK. C MINDEG = N DO 10 J = 1, N IWA1(J-1) = 0 LISTP(J) = (NPNTRP(J) - NPNTRP(J+1) + 1) + * (NPNTRS(J) - NPNTRS(J+1) + 1) MINDEG = MIN(MINDEG,-LISTP(J)) 10 CONTINUE C C CREATE A DOUBLY-LINKED LIST TO ACCESS THE DEGREES OF THE C VERTICES. THE POINTERS FOR THE LINKED LIST ARE AS FOLLOWS. C C EACH UN-ORDERED VERTEX I IS IN A LIST (THE DEGREE LIST) C OF VERTICES WITH THE SAME DEGREE. C C IWA1(NUMDEG) IS THE FIRST VERTEX IN THE NUMDEG LIST C UNLESS IWA1(NUMDEG) = 0. IN THIS CASE THERE ARE C NO VERTICES IN THE NUMDEG LIST. C C IWA2(I) IS THE VERTEX BEFORE I IN THE DEGREE LIST C UNLESS IWA2(I) = 0. IN THIS CASE I IS THE FIRST C VERTEX IN THIS DEGREE LIST. C C IWA3(I) IS THE VERTEX AFTER I IN THE DEGREE LIST C UNLESS IWA3(I) = 0. IN THIS CASE I IS THE LAST C VERTEX IN THIS DEGREE LIST. C C IF I IS AN UN-ORDERED VERTEX, THEN -LISTP(I) IS THE C DEGREE OF I IN THE GRAPH INDUCED BY THE UN-ORDERED C VERTICES. IF J IS AN ORDERED VERTEX, THEN LISTP(J) C IS THE SMALLEST-LAST ORDER OF VERTEX J. C DO 20 J = 1, N NUMDEG = -LISTP(J) IWA2(J) = 0 IWA3(J) = IWA1(NUMDEG) IF (IWA1(NUMDEG) .GT. 0) IWA2(IWA1(NUMDEG)) = J IWA1(NUMDEG) = J 20 CONTINUE MAXCLQ = 0 MAXVD = 0 NUMORD = N C C BEGINNING OF ITERATION LOOP. C 30 CONTINUE C C CHOOSE A VERTEX J OF MINIMAL DEGREE MINDEG. C 40 CONTINUE J = IWA1(MINDEG) IF (J .GT. 0) GO TO 50 MINDEG = MINDEG + 1 GO TO 40 50 CONTINUE LISTP(J) = NUMORD MAXVD = MAX(MAXVD,MINDEG) C C MARK THE SIZE OF THE LARGEST CLIQUE C FOUND DURING THE ORDERING. C IF (MINDEG+1 .EQ. NUMORD .AND. MAXCLQ .EQ. 0) * MAXCLQ = NUMORD C C TERMINATION TEST. C NUMORD = NUMORD - 1 IF (NUMORD .EQ. 0) GO TO 80 C C DELETE VERTEX J FROM THE MINDEG LIST. C IWA1(MINDEG) = IWA3(J) IF (IWA3(J) .GT. 0) IWA2(IWA3(J)) = 0 C C DETERMINE ALL THE NEIGHBORS OF VERTEX J WHICH PRECEDE J C IN THE SUBGRAPH SPANNED BY THE UN-ORDERED VERTICES. C DO 60 K = NPNTRP(J), NPNTRP(J+1)-1 I = NGHBRP(K) C C UPDATE THE POINTERS TO THE CURRENT DEGREE LISTS. C NUMDEG = -LISTP(I) IF (NUMDEG .GE. 0) THEN LISTP(I) = LISTP(I) + 1 MINDEG = MIN(MINDEG,-LISTP(I)) C C DELETE VERTEX I FROM THE NUMDEG LIST. C IF (IWA2(I) .EQ. 0) THEN IWA1(NUMDEG) = IWA3(I) ELSE IWA3(IWA2(I)) = IWA3(I) END IF IF (IWA3(I) .GT. 0) IWA2(IWA3(I)) = IWA2(I) C C ADD VERTEX I TO THE NUMDEG-1 LIST. C IWA2(I) = 0 IWA3(I) = IWA1(NUMDEG-1) IF (IWA1(NUMDEG-1) .GT. 0) IWA2(IWA1(NUMDEG-1)) = I IWA1(NUMDEG-1) = I END IF 60 CONTINUE C C DETERMINE ALL THE NEIGHBORS OF VERTEX J WHICH SUCCEED J C IN THE SUBGRAPH SPANNED BY THE UN-ORDERED VERTICES. C DO 70 K = NPNTRS(J), NPNTRS(J+1)-1 I = NGHBRS(K) C C UPDATE THE POINTERS TO THE CURRENT DEGREE LISTS. C NUMDEG = -LISTP(I) IF (NUMDEG .GE. 0) THEN LISTP(I) = LISTP(I) + 1 MINDEG = MIN(MINDEG,-LISTP(I)) C C DELETE VERTEX I FROM THE NUMDEG LIST. C IF (IWA2(I) .EQ. 0) THEN IWA1(NUMDEG) = IWA3(I) ELSE IWA3(IWA2(I)) = IWA3(I) END IF IF (IWA3(I) .GT. 0) IWA2(IWA3(I)) = IWA2(I) C C ADD VERTEX I TO THE NUMDEG-1 LIST. C IWA2(I) = 0 IWA3(I) = IWA1(NUMDEG-1) IF (IWA1(NUMDEG-1) .GT. 0) IWA2(IWA1(NUMDEG-1)) = I IWA1(NUMDEG-1) = I END IF 70 CONTINUE C C END OF ITERATION LOOP. C GO TO 30 80 CONTINUE RETURN C C LAST CARD OF SUBROUTINE SLOG. C END SUBROUTINE FDHS(N,INDROW,JPNTR,INDCOL,IPNTR,LISTP,NGRP, * MAXGRP,NUMGRP,ETA,FHESD,FHES,IWA) INTEGER N,MAXGRP,NUMGRP INTEGER INDROW(*),JPNTR(N+1),INDCOL(*),IPNTR(N+1), * LISTP(N),NGRP(N),IWA(N) CD DOUBLE PRECISION ETA(N),FHESD(N),FHES(*) REAL ETA(N),FHESD(N),FHES(*) C ********** C C SUBROUTINE FDHS C C THIS SUBROUTINE COMPUTES AN APPROXIMATION TO THE (SYMMETRIC) C HESSIAN MATRIX OF A FUNCTION BY A SUBSTITUTION METHOD. C THE LOWER TRIANGULAR PART OF THE APPROXIMATION IS STORED C WITH A COLUMN-ORIENTED DEFINITION OF THE SPARSITY PATTERN. C C THIS SUBROUTINE REQUIRES A SYMMETRIC PERMUTATION OF THE C HESSIAN MATRIX AND A PARTITION OF THE COLUMNS OF THE HESSIAN C MATRIX CONSISTENT WITH THE DETERMINATION OF THE HESSIAN C MATRIX BY A LOWER TRIANGULAR SUBSTITUTION METHOD. C THIS INFORMATION CAN BE PROVIDED BY SUBROUTINE DSSM. C C THE SYMMETRIC PERMUTATION OF THE HESSIAN MATRIX IS DEFINED C BY THE ARRAY LISTP. THIS ARRAY IS ONLY USED INTERNALLY. C C THE PARTITION OF THE HESSIAN MATRIX IS DEFINED BY THE ARRAY C NGRP BY SETTING NGRP(J) TO THE GROUP NUMBER OF COLUMN J. C THE USER MUST PROVIDE AN APPROXIMATION TO THE COLUMNS OF C THE HESSIAN MATRIX IN EACH GROUP BY SPECIFYING A DIFFERENCE C PARAMETER VECTOR ETA AND AN APPROXIMATION TO H*D WHERE H IS C THE HESSIAN MATRIX AND THE VECTOR D IS DEFINED BY THE C FOLLOWING SECTION OF CODE. C C DO 10 J = 1, N C D(J) = 0.0 C IF (NGRP(J) .EQ. NUMGRP) D(J) = ETA(J) C 10 CONTINUE C C IN THE ABOVE CODE NUMGRP IS A GROUP NUMBER AND ETA(J) IS THE C DIFFERENCE PARAMETER USED TO APPROXIMATE COLUMN J OF THE C HESSIAN MATRIX. SUITABLE VALUES FOR ETA(J) MUST BE PROVIDED. C C AS MENTIONED ABOVE, AN APPROXIMATION TO H*D MUST BE PROVIDED. C FOR EXAMPLE, IF GRAD(X) IS THE GRADIENT OF THE FUNCTION AT X, C THEN C C GRAD(X+D) - GRAD(X) C C CORRESPONDS TO THE FORWARD DIFFERENCE APPROXIMATION. C C THE LOWER TRIANGULAR SUBSTITUTION METHOD REQUIRES THAT THE C APPROXIMATIONS TO H*D FOR ALL THE GROUPS BE STORED IN SPECIAL C LOCATIONS OF THE ARRAY FHES. THIS IS DONE BY CALLING FDHS C SUCCESSIVELY WITH NUMGRP = 1,2,...,MAXGRP. ON THE CALL WITH C NUMGRP = MAXGRP, THE SUBROUTINE THEN PROCEEDS TO OVERWRITE C FHES WITH THE APPROXIMATION TO THE LOWER TRIANGULAR PART OF C THE HESSIAN MATRIX. C C THE SUBROUTINE STATEMENT IS C C SUBROUTINE FDHS(N,INDROW,JPNTR,INDCOL,IPNTR,LISTP,NGRP, C MAXGRP,NUMGRP,ETA,FHESD,FHES,IWA) C C WHERE C C N IS A POSITIVE INTEGER INPUT VARIABLE SET TO THE ORDER C OF THE HESSIAN MATRIX. C C INDROW IS AN INTEGER INPUT ARRAY WHICH CONTAINS THE ROW C INDICES FOR THE NON-ZEROES IN THE LOWER TRIANGULAR PART C OF THE HESSIAN MATRIX. C C JPNTR IS AN INTEGER INPUT ARRAY OF LENGTH N + 1 WHICH C SPECIFIES THE LOCATIONS OF THE ROW INDICES IN INDROW. C THE ROW INDICES FOR COLUMN J ARE C C INDROW(K), K = JPNTR(J),...,JPNTR(J+1)-1. C C NOTE THAT JPNTR(N+1)-1 IS THEN THE NUMBER OF NON-ZEROES C IN THE LOWER TRIANGULAR PART OF THE HESSIAN MATRIX. C C INDCOL IS AN INTEGER INPUT ARRAY WHICH CONTAINS THE COLUMN C INDICES FOR THE NON-ZEROES IN THE LOWER TRIANGULAR PART C OF THE HESSIAN MATRIX. C C IPNTR IS AN INTEGER INPUT ARRAY OF LENGTH N + 1 WHICH C SPECIFIES THE LOCATIONS OF THE COLUMN INDICES IN INDCOL. C THE COLUMN INDICES FOR ROW I ARE C C INDCOL(K), K = IPNTR(I),...,IPNTR(I+1)-1. C C NOTE THAT IPNTR(N+1)-1 IS THEN THE NUMBER OF NON-ZEROES C IN THE LOWER TRIANGULAR PART OF THE HESSIAN MATRIX. C C LISTP IS AN INTEGER INPUT ARRAY OF LENGTH N WHICH SPECIFIES C THE SYMMETRIC PERMUTATION OF THE HESSIAN MATRIX. ELEMENT C (I,J) OF THE HESSIAN MATRIX IS THE (LISTP(I),LISTP(J)) C ELEMENT OF THE PERMUTED HESSIAN. C C NGRP IS AN INTEGER INPUT ARRAY OF LENGTH N WHICH SPECIFIES C THE PARTITION OF THE COLUMNS OF THE HESSIAN MATRIX. C COLUMN J BELONGS TO GROUP NGRP(J). C C MAXGRP IS A POSITIVE INTEGER INPUT VARIABLE WHICH SPECIFIES C THE NUMBER OF GROUPS IN THE PARTITION OF THE COLUMNS OF C THE HESSIAN MATRIX. C C NUMGRP IS A POSITIVE INTEGER INPUT VARIABLE SET TO A GROUP C NUMBER IN THE PARTITION. C C ETA IS AN INPUT ARRAY OF LENGTH N WHICH CONTAINS THE C DIFFERENCE PARAMETER VECTOR. C C FHESD IS AN INPUT ARRAY OF LENGTH N WHICH CONTAINS AN C APPROXIMATION TO H*D, WHERE H IS THE HESSIAN MATRIX C AND D IS THE DIFFERENCE VECTOR FOR GROUP NUMGRP. C C FHES IS AN OUTPUT ARRAY OF LENGTH NNZ, WHERE NNZ IS THE C NUMBER OF NON-ZERO ELEMENTS IN THE LOWER TRIANGULAR PART C OF THE HESSIAN MATRIX. ON OUTPUT WITH NUMGRP LESS THAN C MAXGRP, THE FHESD ARRAY FOR GROUP NUMGRP HAS BEEN STORED C IN FHES. WHEN NUMGRP = MAXGRP THE SUBROUTINE OVERWRITES C FHES WITH AN APPROXIMATION TO THE LOWER TRIANGULAR PART C OF THE HESSIAN MATRIX. THE APPROXIMATION IS STORED IN C FHES WITH A COLUMN-ORIENTED DEFINITION OF THE SPARSITY C PATTERN. THUS THE ELEMENTS IN COLUMN J OF THE LOWER C TRIANGULAR PART OF THE HESSIAN MATRIX ARE C C FHES(K), K = JPNTR(J),...,JPNTR(J+1)-1, C C AND THE ROW INDICES FOR THESE ELEMENTS ARE C C INDROW(K), K = JPNTR(J),...,JPNTR(J+1)-1. C C IWA IS AN INTEGER WORK ARRAY OF LENGTH N. C C SUBPROGRAMS CALLED C C FORTRAN-SUPPLIED ... ABS C C ARGONNE NATIONAL LABORATORY. MINPACK PROJECT. DECEMBER 1984. C THOMAS F. COLEMAN, BURTON S. GARBOW, JORGE J. MORE' C C ********** INTEGER I,IP,IROW,J,JP,K,L,NUMG,NUML CD DOUBLE PRECISION SUM REAL SUM C C STORE THE I-TH ELEMENT OF GRADIENT DIFFERENCE FHESD C CORRESPONDING TO GROUP NUMGRP IF THERE IS A POSITION C (I,J) SUCH THAT NGRP(J) = NUMGRP AND (I,J) IS MAPPED C ONTO THE LOWER TRIANGULAR PART OF THE PERMUTED MATRIX. C DO 50 J = 1, N IF (NGRP(J) .EQ. NUMGRP) THEN NUML = LISTP(J) DO 30 IP = IPNTR(J), IPNTR(J+1)-1 I = INDCOL(IP) IF (LISTP(I) .GT. NUML) THEN DO 10 JP = JPNTR(I), JPNTR(I+1)-1 IF (INDROW(JP) .EQ. J) THEN FHES(JP) = FHESD(I) GO TO 20 END IF 10 CONTINUE 20 CONTINUE END IF 30 CONTINUE DO 40 JP = JPNTR(J), JPNTR(J+1)-1 I = INDROW(JP) IF (LISTP(I) .GE. NUML) FHES(JP) = FHESD(I) 40 CONTINUE END IF 50 CONTINUE C C EXIT IF THIS IS NOT THE LAST GROUP. C IF (NUMGRP .LT. MAXGRP) RETURN C C MARK ALL COLUMN INDICES J SUCH THAT (I,J) IS MAPPED ONTO C THE LOWER TRIANGULAR PART OF THE PERMUTED MATRIX. C DO 80 I = 1, N NUML = LISTP(I) DO 60 IP = IPNTR(I), IPNTR(I+1)-1 J = INDCOL(IP) IF (NUML .GE. LISTP(J)) INDCOL(IP) = -INDCOL(IP) 60 CONTINUE DO 70 JP = JPNTR(I), JPNTR(I+1)-1 J = INDROW(JP) IF (NUML .GT. LISTP(J)) INDROW(JP) = -INDROW(JP) 70 CONTINUE 80 CONTINUE C C INVERT THE ARRAY LISTP. C DO 90 J = 1, N IWA(LISTP(J)) = J 90 CONTINUE DO 100 J = 1, N LISTP(J) = IWA(J) 100 CONTINUE C C DETERMINE THE LOWER TRIANGULAR PART OF THE ORIGINAL MATRIX. C DO 220 IROW = N, 1, -1 I = LISTP(IROW) C C FIND THE POSITIONS OF THE ELEMENTS IN THE I-TH ROW OF THE C LOWER TRIANGULAR PART OF THE ORIGINAL MATRIX THAT HAVE C ALREADY BEEN DETERMINED. C DO 130 IP = IPNTR(I), IPNTR(I+1)-1 J = INDCOL(IP) IF (J .GT. 0) THEN DO 110 JP = JPNTR(J), JPNTR(J+1)-1 IF (INDROW(JP) .EQ. I) THEN IWA(J) = JP GO TO 120 END IF 110 CONTINUE 120 CONTINUE END IF 130 CONTINUE C C DETERMINE THE ELEMENTS IN THE I-TH ROW OF THE LOWER C TRIANGULAR PART OF THE ORIGINAL MATRIX WHICH GET MAPPED C ONTO THE LOWER TRIANGULAR PART OF THE PERMUTED MATRIX. C DO 180 K = IPNTR(I), IPNTR(I+1)-1 J = -INDCOL(K) IF (J .GT. 0) THEN INDCOL(K) = J C C DETERMINE THE (I,J) ELEMENT. C NUMG = NGRP(J) SUM = 0.0 DO 140 IP = IPNTR(I), IPNTR(I+1)-1 L = ABS(INDCOL(IP)) IF (NGRP(L) .EQ. NUMG .AND. L .NE. J) * SUM = SUM + FHES(IWA(L))*ETA(L) 140 CONTINUE DO 150 JP = JPNTR(I), JPNTR(I+1)-1 L = ABS(INDROW(JP)) IF (NGRP(L) .EQ. NUMG .AND. L .NE. J) * SUM = SUM + FHES(JP)*ETA(L) 150 CONTINUE C C STORE THE (I,J) ELEMENT. C DO 160 JP = JPNTR(J), JPNTR(J+1)-1 IF (INDROW(JP) .EQ. I) THEN FHES(JP) = (FHES(JP) - SUM)/ETA(J) GO TO 170 END IF 160 CONTINUE 170 CONTINUE END IF 180 CONTINUE C C DETERMINE THE ELEMENTS IN THE I-TH ROW OF THE STRICT UPPER C TRIANGULAR PART OF THE ORIGINAL MATRIX WHICH GET MAPPED C ONTO THE LOWER TRIANGULAR PART OF THE PERMUTED MATRIX. C DO 210 K = JPNTR(I), JPNTR(I+1)-1 J = -INDROW(K) IF (J .GT. 0) THEN INDROW(K) = J C C DETERMINE THE (I,J) ELEMENT. C NUMG = NGRP(J) SUM = 0.0 DO 190 IP = IPNTR(I), IPNTR(I+1)-1 L = ABS(INDCOL(IP)) IF (NGRP(L) .EQ. NUMG) * SUM = SUM + FHES(IWA(L))*ETA(L) 190 CONTINUE DO 200 JP = JPNTR(I), JPNTR(I+1)-1 L = ABS(INDROW(JP)) IF (NGRP(L) .EQ. NUMG .AND. L .NE. J) * SUM = SUM + FHES(JP)*ETA(L) 200 CONTINUE C C STORE THE (I,J) ELEMENT. C FHES(K) = (FHES(K) - SUM)/ETA(J) END IF 210 CONTINUE 220 CONTINUE C C RE-INVERT THE ARRAY LISTP. C DO 230 J = 1, N IWA(LISTP(J)) = J 230 CONTINUE DO 240 J = 1, N LISTP(J) = IWA(J) 240 CONTINUE RETURN C C LAST CARD OF SUBROUTINE FDHS. C END SUBROUTINE DEGR(N,INDROW,JPNTR,INDCOL,IPNTR,NDEG,IWA) INTEGER N INTEGER INDROW(*),JPNTR(N+1),INDCOL(*),IPNTR(*),NDEG(N),IWA(N) C ********** C C SUBROUTINE DEGR C C GIVEN THE SPARSITY PATTERN OF AN M BY N MATRIX A, C THIS SUBROUTINE DETERMINES THE DEGREE SEQUENCE FOR C THE INTERSECTION GRAPH OF THE COLUMNS OF A. C C IN GRAPH-THEORY TERMINOLOGY, THE INTERSECTION GRAPH OF C THE COLUMNS OF A IS THE LOOPLESS GRAPH G WITH VERTICES C A(J), J = 1,2,...,N WHERE A(J) IS THE J-TH COLUMN OF A C AND WITH EDGE (A(I),A(J)) IF AND ONLY IF COLUMNS I AND J C HAVE A NON-ZERO IN THE SAME ROW POSITION. C C NOTE THAT THE VALUE OF M IS NOT NEEDED BY DEGR AND IS C THEREFORE NOT PRESENT IN THE SUBROUTINE STATEMENT. C C THE SUBROUTINE STATEMENT IS C C SUBROUTINE DEGR(N,INDROW,JPNTR,INDCOL,IPNTR,NDEG,IWA) C C WHERE C C N IS A POSITIVE INTEGER INPUT VARIABLE SET TO THE NUMBER C OF COLUMNS OF A. C C INDROW IS AN INTEGER INPUT ARRAY WHICH CONTAINS THE ROW C INDICES FOR THE NON-ZEROES IN THE MATRIX A. C C JPNTR IS AN INTEGER INPUT ARRAY OF LENGTH N + 1 WHICH C SPECIFIES THE LOCATIONS OF THE ROW INDICES IN INDROW. C THE ROW INDICES FOR COLUMN J ARE C C INDROW(K), K = JPNTR(J),...,JPNTR(J+1)-1. C C NOTE THAT JPNTR(N+1)-1 IS THEN THE NUMBER OF NON-ZERO C ELEMENTS OF THE MATRIX A. C C INDCOL IS AN INTEGER INPUT ARRAY WHICH CONTAINS THE C COLUMN INDICES FOR THE NON-ZEROES IN THE MATRIX A. C C IPNTR IS AN INTEGER INPUT ARRAY OF LENGTH M + 1 WHICH C SPECIFIES THE LOCATIONS OF THE COLUMN INDICES IN INDCOL. C THE COLUMN INDICES FOR ROW I ARE C C INDCOL(K), K = IPNTR(I),...,IPNTR(I+1)-1. C C NOTE THAT IPNTR(M+1)-1 IS THEN THE NUMBER OF NON-ZERO C ELEMENTS OF THE MATRIX A. C C NDEG IS AN INTEGER OUTPUT ARRAY OF LENGTH N WHICH C SPECIFIES THE DEGREE SEQUENCE. THE DEGREE OF THE C J-TH COLUMN OF A IS NDEG(J). C C IWA IS AN INTEGER WORK ARRAY OF LENGTH N. C C ARGONNE NATIONAL LABORATORY. MINPACK PROJECT. JULY 1983. C THOMAS F. COLEMAN, BURTON S. GARBOW, JORGE J. MORE' C C ********** INTEGER IC,IP,IR,JCOL,JP C C INITIALIZATION BLOCK. C DO 10 JP = 1, N NDEG(JP) = 0 IWA(JP) = 0 10 CONTINUE C C COMPUTE THE DEGREE SEQUENCE BY DETERMINING THE CONTRIBUTIONS C TO THE DEGREES FROM THE CURRENT(JCOL) COLUMN AND FURTHER C COLUMNS WHICH HAVE NOT YET BEEN CONSIDERED. C DO 40 JCOL = 2, N IWA(JCOL) = N C C DETERMINE ALL POSITIONS (IR,JCOL) WHICH CORRESPOND C TO NON-ZEROES IN THE MATRIX. C DO 30 JP = JPNTR(JCOL), JPNTR(JCOL+1)-1 IR = INDROW(JP) C C FOR EACH ROW IR, DETERMINE ALL POSITIONS (IR,IC) C WHICH CORRESPOND TO NON-ZEROES IN THE MATRIX. C DO 20 IP = IPNTR(IR), IPNTR(IR+1)-1 IC = INDCOL(IP) C C ARRAY IWA MARKS COLUMNS WHICH HAVE CONTRIBUTED TO C THE DEGREE COUNT OF COLUMN JCOL. UPDATE THE DEGREE C COUNTS OF THESE COLUMNS AS WELL AS COLUMN JCOL. C IF (IWA(IC) .LT. JCOL) THEN IWA(IC) = JCOL NDEG(IC) = NDEG(IC) + 1 NDEG(JCOL) = NDEG(JCOL) + 1 END IF 20 CONTINUE 30 CONTINUE 40 CONTINUE RETURN C C LAST CARD OF SUBROUTINE DEGR. C END SUBROUTINE IDO(M,N,INDROW,JPNTR,INDCOL,IPNTR,NDEG,LIST, * MAXCLQ,IWA1,IWA2,IWA3,IWA4) INTEGER M,N,MAXCLQ INTEGER INDROW(*),JPNTR(N+1),INDCOL(*),IPNTR(M+1),NDEG(N), * LIST(N),IWA1(0:N-1),IWA2(N),IWA3(N),IWA4(N) C ********** C C SUBROUTINE IDO C C GIVEN THE SPARSITY PATTERN OF AN M BY N MATRIX A, THIS C SUBROUTINE DETERMINES AN INCIDENCE-DEGREE ORDERING OF THE C COLUMNS OF A. C C THE INCIDENCE-DEGREE ORDERING IS DEFINED FOR THE LOOPLESS C GRAPH G WITH VERTICES A(J), J = 1,2,...,N WHERE A(J) IS THE C J-TH COLUMN OF A AND WITH EDGE (A(I),A(J)) IF AND ONLY IF C COLUMNS I AND J HAVE A NON-ZERO IN THE SAME ROW POSITION. C C THE INCIDENCE-DEGREE ORDERING IS DETERMINED RECURSIVELY BY C LETTING LIST(K), K = 1,...,N BE A COLUMN WITH MAXIMAL C INCIDENCE TO THE SUBGRAPH SPANNED BY THE ORDERED COLUMNS. C AMONG ALL THE COLUMNS OF MAXIMAL INCIDENCE, IDO CHOOSES A C COLUMN OF MAXIMAL DEGREE. C C THE SUBROUTINE STATEMENT IS C C SUBROUTINE IDO(M,N,INDROW,JPNTR,INDCOL,IPNTR,NDEG,LIST, C MAXCLQ,IWA1,IWA2,IWA3,IWA4) C C WHERE C C M IS A POSITIVE INTEGER INPUT VARIABLE SET TO THE NUMBER C OF ROWS OF A. C C N IS A POSITIVE INTEGER INPUT VARIABLE SET TO THE NUMBER C OF COLUMNS OF A. C C INDROW IS AN INTEGER INPUT ARRAY WHICH CONTAINS THE ROW C INDICES FOR THE NON-ZEROES IN THE MATRIX A. C C JPNTR IS AN INTEGER INPUT ARRAY OF LENGTH N + 1 WHICH C SPECIFIES THE LOCATIONS OF THE ROW INDICES IN INDROW. C THE ROW INDICES FOR COLUMN J ARE C C INDROW(K), K = JPNTR(J),...,JPNTR(J+1)-1. C C NOTE THAT JPNTR(N+1)-1 IS THEN THE NUMBER OF NON-ZERO C ELEMENTS OF THE MATRIX A. C C INDCOL IS AN INTEGER INPUT ARRAY WHICH CONTAINS THE C COLUMN INDICES FOR THE NON-ZEROES IN THE MATRIX A. C C IPNTR IS AN INTEGER INPUT ARRAY OF LENGTH M + 1 WHICH C SPECIFIES THE LOCATIONS OF THE COLUMN INDICES IN INDCOL. C THE COLUMN INDICES FOR ROW I ARE C C INDCOL(K), K = IPNTR(I),...,IPNTR(I+1)-1. C C NOTE THAT IPNTR(M+1)-1 IS THEN THE NUMBER OF NON-ZERO C ELEMENTS OF THE MATRIX A. C C NDEG IS AN INTEGER INPUT ARRAY OF LENGTH N WHICH SPECIFIES C THE DEGREE SEQUENCE. THE DEGREE OF THE J-TH COLUMN C OF A IS NDEG(J). C C LIST IS AN INTEGER OUTPUT ARRAY OF LENGTH N WHICH SPECIFIES C THE INCIDENCE-DEGREE ORDERING OF THE COLUMNS OF A. THE J-TH C COLUMN IN THIS ORDER IS LIST(J). C C MAXCLQ IS AN INTEGER OUTPUT VARIABLE SET TO THE SIZE C OF THE LARGEST CLIQUE FOUND DURING THE ORDERING. C C IWA1,IWA2,IWA3, AND IWA4 ARE INTEGER WORK ARRAYS OF LENGTH N. C C SUBPROGRAMS CALLED C C MINPACK-SUPPLIED ... NUMSRT C C FORTRAN-SUPPLIED ... MAX C C ARGONNE NATIONAL LABORATORY. MINPACK PROJECT. AUGUST 1984. C THOMAS F. COLEMAN, BURTON S. GARBOW, JORGE J. MORE' C C ********** INTEGER IC,IP,IR,JCOL,JP, * MAXINC,MAXLST,NCOMP,NUMINC,NUMLST,NUMORD,NUMWGT C C SORT THE DEGREE SEQUENCE. C CALL NUMSRT(N,N-1,NDEG,-1,IWA4,IWA2,IWA3) C C INITIALIZATION BLOCK. C C CREATE A DOUBLY-LINKED LIST TO ACCESS THE INCIDENCES OF THE C COLUMNS. THE POINTERS FOR THE LINKED LIST ARE AS FOLLOWS. C C EACH UN-ORDERED COLUMN IC IS IN A LIST (THE INCIDENCE LIST) C OF COLUMNS WITH THE SAME INCIDENCE. C C IWA1(NUMINC) IS THE FIRST COLUMN IN THE NUMINC LIST C UNLESS IWA1(NUMINC) = 0. IN THIS CASE THERE ARE C NO COLUMNS IN THE NUMINC LIST. C C IWA2(IC) IS THE COLUMN BEFORE IC IN THE INCIDENCE LIST C UNLESS IWA2(IC) = 0. IN THIS CASE IC IS THE FIRST C COLUMN IN THIS INCIDENCE LIST. C C IWA3(IC) IS THE COLUMN AFTER IC IN THE INCIDENCE LIST C UNLESS IWA3(IC) = 0. IN THIS CASE IC IS THE LAST C COLUMN IN THIS INCIDENCE LIST. C C IF IC IS AN UN-ORDERED COLUMN, THEN LIST(IC) IS THE C INCIDENCE OF IC TO THE GRAPH INDUCED BY THE ORDERED C COLUMNS. IF JCOL IS AN ORDERED COLUMN, THEN LIST(JCOL) C IS THE INCIDENCE-DEGREE ORDER OF COLUMN JCOL. C MAXINC = 0 DO 10 JP = N, 1, -1 IC = IWA4(JP) IWA1(N-JP) = 0 IWA2(IC) = 0 IWA3(IC) = IWA1(0) IF (IWA1(0) .GT. 0) IWA2(IWA1(0)) = IC IWA1(0) = IC IWA4(JP) = 0 LIST(JP) = 0 10 CONTINUE C C DETERMINE THE MAXIMAL SEARCH LENGTH FOR THE LIST C OF COLUMNS OF MAXIMAL INCIDENCE. C MAXLST = 0 DO 20 IR = 1, M MAXLST = MAXLST + (IPNTR(IR+1) - IPNTR(IR))**2 20 CONTINUE MAXLST = MAXLST/N MAXCLQ = 0 NUMORD = 1 C C BEGINNING OF ITERATION LOOP. C 30 CONTINUE C C CHOOSE A COLUMN JCOL OF MAXIMAL DEGREE AMONG THE C COLUMNS OF MAXIMAL INCIDENCE MAXINC. C 40 CONTINUE JP = IWA1(MAXINC) IF (JP .GT. 0) GO TO 50 MAXINC = MAXINC - 1 GO TO 40 50 CONTINUE NUMWGT = -1 DO 60 NUMLST = 1, MAXLST IF (NDEG(JP) .GT. NUMWGT) THEN NUMWGT = NDEG(JP) JCOL = JP END IF JP = IWA3(JP) IF (JP .LE. 0) GO TO 70 60 CONTINUE 70 CONTINUE LIST(JCOL) = NUMORD C C UPDATE THE SIZE OF THE LARGEST CLIQUE C FOUND DURING THE ORDERING. C IF (MAXINC .EQ. 0) NCOMP = 0 NCOMP = NCOMP + 1 IF (MAXINC+1 .EQ. NCOMP) MAXCLQ = MAX(MAXCLQ,NCOMP) C C TERMINATION TEST. C NUMORD = NUMORD + 1 IF (NUMORD .GT. N) GO TO 100 C C DELETE COLUMN JCOL FROM THE MAXINC LIST. C IF (IWA2(JCOL) .EQ. 0) THEN IWA1(MAXINC) = IWA3(JCOL) ELSE IWA3(IWA2(JCOL)) = IWA3(JCOL) END IF IF (IWA3(JCOL) .GT. 0) IWA2(IWA3(JCOL)) = IWA2(JCOL) C C FIND ALL COLUMNS ADJACENT TO COLUMN JCOL. C IWA4(JCOL) = N C C DETERMINE ALL POSITIONS (IR,JCOL) WHICH CORRESPOND C TO NON-ZEROES IN THE MATRIX. C DO 90 JP = JPNTR(JCOL), JPNTR(JCOL+1)-1 IR = INDROW(JP) C C FOR EACH ROW IR, DETERMINE ALL POSITIONS (IR,IC) C WHICH CORRESPOND TO NON-ZEROES IN THE MATRIX. C DO 80 IP = IPNTR(IR), IPNTR(IR+1)-1 IC = INDCOL(IP) C C ARRAY IWA4 MARKS COLUMNS WHICH ARE ADJACENT TO C COLUMN JCOL. C IF (IWA4(IC) .LT. NUMORD) THEN IWA4(IC) = NUMORD C C UPDATE THE POINTERS TO THE CURRENT INCIDENCE LISTS. C NUMINC = LIST(IC) LIST(IC) = LIST(IC) + 1 MAXINC = MAX(MAXINC,LIST(IC)) C C DELETE COLUMN IC FROM THE NUMINC LIST. C IF (IWA2(IC) .EQ. 0) THEN IWA1(NUMINC) = IWA3(IC) ELSE IWA3(IWA2(IC)) = IWA3(IC) END IF IF (IWA3(IC) .GT. 0) IWA2(IWA3(IC)) = IWA2(IC) C C ADD COLUMN IC TO THE NUMINC+1 LIST. C IWA2(IC) = 0 IWA3(IC) = IWA1(NUMINC+1) IF (IWA1(NUMINC+1) .GT. 0) IWA2(IWA1(NUMINC+1)) = IC IWA1(NUMINC+1) = IC END IF 80 CONTINUE 90 CONTINUE C C END OF ITERATION LOOP. C GO TO 30 100 CONTINUE C C INVERT THE ARRAY LIST. C DO 110 JCOL = 1, N IWA2(LIST(JCOL)) = JCOL 110 CONTINUE DO 120 JP = 1, N LIST(JP) = IWA2(JP) 120 CONTINUE RETURN C C LAST CARD OF SUBROUTINE IDO. C END SUBROUTINE NUMSRT(N,NMAX,NUM,MODE,INDEX,LAST,NEXT) INTEGER N,NMAX,MODE INTEGER NUM(N),INDEX(N),LAST(0:NMAX),NEXT(N) C **********. C C SUBROUTINE NUMSRT C C GIVEN A SEQUENCE OF INTEGERS, THIS SUBROUTINE GROUPS C TOGETHER THOSE INDICES WITH THE SAME SEQUENCE VALUE C AND, OPTIONALLY, SORTS THE SEQUENCE INTO EITHER C ASCENDING OR DESCENDING ORDER. C C THE SEQUENCE OF INTEGERS IS DEFINED BY THE ARRAY NUM, C AND IT IS ASSUMED THAT THE INTEGERS ARE EACH FROM THE SET C 0,1,...,NMAX. ON OUTPUT THE INDICES K SUCH THAT NUM(K) = L C FOR ANY L = 0,1,...,NMAX CAN BE OBTAINED FROM THE ARRAYS C LAST AND NEXT AS FOLLOWS. C C K = LAST(L) C WHILE (K .NE. 0) K = NEXT(K) C C OPTIONALLY, THE SUBROUTINE PRODUCES AN ARRAY INDEX SO THAT C THE SEQUENCE NUM(INDEX(I)), I = 1,2,...,N IS SORTED. C C THE SUBROUTINE STATEMENT IS C C SUBROUTINE NUMSRT(N,NMAX,NUM,MODE,INDEX,LAST,NEXT) C C WHERE C C N IS A POSITIVE INTEGER INPUT VARIABLE. C C NMAX IS A POSITIVE INTEGER INPUT VARIABLE. C C NUM IS AN INPUT ARRAY OF LENGTH N WHICH CONTAINS THE C SEQUENCE OF INTEGERS TO BE GROUPED AND SORTED. IT C IS ASSUMED THAT THE INTEGERS ARE EACH FROM THE SET C 0,1,...,NMAX. C C MODE IS AN INTEGER INPUT VARIABLE. THE SEQUENCE NUM IS C SORTED IN ASCENDING ORDER IF MODE IS POSITIVE AND IN C DESCENDING ORDER IF MODE IS NEGATIVE. IF MODE IS 0, C NO SORTING IS DONE. C C INDEX IS AN INTEGER OUTPUT ARRAY OF LENGTH N SET SO C THAT THE SEQUENCE C C NUM(INDEX(I)), I = 1,2,...,N C C IS SORTED ACCORDING TO THE SETTING OF MODE. IF MODE C IS 0, INDEX IS NOT REFERENCED. C C LAST IS AN INTEGER OUTPUT ARRAY OF LENGTH NMAX + 1. THE C INDEX OF NUM FOR THE LAST OCCURRENCE OF L IS LAST(L) C FOR ANY L = 0,1,...,NMAX UNLESS LAST(L) = 0. IN C THIS CASE L DOES NOT APPEAR IN NUM. C C NEXT IS AN INTEGER OUTPUT ARRAY OF LENGTH N. IF C NUM(K) = L, THEN THE INDEX OF NUM FOR THE PREVIOUS C OCCURRENCE OF L IS NEXT(K) FOR ANY L = 0,1,...,NMAX C UNLESS NEXT(K) = 0. IN THIS CASE THERE IS NO PREVIOUS C OCCURRENCE OF L IN NUM. C C ARGONNE NATIONAL LABORATORY. MINPACK PROJECT. JULY 1983. C THOMAS F. COLEMAN, BURTON S. GARBOW, JORGE J. MORE' C C ********** INTEGER I,J,JINC,JL,JU,K,L C C DETERMINE THE ARRAYS NEXT AND LAST. C DO 10 I = 0, NMAX LAST(I) = 0 10 CONTINUE DO 20 K = 1, N L = NUM(K) NEXT(K) = LAST(L) LAST(L) = K 20 CONTINUE IF (MODE .EQ. 0) RETURN C C STORE THE POINTERS TO THE SORTED ARRAY IN INDEX. C I = 1 IF (MODE .GT. 0) THEN JL = 0 JU = NMAX JINC = 1 ELSE JL = NMAX JU = 0 JINC = -1 END IF DO 50 J = JL, JU, JINC K = LAST(J) 30 CONTINUE IF (K .EQ. 0) GO TO 40 INDEX(I) = K I = I + 1 K = NEXT(K) GO TO 30 40 CONTINUE 50 CONTINUE RETURN C C LAST CARD OF SUBROUTINE NUMSRT. C END SUBROUTINE SEQ(N,INDROW,JPNTR,INDCOL,IPNTR,LIST,NGRP,MAXGRP, * IWA) INTEGER N,MAXGRP INTEGER INDROW(*),JPNTR(N+1),INDCOL(*),IPNTR(*),LIST(N), * NGRP(N),IWA(N) C ********** C C SUBROUTINE SEQ C C GIVEN THE SPARSITY PATTERN OF AN M BY N MATRIX A, THIS C SUBROUTINE DETERMINES A CONSISTENT PARTITION OF THE C COLUMNS OF A BY A SEQUENTIAL ALGORITHM. C C A CONSISTENT PARTITION IS DEFINED IN TERMS OF THE LOOPLESS C GRAPH G WITH VERTICES A(J), J = 1,2,...,N WHERE A(J) IS THE C J-TH COLUMN OF A AND WITH EDGE (A(I),A(J)) IF AND ONLY IF C COLUMNS I AND J HAVE A NON-ZERO IN THE SAME ROW POSITION. C C A PARTITION OF THE COLUMNS OF A INTO GROUPS IS CONSISTENT C IF THE COLUMNS IN ANY GROUP ARE NOT ADJACENT IN THE GRAPH G. C IN GRAPH-THEORY TERMINOLOGY, A CONSISTENT PARTITION OF THE C COLUMNS OF A CORRESPONDS TO A COLORING OF THE GRAPH G. C C THE SUBROUTINE EXAMINES THE COLUMNS IN THE ORDER SPECIFIED C BY THE ARRAY LIST, AND ASSIGNS THE CURRENT COLUMN TO THE C GROUP WITH THE SMALLEST POSSIBLE NUMBER. C C NOTE THAT THE VALUE OF M IS NOT NEEDED BY SEQ AND IS C THEREFORE NOT PRESENT IN THE SUBROUTINE STATEMENT. C C THE SUBROUTINE STATEMENT IS C C SUBROUTINE SEQ(N,INDROW,JPNTR,INDCOL,IPNTR,LIST,NGRP,MAXGRP, C IWA) C C WHERE C C N IS A POSITIVE INTEGER INPUT VARIABLE SET TO THE NUMBER C OF COLUMNS OF A. C C INDROW IS AN INTEGER INPUT ARRAY WHICH CONTAINS THE ROW C INDICES FOR THE NON-ZEROES IN THE MATRIX A. C C JPNTR IS AN INTEGER INPUT ARRAY OF LENGTH N + 1 WHICH C SPECIFIES THE LOCATIONS OF THE ROW INDICES IN INDROW. C THE ROW INDICES FOR COLUMN J ARE C C INDROW(K), K = JPNTR(J),...,JPNTR(J+1)-1. C C NOTE THAT JPNTR(N+1)-1 IS THEN THE NUMBER OF NON-ZERO C ELEMENTS OF THE MATRIX A. C C INDCOL IS AN INTEGER INPUT ARRAY WHICH CONTAINS THE C COLUMN INDICES FOR THE NON-ZEROES IN THE MATRIX A. C C IPNTR IS AN INTEGER INPUT ARRAY OF LENGTH M + 1 WHICH C SPECIFIES THE LOCATIONS OF THE COLUMN INDICES IN INDCOL. C THE COLUMN INDICES FOR ROW I ARE C C INDCOL(K), K = IPNTR(I),...,IPNTR(I+1)-1. C C NOTE THAT IPNTR(M+1)-1 IS THEN THE NUMBER OF NON-ZERO C ELEMENTS OF THE MATRIX A. C C LIST IS AN INTEGER INPUT ARRAY OF LENGTH N WHICH SPECIFIES C THE ORDER TO BE USED BY THE SEQUENTIAL ALGORITHM. C THE J-TH COLUMN IN THIS ORDER IS LIST(J). C C NGRP IS AN INTEGER OUTPUT ARRAY OF LENGTH N WHICH SPECIFIES C THE PARTITION OF THE COLUMNS OF A. COLUMN JCOL BELONGS C TO GROUP NGRP(JCOL). C C MAXGRP IS AN INTEGER OUTPUT VARIABLE WHICH SPECIFIES THE C NUMBER OF GROUPS IN THE PARTITION OF THE COLUMNS OF A. C C IWA IS AN INTEGER WORK ARRAY OF LENGTH N. C C ARGONNE NATIONAL LABORATORY. MINPACK PROJECT. JULY 1983. C THOMAS F. COLEMAN, BURTON S. GARBOW, JORGE J. MORE' C C ********** INTEGER IC,IP,IR,J,JCOL,JP C C INITIALIZATION BLOCK. C MAXGRP = 0 DO 10 JP = 1, N NGRP(JP) = N IWA(JP) = 0 10 CONTINUE C C BEGINNING OF ITERATION LOOP. C DO 60 J = 1, N JCOL = LIST(J) C C FIND ALL COLUMNS ADJACENT TO COLUMN JCOL. C C DETERMINE ALL POSITIONS (IR,JCOL) WHICH CORRESPOND C TO NON-ZEROES IN THE MATRIX. C DO 30 JP = JPNTR(JCOL), JPNTR(JCOL+1)-1 IR = INDROW(JP) C C FOR EACH ROW IR, DETERMINE ALL POSITIONS (IR,IC) C WHICH CORRESPOND TO NON-ZEROES IN THE MATRIX. C DO 20 IP = IPNTR(IR), IPNTR(IR+1)-1 IC = INDCOL(IP) C C ARRAY IWA MARKS THE GROUP NUMBERS OF THE C COLUMNS WHICH ARE ADJACENT TO COLUMN JCOL. C IWA(NGRP(IC)) = J 20 CONTINUE 30 CONTINUE C C ASSIGN THE SMALLEST UN-MARKED GROUP NUMBER TO JCOL. C DO 40 JP = 1, MAXGRP IF (IWA(JP) .NE. J) GO TO 50 40 CONTINUE MAXGRP = MAXGRP + 1 50 CONTINUE NGRP(JCOL) = JP 60 CONTINUE C C END OF ITERATION LOOP. C RETURN C C LAST CARD OF SUBROUTINE SEQ. C END SUBROUTINE SETR(M,N,INDROW,JPNTR,INDCOL,IPNTR,IWA) INTEGER M,N INTEGER INDROW(*),JPNTR(N+1),INDCOL(*),IPNTR(M+1),IWA(M) C ********** C C SUBROUTINE SETR C C GIVEN A COLUMN-ORIENTED DEFINITION OF THE SPARSITY PATTERN C OF AN M BY N MATRIX A, THIS SUBROUTINE DETERMINES A C ROW-ORIENTED DEFINITION OF THE SPARSITY PATTERN OF A. C C ON INPUT THE COLUMN-ORIENTED DEFINITION IS SPECIFIED BY C THE ARRAYS INDROW AND JPNTR. ON OUTPUT THE ROW-ORIENTED C DEFINITION IS SPECIFIED BY THE ARRAYS INDCOL AND IPNTR. C C THE SUBROUTINE STATEMENT IS C C SUBROUTINE SETR(M,N,INDROW,JPNTR,INDCOL,IPNTR,IWA) C C WHERE C C M IS A POSITIVE INTEGER INPUT VARIABLE SET TO THE NUMBER C OF ROWS OF A. C C N IS A POSITIVE INTEGER INPUT VARIABLE SET TO THE NUMBER C OF COLUMNS OF A. C C INDROW IS AN INTEGER INPUT ARRAY WHICH CONTAINS THE ROW C INDICES FOR THE NON-ZEROES IN THE MATRIX A. C C JPNTR IS AN INTEGER INPUT ARRAY OF LENGTH N + 1 WHICH C SPECIFIES THE LOCATIONS OF THE ROW INDICES IN INDROW. C THE ROW INDICES FOR COLUMN J ARE C C INDROW(K), K = JPNTR(J),...,JPNTR(J+1)-1. C C NOTE THAT JPNTR(N+1)-1 IS THEN THE NUMBER OF NON-ZERO C ELEMENTS OF THE MATRIX A. C C INDCOL IS AN INTEGER OUTPUT ARRAY WHICH CONTAINS THE C COLUMN INDICES FOR THE NON-ZEROES IN THE MATRIX A. C C IPNTR IS AN INTEGER OUTPUT ARRAY OF LENGTH M + 1 WHICH C SPECIFIES THE LOCATIONS OF THE COLUMN INDICES IN INDCOL. C THE COLUMN INDICES FOR ROW I ARE C C INDCOL(K), K = IPNTR(I),...,IPNTR(I+1)-1. C C NOTE THAT IPNTR(1) IS SET TO 1 AND THAT IPNTR(M+1)-1 IS C THEN THE NUMBER OF NON-ZERO ELEMENTS OF THE MATRIX A. C C IWA IS AN INTEGER WORK ARRAY OF LENGTH M. C C ARGONNE NATIONAL LABORATORY. MINPACK PROJECT. JULY 1983. C THOMAS F. COLEMAN, BURTON S. GARBOW, JORGE J. MORE' C C ********** INTEGER IR,JCOL,JP C C STORE IN ARRAY IWA THE COUNTS OF NON-ZEROES IN THE ROWS. C DO 10 IR = 1, M IWA(IR) = 0 10 CONTINUE DO 20 JP = 1, JPNTR(N+1)-1 IWA(INDROW(JP)) = IWA(INDROW(JP)) + 1 20 CONTINUE C C SET POINTERS TO THE START OF THE ROWS IN INDCOL. C IPNTR(1) = 1 DO 30 IR = 1, M IPNTR(IR+1) = IPNTR(IR) + IWA(IR) IWA(IR) = IPNTR(IR) 30 CONTINUE C C FILL INDCOL. C DO 50 JCOL = 1, N DO 40 JP = JPNTR(JCOL), JPNTR(JCOL+1)-1 IR = INDROW(JP) INDCOL(IWA(IR)) = JCOL IWA(IR) = IWA(IR) + 1 40 CONTINUE 50 CONTINUE RETURN C C LAST CARD OF SUBROUTINE SETR. C END SUBROUTINE SLO(N,INDROW,JPNTR,INDCOL,IPNTR,NDEG,LIST, * MAXCLQ,IWA1,IWA2,IWA3,IWA4) INTEGER N,MAXCLQ INTEGER INDROW(*),JPNTR(N+1),INDCOL(*),IPNTR(*),NDEG(N), * LIST(N),IWA1(0:N-1),IWA2(N),IWA3(N),IWA4(N) C ********** C C SUBROUTINE SLO C C GIVEN THE SPARSITY PATTERN OF AN M BY N MATRIX A, THIS C SUBROUTINE DETERMINES THE SMALLEST-LAST ORDERING OF THE C COLUMNS OF A. C C THE SMALLEST-LAST ORDERING IS DEFINED FOR THE LOOPLESS C GRAPH G WITH VERTICES A(J), J = 1,2,...,N WHERE A(J) IS THE C J-TH COLUMN OF A AND WITH EDGE (A(I),A(J)) IF AND ONLY IF C COLUMNS I AND J HAVE A NON-ZERO IN THE SAME ROW POSITION. C C THE SMALLEST-LAST ORDERING IS DETERMINED RECURSIVELY BY C LETTING LIST(K), K = N,...,1 BE A COLUMN WITH LEAST DEGREE C IN THE SUBGRAPH SPANNED BY THE UN-ORDERED COLUMNS. C C NOTE THAT THE VALUE OF M IS NOT NEEDED BY SLO AND IS C THEREFORE NOT PRESENT IN THE SUBROUTINE STATEMENT. C C THE SUBROUTINE STATEMENT IS C C SUBROUTINE SLO(N,INDROW,JPNTR,INDCOL,IPNTR,NDEG,LIST, C MAXCLQ,IWA1,IWA2,IWA3,IWA4) C C WHERE C C N IS A POSITIVE INTEGER INPUT VARIABLE SET TO THE NUMBER C OF COLUMNS OF A. C C INDROW IS AN INTEGER INPUT ARRAY WHICH CONTAINS THE ROW C INDICES FOR THE NON-ZEROES IN THE MATRIX A. C C JPNTR IS AN INTEGER INPUT ARRAY OF LENGTH N + 1 WHICH C SPECIFIES THE LOCATIONS OF THE ROW INDICES IN INDROW. C THE ROW INDICES FOR COLUMN J ARE C C INDROW(K), K = JPNTR(J),...,JPNTR(J+1)-1. C C NOTE THAT JPNTR(N+1)-1 IS THEN THE NUMBER OF NON-ZERO C ELEMENTS OF THE MATRIX A. C C INDCOL IS AN INTEGER INPUT ARRAY WHICH CONTAINS THE C COLUMN INDICES FOR THE NON-ZEROES IN THE MATRIX A. C C IPNTR IS AN INTEGER INPUT ARRAY OF LENGTH M + 1 WHICH C SPECIFIES THE LOCATIONS OF THE COLUMN INDICES IN INDCOL. C THE COLUMN INDICES FOR ROW I ARE C C INDCOL(K), K = IPNTR(I),...,IPNTR(I+1)-1. C C NOTE THAT IPNTR(M+1)-1 IS THEN THE NUMBER OF NON-ZERO C ELEMENTS OF THE MATRIX A. C C NDEG IS AN INTEGER INPUT ARRAY OF LENGTH N WHICH SPECIFIES C THE DEGREE SEQUENCE. THE DEGREE OF THE J-TH COLUMN C OF A IS NDEG(J). C C LIST IS AN INTEGER OUTPUT ARRAY OF LENGTH N WHICH SPECIFIES C THE SMALLEST-LAST ORDERING OF THE COLUMNS OF A. THE J-TH C COLUMN IN THIS ORDER IS LIST(J). C C MAXCLQ IS AN INTEGER OUTPUT VARIABLE SET TO THE SIZE C OF THE LARGEST CLIQUE FOUND DURING THE ORDERING. C C IWA1,IWA2,IWA3, AND IWA4 ARE INTEGER WORK ARRAYS OF LENGTH N. C C SUBPROGRAMS CALLED C C FORTRAN-SUPPLIED ... MIN C C ARGONNE NATIONAL LABORATORY. MINPACK PROJECT. AUGUST 1984. C THOMAS F. COLEMAN, BURTON S. GARBOW, JORGE J. MORE' C C ********** INTEGER IC,IP,IR,JCOL,JP,MINDEG,NUMDEG,NUMORD C C INITIALIZATION BLOCK. C MINDEG = N DO 10 JP = 1, N IWA1(JP-1) = 0 IWA4(JP) = N LIST(JP) = NDEG(JP) MINDEG = MIN(MINDEG,NDEG(JP)) 10 CONTINUE C C CREATE A DOUBLY-LINKED LIST TO ACCESS THE DEGREES OF THE C COLUMNS. THE POINTERS FOR THE LINKED LIST ARE AS FOLLOWS. C C EACH UN-ORDERED COLUMN IC IS IN A LIST (THE DEGREE LIST) C OF COLUMNS WITH THE SAME DEGREE. C C IWA1(NUMDEG) IS THE FIRST COLUMN IN THE NUMDEG LIST C UNLESS IWA1(NUMDEG) = 0. IN THIS CASE THERE ARE C NO COLUMNS IN THE NUMDEG LIST. C C IWA2(IC) IS THE COLUMN BEFORE IC IN THE DEGREE LIST C UNLESS IWA2(IC) = 0. IN THIS CASE IC IS THE FIRST C COLUMN IN THIS DEGREE LIST. C C IWA3(IC) IS THE COLUMN AFTER IC IN THE DEGREE LIST C UNLESS IWA3(IC) = 0. IN THIS CASE IC IS THE LAST C COLUMN IN THIS DEGREE LIST. C C IF IC IS AN UN-ORDERED COLUMN, THEN LIST(IC) IS THE C DEGREE OF IC IN THE GRAPH INDUCED BY THE UN-ORDERED C COLUMNS. IF JCOL IS AN ORDERED COLUMN, THEN LIST(JCOL) C IS THE SMALLEST-LAST ORDER OF COLUMN JCOL. C DO 20 JP = 1, N NUMDEG = NDEG(JP) IWA2(JP) = 0 IWA3(JP) = IWA1(NUMDEG) IF (IWA1(NUMDEG) .GT. 0) IWA2(IWA1(NUMDEG)) = JP IWA1(NUMDEG) = JP 20 CONTINUE MAXCLQ = 0 NUMORD = N C C BEGINNING OF ITERATION LOOP. C 30 CONTINUE C C CHOOSE A COLUMN JCOL OF MINIMAL DEGREE MINDEG. C 40 CONTINUE JCOL = IWA1(MINDEG) IF (JCOL .GT. 0) GO TO 50 MINDEG = MINDEG + 1 GO TO 40 50 CONTINUE LIST(JCOL) = NUMORD C C MARK THE SIZE OF THE LARGEST CLIQUE C FOUND DURING THE ORDERING. C IF (MINDEG+1 .EQ. NUMORD .AND. MAXCLQ .EQ. 0) * MAXCLQ = NUMORD C C TERMINATION TEST. C NUMORD = NUMORD - 1 IF (NUMORD .EQ. 0) GO TO 80 C C DELETE COLUMN JCOL FROM THE MINDEG LIST. C IWA1(MINDEG) = IWA3(JCOL) IF (IWA3(JCOL) .GT. 0) IWA2(IWA3(JCOL)) = 0 C C FIND ALL COLUMNS ADJACENT TO COLUMN JCOL. C IWA4(JCOL) = 0 C C DETERMINE ALL POSITIONS (IR,JCOL) WHICH CORRESPOND C TO NON-ZEROES IN THE MATRIX. C DO 70 JP = JPNTR(JCOL), JPNTR(JCOL+1)-1 IR = INDROW(JP) C C FOR EACH ROW IR, DETERMINE ALL POSITIONS (IR,IC) C WHICH CORRESPOND TO NON-ZEROES IN THE MATRIX. C DO 60 IP = IPNTR(IR), IPNTR(IR+1)-1 IC = INDCOL(IP) C C ARRAY IWA4 MARKS COLUMNS WHICH ARE ADJACENT TO C COLUMN JCOL. C IF (IWA4(IC) .GT. NUMORD) THEN IWA4(IC) = NUMORD C C UPDATE THE POINTERS TO THE CURRENT DEGREE LISTS. C NUMDEG = LIST(IC) LIST(IC) = LIST(IC) - 1 MINDEG = MIN(MINDEG,LIST(IC)) C C DELETE COLUMN IC FROM THE NUMDEG LIST. C IF (IWA2(IC) .EQ. 0) THEN IWA1(NUMDEG) = IWA3(IC) ELSE IWA3(IWA2(IC)) = IWA3(IC) END IF IF (IWA3(IC) .GT. 0) IWA2(IWA3(IC)) = IWA2(IC) C C ADD COLUMN IC TO THE NUMDEG-1 LIST. C IWA2(IC) = 0 IWA3(IC) = IWA1(NUMDEG-1) IF (IWA1(NUMDEG-1) .GT. 0) IWA2(IWA1(NUMDEG-1)) = IC IWA1(NUMDEG-1) = IC END IF 60 CONTINUE 70 CONTINUE C C END OF ITERATION LOOP. C GO TO 30 80 CONTINUE C C INVERT THE ARRAY LIST. C DO 90 JCOL = 1, N IWA2(LIST(JCOL)) = JCOL 90 CONTINUE DO 100 JP = 1, N LIST(JP) = IWA2(JP) 100 CONTINUE RETURN C C LAST CARD OF SUBROUTINE SLO. C END SUBROUTINE SRTDAT(N,NNZ,INDROW,INDCOL,JPNTR,IWA) INTEGER N,NNZ INTEGER INDROW(NNZ),INDCOL(NNZ),JPNTR(N+1),IWA(N) C ********** C C SUBROUTINE SRTDAT C C GIVEN THE NON-ZERO ELEMENTS OF AN M BY N MATRIX A IN C ARBITRARY ORDER AS SPECIFIED BY THEIR ROW AND COLUMN C INDICES, THIS SUBROUTINE PERMUTES THESE ELEMENTS SO C THAT THEIR COLUMN INDICES ARE IN NON-DECREASING ORDER. C C ON INPUT IT IS ASSUMED THAT THE ELEMENTS ARE SPECIFIED IN C C INDROW(K),INDCOL(K), K = 1,...,NNZ. C C ON OUTPUT THE ELEMENTS ARE PERMUTED SO THAT INDCOL IS C IN NON-DECREASING ORDER. IN ADDITION, THE ARRAY JPNTR C IS SET SO THAT THE ROW INDICES FOR COLUMN J ARE C C INDROW(K), K = JPNTR(J),...,JPNTR(J+1)-1. C C NOTE THAT THE VALUE OF M IS NOT NEEDED BY SRTDAT AND IS C THEREFORE NOT PRESENT IN THE SUBROUTINE STATEMENT. C C THE SUBROUTINE STATEMENT IS C C SUBROUTINE SRTDAT(N,NNZ,INDROW,INDCOL,JPNTR,IWA) C C WHERE C C N IS A POSITIVE INTEGER INPUT VARIABLE SET TO THE NUMBER C OF COLUMNS OF A. C C NNZ IS A POSITIVE INTEGER INPUT VARIABLE SET TO THE NUMBER C OF NON-ZERO ELEMENTS OF A. C C INDROW IS AN INTEGER ARRAY OF LENGTH NNZ. ON INPUT INDROW C MUST CONTAIN THE ROW INDICES OF THE NON-ZERO ELEMENTS OF A. C ON OUTPUT INDROW IS PERMUTED SO THAT THE CORRESPONDING C COLUMN INDICES OF INDCOL ARE IN NON-DECREASING ORDER. C C INDCOL IS AN INTEGER ARRAY OF LENGTH NNZ. ON INPUT INDCOL C MUST CONTAIN THE COLUMN INDICES OF THE NON-ZERO ELEMENTS C OF A. ON OUTPUT INDCOL IS PERMUTED SO THAT THESE INDICES C ARE IN NON-DECREASING ORDER. C C JPNTR IS AN INTEGER OUTPUT ARRAY OF LENGTH N + 1 WHICH C SPECIFIES THE LOCATIONS OF THE ROW INDICES IN THE OUTPUT C INDROW. THE ROW INDICES FOR COLUMN J ARE C C INDROW(K), K = JPNTR(J),...,JPNTR(J+1)-1. C C NOTE THAT JPNTR(1) IS SET TO 1 AND THAT JPNTR(N+1)-1 C IS THEN NNZ. C C IWA IS AN INTEGER WORK ARRAY OF LENGTH N. C C SUBPROGRAMS CALLED C C FORTRAN-SUPPLIED ... MAX C C ARGONNE NATIONAL LABORATORY. MINPACK PROJECT. JULY 1983. C THOMAS F. COLEMAN, BURTON S. GARBOW, JORGE J. MORE' C C ********** INTEGER I,J,K,L C C STORE IN ARRAY IWA THE COUNTS OF NON-ZEROES IN THE COLUMNS. C DO 10 J = 1, N IWA(J) = 0 10 CONTINUE DO 20 K = 1, NNZ IWA(INDCOL(K)) = IWA(INDCOL(K)) + 1 20 CONTINUE C C SET POINTERS TO THE START OF THE COLUMNS IN INDROW. C JPNTR(1) = 1 DO 30 J = 1, N JPNTR(J+1) = JPNTR(J) + IWA(J) IWA(J) = JPNTR(J) 30 CONTINUE K = 1 C C BEGIN IN-PLACE SORT. C 40 CONTINUE J = INDCOL(K) IF (K .GE. JPNTR(J)) THEN C C CURRENT ELEMENT IS IN POSITION. NOW EXAMINE THE C NEXT ELEMENT OR THE FIRST UN-SORTED ELEMENT IN C THE J-TH GROUP. C K = MAX(K+1,IWA(J)) ELSE C C CURRENT ELEMENT IS NOT IN POSITION. PLACE ELEMENT C IN POSITION AND MAKE THE DISPLACED ELEMENT THE C CURRENT ELEMENT. C L = IWA(J) IWA(J) = IWA(J) + 1 I = INDROW(K) INDROW(K) = INDROW(L) INDCOL(K) = INDCOL(L) INDROW(L) = I INDCOL(L) = J END IF IF (K .LE. NNZ) GO TO 40 RETURN C C LAST CARD OF SUBROUTINE SRTDAT. C END SHAR_EOF fi # end of overwriting check if test -f 'blas1.f' then echo shar: will not over-write existing file "'blas1.f'" else cat << \SHAR_EOF > 'blas1.f' SUBROUTINE SCOPY(N,DX,INCX,DY,INCY) CD SUBROUTINE DCOPY(N,DX,INCX,DY,INCY) C C COPIES A VECTOR, X, TO A VECTOR, Y. C USES UNROLLED LOOPS FOR INCREMENTS EQUAL TO ONE. C JACK DONGARRA, LINPACK, 3/11/78. C INTEGER I,INCX,INCY,IX,IY,M,MP1,N CD DOUBLE PRECISION DX(*),DY(*) REAL DX(*),DY(*) C IF(N.LE.0)RETURN IF(INCX.EQ.1.AND.INCY.EQ.1)GO TO 20 C C CODE FOR UNEQUAL INCREMENTS OR EQUAL INCREMENTS C NOT EQUAL TO 1 C IX = 1 IY = 1 IF(INCX.LT.0)IX = (-N+1)*INCX + 1 IF(INCY.LT.0)IY = (-N+1)*INCY + 1 DO 10 I = 1,N DY(IY) = DX(IX) IX = IX + INCX IY = IY + INCY 10 CONTINUE RETURN C C CODE FOR BOTH INCREMENTS EQUAL TO 1 C C C CLEAN-UP LOOP C 20 M = MOD(N,7) IF( M .EQ. 0 ) GO TO 40 DO 30 I = 1,M DY(I) = DX(I) 30 CONTINUE IF( N .LT. 7 ) RETURN 40 MP1 = M + 1 DO 50 I = MP1,N,7 DY(I) = DX(I) DY(I + 1) = DX(I + 1) DY(I + 2) = DX(I + 2) DY(I + 3) = DX(I + 3) DY(I + 4) = DX(I + 4) DY(I + 5) = DX(I + 5) DY(I + 6) = DX(I + 6) 50 CONTINUE RETURN END REAL FUNCTION SDOT(N,DX,INCX,DY,INCY) CD DOUBLE PRECISION FUNCTION DDOT(N,DX,INCX,DY,INCY) C C FORMS THE DOT PRODUCT OF TWO VECTORS. C USES UNROLLED LOOPS FOR INCREMENTS EQUAL TO ONE. C JACK DONGARRA, LINPACK, 3/11/78. C INTEGER I,INCX,INCY,IX,IY,M,MP1,N CD DOUBLE PRECISION DX(*),DY(*),DTEMP,ZERO REAL DX(*),DY(*),DTEMP,ZERO PARAMETER ( ZERO = 0.0E+0 ) CD PARAMETER ( ZERO = 0.0D+0 ) C SDOT = ZERO CD DDOT = ZERO DTEMP = ZERO IF(N.LE.0)RETURN IF(INCX.EQ.1.AND.INCY.EQ.1)GO TO 20 C C CODE FOR UNEQUAL INCREMENTS OR EQUAL INCREMENTS C NOT EQUAL TO 1 C IX = 1 IY = 1 IF(INCX.LT.0)IX = (-N+1)*INCX + 1 IF(INCY.LT.0)IY = (-N+1)*INCY + 1 DO 10 I = 1,N DTEMP = DTEMP + DX(IX)*DY(IY) IX = IX + INCX IY = IY + INCY 10 CONTINUE SDOT = DTEMP CD DDOT = DTEMP RETURN C C CODE FOR BOTH INCREMENTS EQUAL TO 1 C C C CLEAN-UP LOOP C 20 M = MOD(N,5) IF( M .EQ. 0 ) GO TO 40 DO 30 I = 1,M DTEMP = DTEMP + DX(I)*DY(I) 30 CONTINUE IF( N .LT. 5 ) GO TO 60 40 MP1 = M + 1 DO 50 I = MP1,N,5 DTEMP = DTEMP + DX(I)*DY(I) + DX(I + 1)*DY(I + 1) + * DX(I + 2)*DY(I + 2) + DX(I + 3)*DY(I + 3) + DX(I + 4)*DY(I + 4) 50 CONTINUE 60 CONTINUE SDOT = DTEMP CD DDOT = DTEMP RETURN END REAL FUNCTION SNRM2 ( N, DX, INCX) CD DOUBLE PRECISION FUNCTION DNRM2 ( N, DX, INCX) INTEGER N, INCX, NEXT INTEGER I, J, NN CD DOUBLE PRECISION DX( * ), CUTLO, CUTHI, HITEST, SUM, CD * XMAX, ZERO, ONE REAL DX( * ), CUTLO, CUTHI, HITEST, SUM, * XMAX, ZERO, ONE INTRINSIC ABS, SQRT PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 ) CD PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) C C EUCLIDEAN NORM OF THE N-VECTOR STORED IN DX() WITH STORAGE C INCREMENT INCX . C IF N .LE. 0 RETURN WITH RESULT = 0. C IF N .GE. 1 THEN INCX MUST BE .GE. 1 C C C.L.LAWSON, 1978 JAN 08 C C FOUR PHASE METHOD USING TWO BUILT-IN CONSTANTS THAT ARE C HOPEFULLY APPLICABLE TO ALL MACHINES. C CUTLO = MAXIMUM OF DSQRT(U/EPS) OVER ALL KNOWN MACHINES. C CUTHI = MINIMUM OF DSQRT(V) OVER ALL KNOWN MACHINES. C WHERE C EPS = SMALLEST NO. SUCH THAT EPS + 1. .GT. 1. C U = SMALLEST POSITIVE NO. (UNDERFLOW LIMIT) C V = LARGEST NO. (OVERFLOW LIMIT) C C BRIEF OUTLINE OF ALGORITHM.. C C PHASE 1 SCANS ZERO COMPONENTS. C MOVE TO PHASE 2 WHEN A COMPONENT IS NONZERO AND .LE. CUTLO C MOVE TO PHASE 3 WHEN A COMPONENT IS .GT. CUTLO C MOVE TO PHASE 4 WHEN A COMPONENT IS .GE. CUTHI/M C WHERE M = N FOR X() REAL AND M = 2*N FOR COMPLEX. C C VALUES FOR CUTLO AND CUTHI.. C FROM THE ENVIRONMENTAL PARAMETERS LISTED IN THE IMSL CONVERTER C DOCUMENT THE LIMITING VALUES ARE AS FOLLOWS.. C CUTLO, S.P. U/EPS = 2**(-102) FOR HONEYWELL. CLOSE SECONDS ARE C UNIVAC AND DEC AT 2**(-103) C THUS CUTLO = 2**(-51) = 4.44089E-16 C CUTHI, S.P. V = 2**127 FOR UNIVAC, HONEYWELL, AND DEC. C THUS CUTHI = 2**(63.5) = 1.30438E19 C CUTLO, D.P. U/EPS = 2**(-67) FOR HONEYWELL AND DEC. C THUS CUTLO = 2**(-33.5) = 8.23181D-11 C CUTHI, D.P. SAME AS S.P. CUTHI = 1.30438D19 DATA CUTLO, CUTHI / 4.441E-16, 1.304E19 / CD DATA CUTLO, CUTHI / 8.232D-11, 1.304D19 / C IF ( N .LE. 0) THEN SNRM2 = ZERO CD DNRM2 = ZERO ELSE NEXT = 1 SUM = ZERO NN = N * INCX C C BEGIN MAIN LOOP C I = 1 20 CONTINUE GO TO ( 30, 50, 70, 110 ), NEXT 30 CONTINUE IF( ABS( DX( I ) ) .GT. CUTLO ) GO TO 85 NEXT = 2 XMAX = ZERO C C PHASE 1. SUM IS ZERO C 50 CONTINUE IF ( DX( I ) .EQ. ZERO ) GO TO 200 IF ( ABS( DX( I ) ) .GT. CUTLO ) GO TO 85 C C PREPARE FOR PHASE 2. C NEXT = 3 GO TO 105 C C PREPARE FOR PHASE 4. C 100 CONTINUE I = J NEXT = 4 SUM = ( SUM / DX( I ) ) / DX( I ) 105 CONTINUE XMAX = ABS( DX( I ) ) SUM = SUM + ( DX( I ) / XMAX ) ** 2 GO TO 200 C C PHASE 2. SUM IS SMALL. SCALE TO AVOID DESTRUCTIVE UNDERFLOW. C 70 CONTINUE IF ( ABS( DX( I ) ) .GT. CUTLO ) THEN C C PREPARE FOR PHASE 3. C SUM = ( SUM * XMAX) * XMAX GO TO 85 ENDIF C C COMMON CODE FOR PHASES 2 AND 4. C IN PHASE 4 SUM IS LARGE. SCALE TO AVOID OVERFLOW. C 110 CONTINUE IF ( ABS( DX( I ) ) .GT. XMAX ) THEN SUM = ONE + SUM * ( XMAX / DX( I ) ) ** 2 XMAX = ABS( DX( I ) ) ELSE SUM = SUM + ( DX( I ) / XMAX ) ** 2 ENDIF 200 CONTINUE I = I + INCX IF ( I .LE. NN ) GO TO 20 C C END OF MAIN LOOP. C C COMPUTE SQUARE ROOT AND ADJUST FOR SCALING. C SNRM2 = XMAX * SQRT(SUM) CD DNRM2 = XMAX * SQRT(SUM) GO TO 300 C C FOR REAL OR D.P. SET HITEST = CUTHI/N C 85 CONTINUE HITEST = CUTHI/FLOAT( N ) C C PHASE 3. SUM IS MID-RANGE. NO SCALING. C DO 95 J = I, NN, INCX IF( ABS( DX( J ) ) .GE. HITEST ) GO TO 100 SUM = SUM + DX( J ) ** 2 95 CONTINUE SNRM2 = SQRT( SUM ) CD DNRM2 = SQRT( SUM ) ENDIF 300 CONTINUE RETURN END SUBROUTINE SSCAL(N,DA,DX,INCX) CD SUBROUTINE DSCAL(N,DA,DX,INCX) C C SCALES A VECTOR BY A CONSTANT. C USES UNROLLED LOOPS FOR INCREMENT EQUAL TO ONE. C JACK DONGARRA, LINPACK, 3/11/78. C INTEGER I,INCX,M,MP1,N,NINCX CD DOUBLE PRECISION DA,DX(1) REAL DA,DX(1) C IF(N.LE.0)RETURN IF(INCX.EQ.1)GO TO 20 C C CODE FOR INCREMENT NOT EQUAL TO 1 C NINCX = N*INCX DO 10 I = 1,NINCX,INCX DX(I) = DA*DX(I) 10 CONTINUE RETURN C C CODE FOR INCREMENT EQUAL TO 1 C C C CLEAN-UP LOOP C 20 M = MOD(N,5) IF( M .EQ. 0 ) GO TO 40 DO 30 I = 1,M DX(I) = DA*DX(I) 30 CONTINUE IF( N .LT. 5 ) RETURN 40 MP1 = M + 1 DO 50 I = MP1,N,5 DX(I) = DA*DX(I) DX(I + 1) = DA*DX(I + 1) DX(I + 2) = DA*DX(I + 2) DX(I + 3) = DA*DX(I + 3) DX(I + 4) = DA*DX(I + 4) 50 CONTINUE RETURN END SHAR_EOF fi # end of overwriting check cd .. if test ! -d 'Dp' then mkdir 'Dp' fi cd 'Dp' if test -f 'stenmin.f' then echo shar: will not over-write existing file "'stenmin.f'" else cat << \SHAR_EOF > 'stenmin.f' * * STENMIN: A SOFTWARE PACKAGE FOR LARGE, SPARSE UNCONSTRAINED * OPTIMIZATION USING TENSOR METHODS. * * * AUTHOR: * ------ * ALI BOUARICHA * ARGONNE NATIONAL LABORATORY * MCS DIVISION * 9700 SOUTH CASS AVENUE * ARGONNE, IL 60439 * E-MAIL: BOUARICH@MCS.ANL.GOV * * DATE: OCTOBER, 1994 * ---- * * PURPOSE: * ------- * * STENMIN MINIMIZES AN UNCONSTRAINED NONLINEAR FUNCTION IN N UNKNOWNS * WHERE THE HESSIAN IS LARGE AND SPARSE USING A NEW CLASS OF METHODS * CALLED TENSOR METHODS. THE SOFTWARE ALLOWS THE USER TO SELECT BETWEEN * A TENSOR METHOD AND A STANDARD METHOD BASED UPON A QUADRATIC MODEL. * THE TENSOR METHOD MODELS THE OBJECTIVE FUNCTION BY A FOURTH-ORDER * MODEL, WHERE THE THIRD- AND FOURTH-ORDER TERMS ARE CHOSEN SUCH THAT * THE EXTRA COST OF FORMING AND SOLVING THE MODEL IS SMALL. THE TEST * RESULTS INDICATE THAT, IN GENERAL, THE TENSOR METHOD IS OFTEN MORE * EFFICIENT AND MORE RELIABLE THAN THE STANDARD nEWTON METHOD FOR * SOLVING LARGE, SPARSE UNCONSTRAINED OPTIMIZATION PROBLEMS. IT IS * ESPECIALLY USEFUL ON PROBLEMS WHERE THE HESSIAN AT THE SOLUTION IS * SINGULAR. * * TWO INTERFACES ARE PROVIDED WITH STENMIN. IF THE USER WISHES TO USE * ALL THE DEFAULTS OPTIONS PROVIDED BY THE PACKAGE, THEN HE (OR SHE) * SHOULD CALL STUMSD (STUMSS IF SINGLE-PRECISION IS USED). THE OTHER * INTERFACE, STUMCD (STUMCS IF SINGLE-PRECISION IS USED), REQUIRES THE * USER TO SUPPLY ALL PARAMETERS. THE USER MAY SPECIFY SELECTED * PARAMETERS ONLY BY FIRST INVOKING THE SUBROUTINE STDFLT, WHICH SETS * ALL PARAMETERS TO THEIR DEFAULT VALUES, AND THEN OVERRIDING ONLY THE * DESIRED VALUES. * * * LIST OF SUBROUTINE AND FUNCTION NAMES CALLED BY STENMIN: * * STCHKG,STCHKH,STCHKI,STCHKS,STCZ3P,STDFLT,STDRUO,STDUGR,STDUSH, * STFDGR,STFDHS,STFTSM,STHMUV,STLSCH,STMA27,STMNEC,STMSLV,STRSLT, * STSORT,STUMCD,STUMSD. * * PACKAGES CALLED BY STENMIN: * * * MA27 (I. S. DUFF AND J. K. REID, "MA27: A SET OF FORTRAN SUBROUTINES * FOR SOLVING SPARSE SYMMETRIC SETS OF LINEAR EQUATIONS", TECH. REP. * R-10533, AERE HARWELL LABORATORY, HARWELL, UK, 1983.) * * DSSM (T. F. COLEMAN, B. S. GARBOW, and J. J. MORE', "SOFTWARE FOR * ESTIMATING SPARSE HESSIAN MATRICES", ACM TRANS. MATH. SOFTW., 11: * 363-377,1985.) * * DSYPRC FROM LANCELOT (A. R. CONN, N. I. M. GOULD, AND C PH. L. TOINT, * "LANCELOT", SPRINGER SERIES IN COMPUTATIONAL MATHEMATICS, * SPRINGER-VERLAG, 1992.) * * D1MACH (FOX P.A., HALL A.D., SCHRYER N.L., FRAMEWORK FOR A PORTABLE * LIBRARY, ACM TRANS. MATH. SOFTW., 4:177-188, 1978.) * * BLAS CALLED BY STENMIN: * * LEVEL 1 BLAS: DCOPY,DDOT,DNRM2,DSCAL * SUBROUTINE STCHKG(N,X,FCN,F,G,TYPX,RNF,ANALTL,WRK1,MSG,IPR) * * PURPOSE: * ------- * * THIS ROUTINE CHECKS THE USER'S ANALYTIC GRADIENT AGAINST A * FINITE-DIFFERENCE GRADIENT. * * PARAMETERS: * ---------- * * N --> DIMENSION OF PROBLEM * X --> CURRENT ITERATE * FCN --> NAME OF SUBROUTINE THAT EVALUATES THE * OPTIMIZATION FUNCTION * F --> FUNCTION VALUE AT X * G --> GRADIENT VALUE AT X * TYPX --> TYPICAL SIZE FOR EACH COMPONENT OF X * RNF --> RELATIVE NOISE IN FCN * ANALTL --> TOLERANCE FOR COMPARISON OF ESTIMATED AND ANALYTICAL * GRADIENTS * WRK1 --> WORKSPACE. IT IS USED TO HOLD THE FINITE-DIFFERENCE * APPROXIMATION OF THE GRADIENT AT X * MSG <-- MESSAGE OR ERROR CODE. IF PROBABLE CODING ERROR OF * GRADIENT IS DETECTED THEN MSG = -8 * IPR --> DEVICE TO WHICH TO SEND OUTPUT * INTEGER N,MSG,IPR DOUBLE PRECISION X(N),F,G(N),TYPX(N),RNF,ANALTL,WRK1(N) * * LOCAL VARIABLES * INTEGER KER,I DOUBLE PRECISION GS,ONE EXTERNAL FCN INTRINSIC ABS,MAX DATA ONE/1.0D0/ * * COMPUTE FIRST-ORDER FINITE DIFFERENCE GRADIENT AND COMPARE TO * ANALYTIC GRADIENT * CALL STFDGR(N,X,FCN,F,RNF,WRK1) KER = 0 DO 10 I = 1,N GS = MAX(ABS(F),ONE)/MAX(ABS(X(I)),ONE/TYPX(I)) IF (ABS(G(I)-WRK1(I)).GT.MAX(ABS(G(I)),GS)*ANALTL) KER = 1 10 CONTINUE IF (KER.EQ.0) GO TO 20 WRITE (IPR,9000) WRITE (IPR,9010) (I,G(I),WRK1(I),I=1,N) MSG = -8 20 CONTINUE RETURN 9000 FORMAT (' STCHKG PROBABLE ERROR IN CODING OF ANALYTIC', + ' GRADIENT FUNCTION.'/' STCHKG COMP',12X,'ANALYTIC', + 12X,'ESTIMATE') 9010 FORMAT (' STCHKG ',I5,3X,E20.13,3X,E20.13) END SUBROUTINE STCHKH(N,X,NPAIRS,NZ,IRN,LIRN,ICN,LICN,FCN,UGR,SCALE, + TYPX,GRDFLG,F,G,RNF,ANALTL,FCALL,CHECKH,LISTP, + NGRP,IPNTR,JPNTR,IWA,LIWA,FHESD,XD,ETA,HESS,IPR, + MSG) * * PURPOSE * ------- * * THIS ROUTINE CHECKS THE USER'S ANALYTIC HESSIAN AGAINST A * FINITE-DIFFERENCE HESSIAN. * * PARAMETERS: * ---------- * * N --> NUMBER OF VARIABLES IN THE PROBLEM * X --> CURRENT ITERATE * NPAIRS --> A POSITIVE INTEGER INPUT VARIABLE SET TO THE NUMBER * OF (IRN,ICN) PAIRS USED TO DESCRIBE THE SPARSITY * PATTERN OF THE HESSIAN MATRIX * NZ <-- ACTUAL NUMBER OF NONZEROS STORED IN THE LOWER * HALF OF THE HESSIAN MATRIX * IRN <-->ROW INDEX OF EACH NONZERO STORED IN THE LOWER * OR UPPER HALF OF THE HESSIAN MATRIX ON ENTRY. ON EXIT, * IT HOLDS THE ROW INDEX OF EACH NONZERO STORED IN THE * LOWER HALF OF THE HESSIAN MATRIX * ICN <-->COLUMN INDEX OF THE NONZEROS STORED IN THE LOWER * OR UPPER HALF OF THE HESSIAN MATRIX ON ENTRY. ON EXIT, * IT HOLDS THE COLUMN INDEX OF EACH NONZERO STORED IN THE * LOWER HALF OF THE HESSIAN MATRIX * FCN --> THE NAME OF A USER SUPPLIED SUBROUTINE THAT EVALUATES * THE FUNCTION F AT AN ARBITRARY VECTOR X * UGR --> THE NAME OF A USER SUPPLIED SUBROUTINE THAT RETURNS IN * THE VALUE OF THE GRADIENT * SCALE --> LOGICAL VARIABLE INDICATING WHETHER OR NOT SCALING * OF THE VARIABLES IS PERFORMED * TYPX --> TYPICAL SIZE OF THE COMPONENTS OF X * GRDFLG --> INTEGER FLAG DESIGNATING WHETHER OR NOT ANALYTIC * GRADIENT HAS BEEN SUPPLIED BY THE USER * F --> FUNCTION VALUE AT CURRENT ITERATE * G --> GRADIENT VALUE AT CURRENT ITERATE * RNF --> RELATIVE NOISE IN FUNCTION FCN * ANALTL --> TOLERANCE FOR COMPARISON OF ESTIMATED AND ANALYTICAL * GRADIENTS * FCALL --> A LOGICAL FLAG. IF THE HESSIAN MATRIX IS EVALUATED FOR * THE FIRST TIME, THEN FCALL IS SET TO TRUE; OTHERWISE * IT IS SET TO FALSE * CHECKH --> A LOGICAL FLAG. IF IT IS SET TO TRUE THEN THE USER'S * ANALYTIC HESSIAN IS CHECKED AGAINST THE FINITE * DIFFERENCE APPROXIMATION ONE * LISTP,NGRP,IPNTR,JPNTR,IWA --> WORSPACE FOR THE STFDHS SUBROUTINE * (SEE SUBROUTINE STFDHS FOR MORE DETAIL) * LIWA --> LENGTH OF ARRAY IWA * FHESD,XD,ETA --> WORKSPACE FOR THE STFDHS SUBROUTINE * HESS <--> ANALYTIC HESSIAN ON INPUT. ON OUTPUT, IT ALSO HOLDS * A FINITE-DIFFERENCE APPROXIMATION OF THE HESSIAN * (FROM NZ+1 TO 2*NZ) * IPR --> DEVICE TO WHICH TO SEND OUTPUT * MSG <-- MESSAGE OR ERROR CODE. IF PROBABLE CODING ERROR OF * HESSIAN IS DETECTED THEN MSG = -9 * INTEGER N,NPAIRS,NZ,GRDFLG,LIWA,IPR,MSG,LIRN,LICN INTEGER IRN(LIRN),ICN(LICN),LISTP(N),NGRP(N),IPNTR(N+1) INTEGER JPNTR(N+1),IWA(LIWA) DOUBLE PRECISION X(N),TYPX(N),F,G(N),RNF,ANALTL,FHESD(N) DOUBLE PRECISION XD(N),ETA(N),HESS(LICN) LOGICAL SCALE,FCALL,CHECKH * * LOCAL VARIABLES * INTEGER IST,IEND,KER,I,IR,J,JP,K,MAXGRP,MINGRP DOUBLE PRECISION HS,ONE EXTERNAL FCN,UGR INTRINSIC ABS,MAX,MIN DATA ONE/1.0D+0/ * * CHECK USER'S INPUT DATA. * * IF NPAIRS IS NOT POSITIVE THEN SET MSG TO -2 AND ABORT. * IF (NPAIRS.LE.0) THEN WRITE (IPR,9000) NPAIRS MSG = -4 RETURN END IF * * IF THE K-TH ELEMENT OF IRN OR THE K-TH ELEMENT OF ICN IS * NOT AN INTEGER BETWEEN 1 AND N THEN SET MSG TO -4 AND ABORT. * DO 10 K = 1,NPAIRS IF (IRN(K).LT.1 .OR. IRN(K).GT.N .OR. ICN(K).LT.1 .OR. + ICN(K).GT.N) THEN WRITE (IPR,9010) K,IRN(K),K,ICN(K) MSG = -5 RETURN END IF 10 CONTINUE * * GENERATE THE SPARSITY PATTERN FOR THE LOWER * TRIANGULAR PART OF HESSIAN. * DO 20 K = 1,NPAIRS I = IRN(K) J = ICN(K) IRN(K) = MAX(I,J) ICN(K) = MIN(I,J) 20 CONTINUE * * SORT THE DATA STRUCTURE BY COLUMNS * CALL STSORT(N,NPAIRS,HESS,IRN,ICN,JPNTR,IWA) * * COMPRESS THE DATA AND DETERMINE THE NUMBER OF NONZERO * ELEMENTS IN THE LOWER TRIANGULAR PART OF HESSIAN. * DO 30 I = 1,N IWA(I) = 0 30 CONTINUE NZ = 0 DO 50 J = 1,N K = NZ DO 40 JP = JPNTR(J),JPNTR(J+1) - 1 IR = IRN(JP) IF (IWA(IR).NE.J) THEN NZ = NZ + 1 IRN(NZ) = IR ICN(NZ) = J IWA(IR) = J END IF 40 CONTINUE JPNTR(J) = K + 1 50 CONTINUE JPNTR(N+1) = NZ + 1 IF (NZ.NE.NPAIRS) THEN MSG = -7 WRITE (IPR,9020) NPAIRS - NZ RETURN END IF * IF (.NOT.CHECKH) RETURN * * COMPUTE FINITE-DIFFERENCE APPROXIMATION OF THE HESSIAN * CALL STFDHS(N,NZ,X,IRN,ICN,G,RNF,GRDFLG,FCN,UGR,SCALE,TYPX,FCALL, + LISTP,NGRP,IPNTR,JPNTR,IWA,LIWA,FHESD,XD,ETA,MAXGRP, + MINGRP,HESS(NZ+1)) * * COMPARE USER'S ANALYTIC HESSIAN TO THE FINITE-DIFFERENCE * APPROXIMATION * KER = 0 DO 70 J = 1,N HS = MAX(ABS(G(J)),ONE)/MAX(ABS(X(J)),ONE/TYPX(J)) IST = JPNTR(J) IEND = JPNTR(J+1) DO 60 I = IST,IEND - 1 IF (ABS(HESS(I)-HESS(NZ+I)).GT. + MAX(ABS(HESS(I)),HS)*ANALTL) KER = 1 60 CONTINUE 70 CONTINUE * IF (KER.EQ.0) GO TO 90 WRITE (IPR,9030) DO 80 I = 1,NZ WRITE (IPR,9040) IRN(I),ICN(I),HESS(I),HESS(NZ+I) 80 CONTINUE MSG = -9 90 CONTINUE RETURN * 9000 FORMAT (' STCHKH ILLEGAL NUMBER OF NONZEROS NZ=',I10) 9010 FORMAT (' STCHKH ILLEGAL ROW OR COLUMN INDEX IN ANALYTIC', + ' HESSIAN'/,' STCHKH IRN( ',I6,' )=',I6,2X,' ICN( ',I6, + ' )=',I6) 9020 FORMAT (' STCHKH ',I5,' REDUNDANT ENTRIES IN SPARSITY', + ' PATTERN ENCOUNTERED') 9030 FORMAT (' STCHKH PROBABLE ERROR IN CODING OF ANALYTIC', + ' HESSIAN FUNCTION'/' STCHKH IRN ICN',14X,'ANALYTIC', + 14X,'(ESTIMATE)') 9040 FORMAT (' STCHKH ',2I5,2X,E20.13,2X,'(',E20.13,')') END SUBROUTINE STCHKI(N,NPAIRS,NZ,X,IRN,LIRN,ICN,LICN,LIWRK,LWRK, + SCALE,TYPX,FSCALE,GRADTL,STEPTL,ILIM,NDIGIT,EPS, + METHOD,GRDFLG,HSNFLG,STEPMX,IWA,LIWA,JPNTR,MSG, + IPR) * * PURPOSE: * ------- * * THIS ROUTINE CHECKS THE INPUT FOR REASONABLENESS. * * PARAMETERS: * ---------- * N --> DIMENSION OF PROBLEM * NPAIRS --> A POSITIVE INTEGER INPUT VARIABLE SET TO THE NUMBER * OF (IRN,ICN) PAIRS USED TO DESCRIBE THE SPARSITY * PATTERN OF THE HESSIAN MATRIX * NZ <-- NUMBER OF NONZEROS IN THE LOWER TRIANGULAR PART * OF THE HESSIAN MATRIX * X --> ON ENTRY, ESTIMATE TO ROOT OF FCN * IRN <--> AN INTEGER ARRAY OF LENGTH NPAIRS. ON INPUT IRN MUST * CONTAIN THE ROW INDICES OF THE NONZERO ELEMENTS IN * THE LOWER TRIANGULAR PART OF THE HESSIAN. ON OUTPUT * IRN IS PERMUTED SO THAT THE CORRESPONDING COLUMN * INDICES ARE IN NON-DECREASING ORDER. THE COLUMN * INDICES CAN BE RECOVERED FROM THE ARRAY JPNTR * LIRN --> LENGTH OF ARRAY IRN * ICN <--> AN INTEGER ARRAY OF LENGTH NPAIRS. ON INPUT ICN MUST * CONTAIN THE COLUMN INDICES OF THE NONZERO ELEMENTS * IN THE LOWER TRIANGULAR PART OF THE HESSIAN. ON * OUTPUT ICN IS PERMUTED SO THAT THE CORRESPONDING * ROW INDICES ARE IN NON-DECREASING ORDER * LICN --> LENGTH OF ARRAY ICN * LIWRK --> LENGTH OF WORKSPACE ARRAY IWRK * LWRK --> LENGTH OF WORKSPACE ARRAY WRK * SCALE <-- LOGICAL VARIABLE INDICATING WHETHER OR NOT SCALING * OF THE X VARIABLES IS PERFORMED * TYPX <--> TYPICAL SIZE OF EACH COMPONENT OF X * FSCALE <--> ESTIMATE OF SCALE OF OBJECTIVE FUNCTION FCN * GRADTL <--> TOLERANCE AT WHICH GRADIENT CONSIDERED CLOSE * ENOUGH TO ZERO TO TERMINATE ALGORITHM * STEPTL <--> TOLERANCE AT WHICH STEP LENGTH CONSIDERED CLOSE * ENOUGH TO ZERO TO TERMINATE ALGORITHM * ILIM <--> MAXIMUM NUMBER OF ALLOWABLE ITERATIONS * NDIGIT <--> NUMBER OF GOOD DIGITS IN OPTIMIZATION FUNCTION FCN * EPS --> MACHINE PRECISION * METHOD <--> METHOD TO BE USED (NEWTON OR TENSOR) * GRDFLG <--> =1 IF ANALYTIC GRADIENT SUPPLIED * HSNFLG <--> =1 IF ANALYTIC HESSIAN SUPPLIED * STEPMX <--> MAXIMUM STEP SIZE * IWA --> WORKSPACE ARRAY * LIWA <-- ACTUAL LENGTH OF ARRAY IWA ON EXIT * JPNTR <-- AN INTEGER OUTPUT ARRAY OF LENGTH N + 1 WHICH * SPECIFIES THE LOCATIONS OF THE ROW INDICES IN IRN. * THE ROW INDICES FOR COLUMN J ARE * IRN(K), K = JPNTR(J),...,JPNTR(J+1)-1. NOTE THAT * JPNTR(N+1)-1 IS THEN THE NUMBER OF NONZERO ELEMENTS * IN THE LOWER TRIANGULAR PART OF THE MATRIX HESSIAN * MSG <--> AN INTEGER VARIABLE THAT THE USER MAY SET ON INPUT TO * INHIBIT CERTAIN AUTOMATIC CHECKS OR OVERRIDE CERTAIN * DEFAULT CHARACTERISTICS OF THE PACKAGE. ON INPUT: * MSG = 0 : NO OUTPUT WILL BE PRODUCED * MSG = 1 : PRINT THE INPUT STATE, THE FINAL RESULTS, * AND THE STOPPING CONDITIONS * MSG = 2 : PRINT THE INTERMEDIATE RESULTS, THAT IS, THE * INPUT STATE, THE VALUES OF THE OBJECTIVE * FUNCTION AND THE SCALED GRADIENT AT EACH * ITERATION, AND THE FINAL RESULTS INCLUDING * THE STOPPING CONDITIONS AND THE NUMBER OF * FUNCTION, GRADIENT, AND HESSIAN * * ON OUTPUT, IF THE PROGRAM HAS TERMINATED BECAUSE OF * ERRONEOUS INPUT, MSG CONTAINS AN ERROR CODE INDICATING * THE REASON: * * MSG = -1 : ILLEGAL DIMENSION N; N <= 0. THE PROGRAM ABORTS * MSG = -2 : ILLEGAL LENGTH OF LIRN OR LICN; LIRN <= 0 OR * LICN <= 0. THE PROGRAM ABORTS * MSG = -3 : ILLEGAL LENGTH OF LIWRK OR LWRK; LIWRK < 2*LIRN * +12*N+2 OR LWRK < 7*N. THE PROGRAM ABORTS * MSG = -4 : ILLEGAL NUMBER OF NONZEROS NZ; NZ <= 0. THE * PROGRAM ABORTS * MSG = -5 : THE K-TH ELEMENT OF IRN OR THE K-TH ELEMENT OF * ICN IS NOT AN INTEGER BETWEEN 1 AND N; (IRN(K) * < 1 OR IRN(K) > N) OR (ICN(K) < 1 OR ICN(K) > N). * THE PROGRAM ABORTS * MSG = -6 : THE K-TH DIAGONAL ELEMENT IS NOT IN THE SPARSITY * PATTERN. THIS IS CHECKED ONLY IF HSNFLG = 0. * THE PROGRAM ABORTS * MSG = -7 : REDUNDANT ENTRIES IN SPARSITY PATTERN WAS * ENCOUNTERED. WHEN HSNFLG = 1 OR HSNFLG = 2, THE * PROGRAM ABORTS. WHEN HSNFLG = 0, THE PROGRAM * ELIMINATES THE REDUNDANT ENTRIES AND CONTINUES * THE EXECUTION * MSG = -8 : PROBABLE CODING ERROR IN THE USER'S ANALYTIC * GRADIENT ROUTINE UGR. THE PROGRAM ABORTS. (THIS * CHECK CAN BE OVERRIDDEN BY SETTING GRDFLG = 2.) * (SEE SUBROUTINE STCHKG) * MSG = -9 : PROBABLE CODING ERROR IN THE USER'S ANALYTIC * HESSIAN ROUTINE USH. THE PROGRAM ABORTS. (THIS * CHECK CAN BE OVERRIDDEN BY SETTING HSNFLG = 2.) * (SEE SUBROUTINE STCHKH) * * IPR --> DEVICE TO WHICH TO SEND OUTPUT * INTEGER N,NPAIRS,NZ,LIRN,LICN,LIWRK,LWRK,ILIM,NDIGIT, + METHOD INTEGER GRDFLG,HSNFLG,LIWA,MSG,IPR INTEGER IRN(LIRN),ICN(LICN),IWA(N),JPNTR(N+1) DOUBLE PRECISION X(N),TYPX(N),FSCALE,GRADTL,STEPTL,STEPMX LOGICAL SCALE * * LOCAL VARIABLES * INTEGER I,IR,J,JP,K DOUBLE PRECISION STPSIZ,TEMP,EPS,ZERO,ONE,THREE,TEN,THOUS INTRINSIC LOG10,MAX,MIN,SQRT DATA ZERO,ONE,THREE,TEN,THOUS/0.0D+0,1.0D+0,3.0D+0, + 10.0D+0,1000.0D+0/ * * CHECK THAT PARAMETERS ONLY TAKE ON ACCEPTABLE VALUES. * IF NOT, SET THEM TO DEFAULT VALUES. * * CHECK DIMENSION OF THE PROBLEM * IF (N.LE.0) THEN WRITE (IPR,9000) N MSG = -1 RETURN END IF * IF (LIRN.LE.0 .OR. LICN.LE.0) THEN WRITE (IPR,9010) LIRN,LICN MSG = -2 RETURN END IF * * IF LIWRK IS LESS THAN 2*LIRN+12*N+2 OR LWRK LESS THAN 7*N * SET MSG = -3 AND ABORT. * IF (LIWRK.LT.2*LIRN+12*N+2 .OR. LWRK.LT.7*N) THEN WRITE (IPR,9020) LIWRK,LWRK MSG = -3 RETURN END IF * * SET THE LENGTH OF LIWA * LIWA = 2*LIRN + 8*N * IF (HSNFLG.EQ.0) THEN * * IF NPAIRS IS NOT POSITIVE THEN SET MSG TO -4 AND ABORT. * IF (NPAIRS.LE.0) THEN WRITE (IPR,9030) NPAIRS MSG = -4 RETURN END IF * * IF THE K-TH ELEMENT OF IRN OR THE K-TH ELEMENT OF ICN IS * NOT AN INTEGER BETWEEN 1 AND N THEN SET MSG TO -5 AND ABORT. * DO 10 K = 1,NPAIRS IF (IRN(K).LT.1 .OR. IRN(K).GT.N .OR. ICN(K).LT.1 .OR. + ICN(K).GT.N) THEN WRITE (IPR,9040) K,IRN(K),K,ICN(K) MSG = -5 RETURN END IF 10 CONTINUE * * GENERATE THE SPARSITY PATTERN FOR THE LOWER * TRIANGULAR PART OF HESSIAN. * DO 20 K = 1,NPAIRS I = IRN(K) J = ICN(K) IRN(K) = MAX(I,J) ICN(K) = MIN(I,J) 20 CONTINUE * * IF HSNFLG = 0 AND THE K-TH DIAGONAL ELEMENT IS NOT IN THE * SPARSITY PATTERN THEN ABORT. * I = 0 DO 30 K = 1,N IWA(K) = 0 30 CONTINUE DO 40 K = 1,NPAIRS IF (IRN(K).EQ.ICN(K)) IWA(IRN(K)) = 1 40 CONTINUE DO 50 K = 1,N IF (IWA(K).NE.1) THEN WRITE (IPR,9050) K MSG = -6 RETURN END IF 50 CONTINUE * * SORT THE DATA STRUCTURE BY COLUMNS. * CALL SRTDAT(N,NPAIRS,IRN,ICN,JPNTR,IWA) * * COMPRESS THE DATA AND DETERMINE THE NUMBER OF NONZERO * ELEMENTS IN THE LOWER TRIANGULAR PART OF HESSIAN. * DO 60 I = 1,N IWA(I) = 0 60 CONTINUE NZ = 0 DO 80 J = 1,N K = NZ DO 70 JP = JPNTR(J),JPNTR(J+1) - 1 IR = IRN(JP) IF (IWA(IR).NE.J) THEN NZ = NZ + 1 IRN(NZ) = IR ICN(NZ) = J IWA(IR) = J END IF 70 CONTINUE JPNTR(J) = K + 1 80 CONTINUE JPNTR(N+1) = NZ + 1 * END IF * * CHECK METHOD USED * IF (METHOD.NE.0) METHOD = 1 * * CHECK GRADIENT FLAG * IF (GRDFLG.NE.2 .AND. GRDFLG.NE.1) GRDFLG = 0 * * CHECK HESSIAN FLAG * IF (HSNFLG.NE.2 .AND. HSNFLG.NE.1) HSNFLG = 0 * * CHECK MSG * IF (MSG.NE.2 .AND. MSG.NE.0) MSG = 1 * * CHECK SCALE MATRIX * DO 90 I = 1,N IF (TYPX(I).EQ.ZERO) TYPX(I) = ONE IF (TYPX(I).LT.ZERO) TYPX(I) = -TYPX(I) 90 CONTINUE * * SET THE LOGICAL VARIABLE SCALE * SCALE = .FALSE. DO 100 I = 1,N IF (TYPX(I).NE.ONE) THEN SCALE = .TRUE. GO TO 110 END IF 100 CONTINUE 110 CONTINUE * * CHECK MAXIMUM STEP SIZE * IF (STEPMX.GT.ZERO) GO TO 130 STPSIZ = ZERO DO 120 I = 1,N STPSIZ = STPSIZ + X(I)*X(I)/TYPX(I)*TYPX(I) 120 CONTINUE STPSIZ = SQRT(STPSIZ) STEPMX = MAX(THOUS*STPSIZ,THOUS) 130 CONTINUE * * CHECK FUNCTION SCALE * IF (FSCALE.EQ.ZERO) FSCALE = ONE IF (FSCALE.LT.ZERO) FSCALE = -FSCALE * * CHECK GRADIENT TOLERANCE * IF (GRADTL.LT.ZERO) GRADTL = EPS** (ONE/THREE) * * CHECK STEP TOLERANCE * IF (STEPTL.LT.ZERO) THEN TEMP = EPS** (ONE/THREE) STEPTL = TEMP*TEMP END IF * * CHECK ITERATION LIMIT * IF (ILIM.LE.0) ILIM = 150 * * CHECK NUMBER OF DIGITS OF ACCURACY IN FUNCTION FCN * IF (NDIGIT.LE.0) NDIGIT = -LOG10(EPS) IF (TEN** (-NDIGIT).LE.EPS) NDIGIT = -LOG10(EPS) RETURN * * ERROR EXITS AND WARNINGS * 9000 FORMAT (' STCHKI ILLEGAL DIMENSION N=',I10) 9010 FORMAT (' STCHKI ILLEGAL LENGTH OF LIRN',I10,' OR LICN=',I10) 9020 FORMAT (' STCHKI ILLEGAL LENGTH OF INTEGER WORKSPACE'/, + ' STCHKI IWRK=',I10,' OR REAL WORKSPACE WRK=',I10) 9030 FORMAT (' STCHKI ILLEGAL NUMBER OF NONZEROS NZ=',I10) 9040 FORMAT (' STCHKI ILLEGAL ROW OR COLUMN INDEX'/, + ' STCHKI IRN( ',I6,' )=',I6,2X,' ICN( ',I6,' )=',I5) 9050 FORMAT (' STCHKI DIAGONAL ELEMENT',I6,' IS NOT IN THE', + ' SPARSITY PATTERN'/, + ' STCHKI THE FINITE-DIFFERENCE HESSIAN APPROXIMATION', + ' REQUIRE'/, + ' STCHKI THAT DIAGONAL ELEMENTS BE IN THE SPARSITY', + ' PATTERN') END SUBROUTINE STCHKS(N,XPLS,FPLS,GPLS,X,ITNCNT,ICSCMX,ITRMCD,GRADTL, + STEPTL,FSCALE,ILIM,IRETCD,MXTAKE,RGX,IPR,MSG) * * PURPOSE: * ------- * * THIS ROUTINE CHECKS THE STOPPING CRITERIA AND TERMINATES THE * OPTIMIZATION ALGORITHM IF ANY OF THE FOLLOWING IS SATISFIED: * * 1. THE SCALED GRADIENT IS LESS THAN GRADTL * 2. THE LENGTH OF THE CURRENT STEP IS LESS THAN STEPTL * 3. THE CURRENT GLOBAL STEP FAILED TO LOCATE A POINT LOWER THAN XPLS * 4. THE ITERATION LIMIT HAS BEEN EXCEEDED * 5. FIVE CONSECUTIVE STEPS OF LENGTH STEPMX HAVE BEEN TAKEN * * PARAMETERS: * ---------- * * N --> DIMENSION OF PROBLEM * XPLS --> CURRENT ITERATE * FPLS --> FUNCTION VALUE AT XPLS * GPLS --> GRADIENT AT XPLS * X --> OLD ITERATE * ITNCNT --> CURRENT ITERATION NUMBER * ICSCMX <--> NUMBER CONSECUTIVE STEPS .GE. STEPMX * (RETAIN VALUE BETWEEN SUCCESSIVE CALLS) * ITRMCD <-- TERMINATION CODE * GRADTL --> TOLERANCE AT WHICH RELATIVE GRADIENT CONSIDERED * CLOSE ENOUGH TO ZERO TO TERMINATE ALGORITHM * STEPTL --> RELATIVE STEP SIZE AT WHICH SUCCESSIVE ITERATES * CONSIDERED CLOSE ENOUGH TO TERMINATE ALGORITHM * FSCALE --> ESTIMATE OF SCALE OF OBJECTIVE FUNCTION * ILIM --> MAXIMUM NUMBER OF ALLOWABLE ITERATIONS * IRETCD --> RETURN CODE FROM SUBROUTINE STLSCH * MXTAKE --> BOOLEAN FLAG INDICATING STEP OF MAXIMUM LENGTH * RGX <-- SCALED GRADIENT * IPR --> USED DEVICE TO WHICH TO SEND OUTPUT * MSG --> CONTROL OUTPUT * MSG = 0 MEANS NO INFORMATION IS OUTPUT * MSG >= 1 MEANS INFORMATION ABOUT STOPPING CONDITIONS * IS OUTPUT * INTEGER N,ITNCNT,ICSCMX,ITRMCD,ILIM,IRETCD,IPR,MSG DOUBLE PRECISION XPLS(N),FPLS,GPLS(N),X(N),GRADTL,STEPTL DOUBLE PRECISION FSCALE,RGX * * LOCAL VARIABLES * INTEGER I DOUBLE PRECISION D,RELGRD,RSX,RELSTP,ZERO,ONE LOGICAL MXTAKE INTRINSIC ABS,MAX DATA ZERO,ONE/0.0D+0,1.0D+0/ * ITRMCD = 0 * * LAST GLOBAL STEP FAILED TO LOCATE A POINT LOWER THAN X * IF (IRETCD.EQ.1) THEN ITRMCD = 3 IF (MSG.GE.1) WRITE (IPR,9020) RETURN END IF * * COMPUTE SCALED GRADIENT AND CHECK WHETHER IT IS WITHIN TOLERANCE * D = MAX(ABS(FPLS),FSCALE) * D=ONE RGX = ZERO DO 10 I = 1,N RELGRD = ABS(GPLS(I))*MAX(ABS(XPLS(I)),ONE)/D RGX = MAX(RGX,RELGRD) 10 CONTINUE IF (RGX.LE.GRADTL) THEN ITRMCD = 1 IF (MSG.GE.1) WRITE (IPR,9000) RETURN END IF * IF (ITNCNT.EQ.0) RETURN * * FIND DIRECTION IN WHICH RELATIVE STEPSIZE MAXIMUM * CHECK WHETHER WITHIN TOLERANCE * RSX = 0.0 DO 20 I = 1,N RELSTP = ABS(XPLS(I)-X(I))/MAX(ABS(XPLS(I)),ONE) RSX = MAX(RSX,RELSTP) 20 CONTINUE IF (RSX.LE.STEPTL) THEN ITRMCD = 2 IF (MSG.GE.1) WRITE (IPR,9010) RETURN END IF * * CHECK ITERATION LIMIT * IF (ITNCNT.GE.ILIM) THEN ITRMCD = 4 IF (MSG.GE.1) WRITE (IPR,9030) RETURN END IF * * CHECK NUMBER OF CONSECUTIVE STEPS \ STEPMX * IF (MXTAKE) THEN ICSCMX = ICSCMX + 1 IF (ICSCMX.GE.5) THEN ITRMCD = 5 IF (MSG.GE.1) WRITE (IPR,9040) RETURN END IF ELSE ICSCMX = 0 RETURN END IF * 9000 FORMAT (' STCHKS RELATIVE GRADIENT CLOSE TO ZERO'/ + ' STCHKS CURRENT ITERATE IS PROBABLY SOLUTION') 9010 FORMAT (' STCHKS SUCCESSIVE ITERATES WITHIN TOLERANCE'/ + ' STCHKS CURRENT ITERATE IS PROBABLY SOLUTION') 9020 FORMAT (' STCHKS LAST GLOBAL STEP FAILED TO LOCATE A POINT', + ' LOWER THAN X',/ + ' STCHKS EITHER X IS AN APPROXIMATE LOCAL MINIMUM', + ' OF THE FUNCTION',/ + ' STCHKS THE FUNCTION IS TOO NON-LINEAR FOR THIS', + ' ALGORITHM',/' STCHKS OR STEPTL IS TOO LARGE') 9030 FORMAT (' STCHKS ITERATION LIMIT EXCEEDED', + /' STCHKS ALGORITHM FAILED') 9040 FORMAT (' STCHKS MAXIMUM STEP SIZE EXCEEDED 5', + ' CONSECUTIVE TIMES',/ + ' STCHKS EITHER THE FUNCTION IS UNBOUNDED BELOW', + /' STCHKS BECOMES ASYMPTOTIC TO A FINITE VALUE', + ' FROM ABOVE IN SOME DIRECTION', + /' STCHKS OR STEPMX IS TOO SMALL') END SUBROUTINE STCZ3P(A1,A2,A3,XP,NZEROS) * * PURPOSE: * -------- * * THIS ROUTINE COMPUTES THE ROOTS OF A THIRD DEGREE * POLYNOMIAL OF THE FORM : X**3 + A1*X**2 + A2*X + A3. * * PARAMETERS: * ---------- * * A1, A2, A3 ---> COEFFICIENTS OF THE CUBIC POLYNOMIAL * XP <--- ZEROS OF THE CUBIC POLYNOMIAL * NZEROS <--- NUMBER OF ZEROS FOUND * * INTEGER NZEROS DOUBLE PRECISION A1,A2,A3,XP(3) * * LOCAL VARIABLES * DOUBLE PRECISION CST1,CST2,Q,R,D,S,T,THETA,PI,CONST DOUBLE PRECISION ZERO,HALF,ONE,TWO,THREE,FOUR,NINE DOUBLE PRECISION TSEVEN,FFOUR INTRINSIC ACOS,COS,SQRT PARAMETER (PI=3.141592653589793D+0) DATA ZERO,HALF,ONE,TWO,THREE,FOUR,NINE,TSEVEN, + FFOUR/0.0D+0,5.0D-1,1.0D+0,2.0D+0,3.0D+0,4.0D+0, + 9.0D+0,27.0D+0,54.0D+0/ * NZEROS = 0 CST1 = ONE/THREE CST2 = CST1*A1 Q = (THREE*A2-A1**TWO)/NINE R = (NINE*A1*A2-TSEVEN*A3-TWO*A1**THREE)/FFOUR D = Q**THREE + R**TWO IF (D.GT.ZERO) THEN S = R + SQRT(D) IF (S.LT.ZERO) THEN S = - ((-S)**CST1) ELSE S = S**CST1 END IF T = R - SQRT(D) IF (T.LT.ZERO) THEN T = - ((-T)**CST1) ELSE T = T**CST1 END IF * * COMPUTE THE ONLY REAL ZERO * NZEROS = 1 XP(1) = S + T - CST2 ELSE IF (D.EQ.ZERO) THEN IF (R.LT.ZERO) THEN S = - ((-R)**CST1) ELSE S = R**CST1 END IF T = S * * COMPUTE THE TWO REAL ZEROS * NZEROS = 2 XP(1) = S + T - CST2 XP(2) = -HALF* (S+T) ELSE THETA = ACOS(R/SQRT(-Q**THREE)) CONST = TWO*SQRT(-Q) THETA = CST1*THETA * * COMPUTE THE THREE REAL ZEROS * NZEROS = 3 XP(1) = CONST*COS(THETA) - CST2 XP(2) = CONST*COS(THETA+TWO*PI*CST1) - CST2 XP(3) = CONST*COS(THETA+FOUR*PI*CST1) - CST2 END IF RETURN END SUBROUTINE STDFLT(N,TYPX,FSCALE,GRADTL,STEPTL,ILIM,STEPMX,IPR, + METHOD,GRDFLG,HSNFLG,NDIGIT,INFORM,MSG) * * PURPOSE: *-------- * * THIS ROUTINE SETS THE DEFAULT VALUES OF THE PACKAGE. * * N --> A POSITIVE INTEGER VARIABLE SPECIFYING THE NUMBER OF * VARIABLES IN THE PROBLEM * TYPX <-- AN ARRAY OF LENGTH N IN WHICH THE TYPICAL SIZE OF THE * COMPONENTS OF X ARE SPECIFIED * FSCALE <-- A POSITIVE REAL NUMBER ESTIMATING THE MAGNITUDE * OF F(X) NEAR THE MINIMIZER * GRADTL <-- POSITIVE SCALAR GIVING THE TOLERANCE AT WHICH THE * SCALED GRADIENT OF F(X) IS CONSIDERED CLOSE ENOUGH TO * ZERO TO TERMINATE THE ALGORITHM * STEPTL <-- A POSITIVE SCALAR PROVIDING THE MINIMUM ALLOWABLE * RELATIVE STEP LENGTH * ILIM <-- POSITIVE INTEGER SPECIFYING THE MAXIMUM ITERATIONS TO BE * PERFORMED BEFORE THE PROGRAM IS TERMINATED * STEPMX <-- A POSITIVE SCALAR PROVIDING THE MAXIMUM ALLOWABLE SCALED * STEP LENGTH * IPR <-- THE UNIT ON WHICH THE ROUTINE OUTPUTS INFORMATION * METHOD <-- AN INTEGER FLAG DESIGNATING WHICH METHOD TO USE * METHOD = 0 : USE NEWTON'S METHOD * METHOD = 1 : USE THE TENSOR METHOD * GRDFLG <-- INTEGER FLAG DESIGNATING WHETHER OR NOT ANALYTIC HESSIAN * HAS BEEN SUPPLIED BY THE USER * GRDFLG = 0 : NO ANALYTIC GRADIENT SUPPLIED * GRDFLG = 1 : ANALYTIC GRADIENT SUPPLIED (WILL BE CHECKED * AGAINST FINITE DIFFERENCE GRADIENT) * GRDFLG = 2 : ANALYTIC GRADIENT SUPPLIED (WILL NOT BE * CHECKED AGAINST FINITE DIFFERENCE GRADIENT) * HSNFLG <-- INTEGER FLAG DESIGNATING WHETHER OR NOT ANALYTIC HESSIAN * HAS BEEN SUPPLIED BY THE USER * HSNFLG = 0 : NO ANALYTIC HESSIAN SUPPLIED * HSNFLG = 1 : ANALYTIC HESSIAN SUPPLIED (WILL BE CHECKED * AGAINST FINITE DIFFERENCE HESSIAN) * HSNFLG = 2 : ANALYTIC HESSIAN SUPPLIED (WILL NOT BE * CHECKED AGAINST FINITE DIFFERENCE HESSIAN) * NDIGIT <-- INTEGER ESTIMATING THE NUMBER OF ACCURATE DIGITS ON THE * OBJECTIVE FUNCTION F(X) * INFORM <-- AN INTEGER VARIABLE. IF IT IS SET TO 1, THE USER MUST * OBTAIN HESS TIMES VECTOR AND RE-ENTER WITH INFORM * UNCHANGED. THE RESULT OF HESS TIMES VECTOR MUST BE * STORED IN VECTOR. THE DEFAULT VALUE OF INFORM IS 0, * MEANING THAT HESS TIMES VECTOR IS COMPUTED BY THE * PACKAGE * MSG <-- AN INTEGER VARIABLE THAT THE USER MAY SET ON INPUT TO * INHIBIT CERTAIN AUTOMATIC CHECKS OR OVERRIDE CERTAIN * DEFAULT CHARACTERISTICS OF THE PACKAGE * INTEGER N,ILIM,IPR,METHOD,GRDFLG,HSNFLG,NDIGIT,INFORM,MSG DOUBLE PRECISION TYPX(N),FSCALE,GRADTL,STEPTL,STEPMX * * LOCAL VARIABLES * INTEGER I DOUBLE PRECISION EPS,TEMP,ZERO,ONE,THREE DOUBLE PRECISION D1MACH INTRINSIC LOG10 DATA ZERO,ONE,THREE/0.0D+0,1.0D+0,3.0D+0/ * EPS = D1MACH(3) METHOD = 1 FSCALE = ONE GRDFLG = 0 HSNFLG = 0 DO 10 I = 1,N TYPX(I) = ONE 10 CONTINUE TEMP = EPS** (ONE/THREE) GRADTL = TEMP STEPTL = TEMP*TEMP NDIGIT = -LOG10(EPS) * * SET ACTUAL DFAULT VALUE OF STEPMX IN STCHKI * STEPMX = ZERO ILIM = 500 IPR = 6 MSG = 1 INFORM = 0 RETURN END SUBROUTINE STDRUO(N,X,NPAIRS,IRN,LIRN,ICN,LICN,FCN,UGR,USH,TYPX, + FSCALE,GRADTL,STEPTL,ILIM,STEPMX,IPR,METHOD, + GRDFLG,HSNFLG,NDIGIT,MSG,XPLS,FPLS,GPLS,HESS,G, + S,D,DN,E,XD,BV,LISTP,NGRP,IPNTR,JPNTR,IWA,LIWRK, + LWRK,TERMCD,VECTOR,INFORM) * * PURPOSE: * ------- * * THIS ROUTINE IS THE DRIVER FOR SOLVING LARGE, SPARSE UNCONSTRAINED * OPTIMIZATION PROBLEMS USING TENSOR METHODS. * * PARAMETERS: * ---------- * * N --> A POSITIVE INTEGER VARIABLE SPECIFYING THE NUMBER OF * VARIABLES IN THE PROBLEM * X --> AN ARRAY OF LENGTH N THAT CONTAINS AN INITIAL * ESTIMATE OF THE MINIMIZER * NPAIRS --> AN INTEGER VARIABLE THAT MUST BE SET BY THE USER TO * THE NUMBER OF NONZEROS STORED IN THE LOWER OR UPPER * HALF OF THE HESSIAN MATRIX * IRN --> AN INTEGER ARRAY OF LENGTH LIRN. ON ENTRY, IT MUST * HOLD THE ROW INDEX OF EACH NONZERO STORED IN THE * LOWER OR UPPER HALF OF THE HESSIAN MATRIX * LIRN --> AN INTEGER VARIABLE THAT MUST BE SET BY THE USER TO * THE LENGTH OF ARRAY IRN. LIRN NEED NOT BE AS LARGE * AS LICN; NORMALLY IT NEED NOT BE VERY MUCH GREATER * THAN NPAIRS * ICN --> AN INTEGER ARRAY OF LENGTH LICN. ON ENTRY, IT MUST * HOLD THE COLUMN INDEX OF THE NONZEROS STORED IN THE * LOWER OR UPPER HALF OF THE HESSIAN MATRIX * LICN --> AN INTEGER VARIABLE THAT MUST BE SET BY THE USER TO THE * LENGTH OF THE HESSIAN ARRAY HESS AND ICN. LICN SHOULD * ORDINARILY BE 2 TO 4 TIMES AS LARGE AS NPAIRS * FCN --> THE NAME OF A USER SUPPLIED SUBROUTINE THAT EVALUATES * THE FUNCTION F AT AN ARBITRARY VECTOR X. THE SUBROUTINE * MUST BE DECLARED EXTERNAL IN THE USER'S CALLING PROGRAM * AND MUST CONFORM TO * CALL FCN(N, X, F) * UGR --> THE NAME OF A USER SUPPLIED SUBROUTINE THAT RETURNS IN * THE VALUE OF THE GRADIENT. UGR MUST BE DECLARED EXTERNAL * IN THE USER'S CALLING PROGRAM AND MUST CONFORM TO THE * USAGE * CALL UGR(N, X, G) * IF NO ANALYTIC GRADIENT IS SUPPLIED (GRDFLG = 0), THE * USER MUST USE THE DUMMY NAME STDUGR * USH --> THE NAME OF A USER SUPPLIED SUBROUTINE THAT RETURNS * IN HESS THE VALUE OF THE HESSIAN AT THE CURRENT POINT X. * USH MUST BE DECLARED EXTERNAL IN THE USER'S CALLING * PROGRAM AND MUST CONFORM TO THE USAGE * CALL USH(N,X,NPAIRS,LICN,HESS,IRN,ICN) * ONLY THE LOWER OR UPPER TRIANGULAR PART AND THE DIAGONAL * OF HESS SHOULD BE GIVEN, WITH THEIR CORRESPONDING ROW * AND COLUMN INDICES. IF NO ANALYTIC GRADIENT IS SUPPLIED * (HSNFLG = 0), THE USER MUST USE THE DUMMY NAME STDUSH * TYPX --> AN ARRAY OF LENGTH N IN WHICH THE TYPICAL SIZE OF THE * COMPONENTS OF X ARE SPECIFIED * FSCALE --> A POSITIVE REAL NUMBER ESTIMATING THE MAGNITUDE * OF F(X) NEAR THE MINIMIZER * GRADTL --> POSITIVE SCALAR GIVING THE TOLERANCE AT WHICH THE * SCALED GRADIENT OF F(X) IS CONSIDERED CLOSE ENOUGH TO * ZERO TO TERMINATE THE ALGORITHM * STEPTL --> A POSITIVE SCALAR PROVIDING THE MINIMUM ALLOWABLE * RELATIVE STEP LENGTH * ILIM --> POSITIVE INTEGER SPECIFYING THE MAXIMUM ITERATIONS * TO BE PERFORMED BEFORE THE PROGRAM IS TERMINATED * STEPMX --> A POSITIVE SCALAR PROVIDING THE MAXIMUM ALLOWABLE * SCALED STEP LENGTH * IPR --> THE UNIT ON WHICH THE ROUTINE OUTPUTS INFORMATION * METHOD --> AN INTEGER FLAG DESIGNATING WHICH METHOD TO USE * METHOD = 0 : USE NEWTON'S METHOD * METHOD = 1 : USE THE TENSOR METHOD * GRDFLG --> INTEGER FLAG DESIGNATING WHETHER OR NOT ANALYTIC * HESSIAN HAS BEEN SUPPLIED BY THE USER * GRDFLG = 0 : NO ANALYTIC GRADIENT SUPPLIED * GRDFLG = 1 : ANALYTIC GRADIENT SUPPLIED (WILL BE CHECKED * AGAINST FINITE DIFFERENCE GRADIENT) * GRDFLG = 2 : ANALYTIC GRADIENT SUPPLIED (WILL NOT BE * CHECKED AGAINST FINITE DIFFERENCE GRADIENT) * HSNFLG --> INTEGER FLAG DESIGNATING WHETHER OR NOT ANALYTIC * HESSIAN HAS BEEN SUPPLIED BY THE USER * HSNFLG = 0 : NO ANALYTIC HESSIAN SUPPLIED * HSNFLG = 1 : ANALYTIC HESSIAN SUPPLIED (WILL BE CHECKED * AGAINST FINITE DIFFERENCE HESSIAN) * HSNFLG = 2 : ANALYTIC HESSIAN SUPPLIED (WILL NOT BE * CHECKED AGAINST FINITE DIFFERENCE HESSIAN) * NDIGIT --> INTEGER ESTIMATING THE NUMBER OF ACCURATE DIGITS IN THE * OBJECTIVE FUNCTION F(X) * MSG <--> AN INTEGER VARIABLE THAT THE USER MAY SET ON INPUT TO * INHIBIT CERTAIN AUTOMATIC CHECKS OR OVERRIDE CERTAIN * DEFAULT CHARACTERISTICS OF THE PACKAGE * XPLS <-- AN ARRAY OF LENGTH N CONTAINING THE BEST APPROXIMATION * TO THE MINIMIZER UPON RETURN. (IF THE ALGORITHM HAS NOT * CONVERGED, THE LAST ITERATE IS RETURNED) * FPLS <-- A SCALAR VARIABLE THAT CONTAINS THE FUNCTION VALUE AT * THE FINAL ITERATE XPLS * GPLS <-- AN ARRAY OF LENGTH N CONTAINING THE GRADIENT VALUE * AT XPLS * HESS <-- AN ARRAY THAT IS USED TO STORE THE HESSIAN MATRIX AT * EACH ITERATION. IT NEEDS TO BE AT LEAST OF DIMENSION * LICN. ON EXIT, HESS CONTAINS THE HESSIAN VALUE AT THE * MINIMIZER * G,S,D,DN,E,XD,BV,LISTP,NGRP,IPNTR,JPNTR,IWA --> WORKSPACE * LIWRK --> AN INTEGER VARIABLE. IT MUST BE SET BY THE USER TO THE * LENGTH OF ARRAY IWRK AND IS NOT ALTERED BY THE PACKAGE * (SEE SUBROUTINE STUMCD (STUMCS)) * LWRK --> AN INTEGER VARIABLE. IT MUST BE SET BY THE USER TO THE * LENGTH OF ARRAY WRK AND IS NOT ALTERED BY THE PACKAGE * (SEE SUBROUTINE STUMCD (STUMCS)) * TERMCD <-- AN INTEGER THAT SPECIFIES THE REASON WHY THE ALGORITHM * HAS TERMINATED * VECTOR<--> AN ARRAY OF LENGTH N. IT NEED NOT BE SET BY THE USER * ON ENTRY. IF INFORM IS SET TO 1, A RE-ENTRY MUST BE * MADE WITH VECTOR SET TO HESS TIMES VECTOR (SEE INFORM) * INFORM<--> AN INTEGER VARIABLE. IF IT IS SET TO 1, THE USER MUST * OBTAIN HESS TIMES VECTOR AND RE-ENTER WITH INFORM * UNCHANGED. THE RESULT OF HESS TIMES VECTOR MUST BE * STORED IN VECTOR. THE DEFAULT VALUE OF INFORM IS 0, * MEANING THAT HESS TIMES VECTOR IS COMPUTED BY THE * PACKAGE * * BLAS SUBROUTINES: DCOPY,DDOT,DNRM2 * INTEGER N,NPAIRS,LIRN,LICN,ILIM,IPR,METHOD,GRDFLG,HSNFLG INTEGER NDIGIT,MSG,LIWRK,LWRK,TERMCD,INFORM INTEGER IRN(LIRN),ICN(LICN),LISTP(N),NGRP(N),IPNTR(N+1) INTEGER JPNTR(N+1),IWA(*) DOUBLE PRECISION X(N),TYPX(N),FSCALE,GRADTL,STEPTL,STEPMX DOUBLE PRECISION XPLS(N),FPLS,GPLS(N),HESS(LICN),G(N) DOUBLE PRECISION S(N),D(N),DN(N),E(N),XD(N),BV(N),VECTOR(N) LOGICAL SCALE * * LOCAL VARIABLES * INTEGER ITNNO,ICSCMX,I,MAXGRP,MINGRP,IFCNT,IGCNT,IHCNT INTEGER IFLAG,IRETCD,NZ,LIWA DOUBLE PRECISION RNF,ANALTL,F,GNORM,FP,SUM1,SUM2 DOUBLE PRECISION ALMBDA,OCENT,ONE,TWO,THREE,TEN DOUBLE PRECISION EPS,GAMMA,ALPHA,BETA,FN,RGX DOUBLE PRECISION DDOT,DNRM2,D1MACH LOGICAL MXTAKE,DSCENT,FCALL,CHECKH COMMON /ALMB/ALMBDA COMMON /COUNT/IFCNT,IGCNT,IHCNT EXTERNAL FCN,UGR,USH EXTERNAL DCOPY,DNRM2,DDOT DATA OCENT,ONE,TWO,THREE,TEN/1.0D-2,1.0D+0,2.0D+0, + 3.0D+0,10.0D+0/ * * IF INFORM = 2 THEN USER MUST SUPPLY VECTOR = HESS * S * IF (INFORM.EQ.2) GO TO 160 * * COMPUTE MACHINE EPSILON * EPS = D1MACH(3) * * CHECK USER'S INPUT DATA * CALL STCHKI(N,NPAIRS,NZ,X,IRN,LIRN,ICN,LICN,LIWRK,LWRK,SCALE,TYPX, + FSCALE,GRADTL,STEPTL,ILIM,NDIGIT,EPS,METHOD,GRDFLG, + HSNFLG,STEPMX,IWA,LIWA,JPNTR,MSG,IPR) IF (MSG.LT.0) RETURN * * PRINT OUT USED PARAMETERS * IF (MSG.GE.1) THEN WRITE (IPR,9000) GRDFLG WRITE (IPR,9010) HSNFLG WRITE (IPR,9020) METHOD WRITE (IPR,9030) ILIM WRITE (IPR,9040) EPS WRITE (IPR,9050) STEPTL WRITE (IPR,9060) GRADTL WRITE (IPR,9070) STEPMX END IF * * INITIALIZATION * ITNNO = 0 ICSCMX = 0 TERMCD = 0 IFCNT = 0 IGCNT = 0 IHCNT = 0 * * SCALE X * IF (SCALE) THEN DO 10 I = 1,N X(I) = X(I)/TYPX(I) 10 CONTINUE END IF * * INITIAL ITERATION * RNF = MAX(TEN** (-NDIGIT),EPS) ANALTL = MAX(OCENT,SQRT(RNF)) * * UNSCALE X AND COMPUTE F AND G * IF (SCALE) THEN DO 20 I = 1,N VECTOR(I) = X(I)*TYPX(I) 20 CONTINUE ELSE DO 30 I = 1,N VECTOR(I) = X(I) 30 CONTINUE END IF IFCNT = IFCNT + 1 CALL FCN(N,VECTOR,F) IGCNT = IGCNT + 1 IF (GRDFLG.EQ.1) THEN CALL UGR(N,VECTOR,G) CALL STCHKG(N,VECTOR,FCN,F,G,TYPX,RNF,ANALTL,XD,MSG,IPR) IF (MSG.LT.0) RETURN ELSE IF (GRDFLG.EQ.2) THEN CALL UGR(N,VECTOR,G) ELSE CALL STFDGR(N,VECTOR,FCN,F,RNF,G) END IF * * SCALE G * IF (SCALE) THEN DO 40 I = 1,N G(I) = G(I)*TYPX(I) 40 CONTINUE END IF * GNORM = DNRM2(N,G,1) * * PRINT OUT INITIAL ITERATION * IF (MSG.GE.1) THEN CALL STRSLT(N,VECTOR,F,G,RGX,ITNNO,TERMCD,IPR) END IF * * TEST WHETHER INITIAL GUESS SATISFIES THE STOPPING CRITERIA * IF (GNORM.LE.GRADTL) THEN TERMCD = 1 RETURN END IF * * ITERATION 1 * ITNNO = ITNNO + 1 * * COMPUTE HESSIAN * IHCNT = IHCNT + 1 IF (HSNFLG.EQ.1) THEN CALL USH(N,VECTOR,NPAIRS,LICN,HESS,IRN,ICN) FCALL = .TRUE. CHECKH = .TRUE. CALL STCHKH(N,VECTOR,NPAIRS,NZ,IRN,LIRN,ICN,LICN,FCN,UGR, + SCALE,TYPX,GRDFLG,F,G,RNF,ANALTL,FCALL,CHECKH, + LISTP,NGRP,IPNTR,JPNTR,IWA,LIWA,D,XD,E,HESS,IPR, + MSG) IF (MSG.LT.0) RETURN FCALL = .FALSE. ELSE IF (HSNFLG.EQ.2) THEN CALL USH(N,VECTOR,NPAIRS,LICN,HESS,IRN,ICN) CHECKH = .FALSE. CALL STCHKH(N,VECTOR,NPAIRS,NZ,IRN,LIRN,ICN,LICN,FCN,UGR, + SCALE,TYPX,GRDFLG,F,G,RNF,ANALTL,FCALL,CHECKH, + LISTP,NGRP,IPNTR,JPNTR,IWA,LIWA,D,XD,E,HESS,IPR, + MSG) IF (MSG.LT.0) RETURN ELSE FCALL = .TRUE. CALL STFDHS(N,NZ,VECTOR,IRN,ICN,G,RNF,GRDFLG,FCN,UGR,SCALE, + TYPX,FCALL,LISTP,NGRP,IPNTR,JPNTR,IWA,LIWA,D,XD,E, + MAXGRP,MINGRP,HESS) FCALL = .FALSE. END IF * * SCALE HESS * IF (SCALE) THEN DO 50 I = 1,NZ HESS(I) = HESS(I)*TYPX(IRN(I))*TYPX(ICN(I)) 50 CONTINUE END IF * * SOLVE FOR NEWTON STEP D * DO 60 I = 1,N D(I) = -G(I) 60 CONTINUE * * SPARSE CHOLESKY DECOMPOSITION FOR HESS (HESS = L D LT) * CALL STMA27(N,NZ,IRN,ICN,HESS,LICN,IWA,LIWA,D,XD,1,.TRUE.,EPS, + IFLAG) * * COOMPUTE NEWTON STEP USING A LINE SEARCH * CALL STLSCH(N,X,F,G,D,XPLS,FPLS,MXTAKE,IRETCD,STEPMX,STEPTL,SCALE, + TYPX,FCN,VECTOR) * * UPDATE GRADIENT AT XPLS * * UNSCALE XPLS AND COMPUTE GPLS * IF (SCALE) THEN DO 70 I = 1,N VECTOR(I) = XPLS(I)*TYPX(I) 70 CONTINUE ELSE DO 80 I = 1,N VECTOR(I) = XPLS(I) 80 CONTINUE END IF IGCNT = IGCNT + 1 IF (GRDFLG.EQ.0) THEN CALL STFDGR(N,VECTOR,FCN,FPLS,RNF,GPLS) ELSE CALL UGR(N,VECTOR,GPLS) END IF * * SCALE GPLS * IF (SCALE) THEN DO 90 I = 1,N GPLS(I) = GPLS(I)*TYPX(I) 90 CONTINUE END IF * * CHECK STOPPING CONDITIONS * CALL STCHKS(N,XPLS,FPLS,GPLS,X,ITNNO,ICSCMX,TERMCD,GRADTL,STEPTL, + FSCALE,ILIM,IRETCD,MXTAKE,RGX,IPR,MSG) * * IF TERMCD > 0 THEN STOPPING CONDITIONS SATISFIED * IF (TERMCD.GT.0) GO TO 300 * * UPDATE X, F AND S FOR TENSOR MODEL * FP = F F = FPLS DO 100 I = 1,N S(I) = X(I) - XPLS(I) X(I) = XPLS(I) 100 CONTINUE * * IF MSG >= 2 THEN PRINT OUT EACH ITERATION * IF (MSG.GE.2) THEN CALL STRSLT(N,XPLS,FPLS,GPLS,RGX,ITNNO,TERMCD,IPR) END IF * * ITERATION > 1 * * UNSCALE X AND COMPUTE HESS * 110 CONTINUE * IF (SCALE) THEN DO 120 I = 1,N VECTOR(I) = X(I)*TYPX(I) 120 CONTINUE ELSE DO 130 I = 1,N VECTOR(I) = X(I) 130 CONTINUE END IF IHCNT = IHCNT + 1 IF (HSNFLG.EQ.0) THEN CALL STFDHS(N,NZ,VECTOR,IRN,ICN,GPLS,RNF,GRDFLG,FCN,UGR,SCALE, + TYPX,FCALL,LISTP,NGRP,IPNTR,JPNTR,IWA,LIWA,D,XD,E, + MAXGRP,MINGRP,HESS) ELSE CALL USH(N,VECTOR,NZ,LICN,HESS,IRN,ICN) END IF * * SCALE HESS * IF (SCALE) THEN DO 140 I = 1,NZ HESS(I) = HESS(I)*TYPX(IRN(I))*TYPX(ICN(I)) 140 CONTINUE END IF * * IF METHOD = 0 THEN USE NEWTON STEP ONLY * IF (METHOD.EQ.0) THEN * * COMPUTE NEWTON STEP * DO 150 I = 1,N D(I) = -GPLS(I) 150 CONTINUE * * CHOLESKY DECOMPOSITION FOR HESS * CALL STMA27(N,NZ,IRN,ICN,HESS,LICN,IWA,LIWA,D,XD,1,.TRUE.,EPS, + IFLAG) GO TO 190 * END IF * IF (INFORM.EQ.1) THEN INFORM = 2 CALL DCOPY(N,S(1),1,VECTOR(1),1) RETURN ELSE CALL STHMUV(N,NZ,IRN,ICN,HESS,S,VECTOR) GO TO 170 END IF * 160 INFORM = 1 170 CONTINUE * * FORM TENSOR MODEL * CALL STFTSM(N,F,FP,GPLS,G,S,ALPHA,BETA,VECTOR,D) * SUM1 = DDOT(N,S(1),1,S(1),1) SUM2 = DDOT(N,S(1),1,D(1),1) * DO 180 I = 1,N BV(I) = (THREE*SUM1*D(I)-TWO*S(I)*SUM2)/ (THREE*SUM1**3) 180 CONTINUE GAMMA = BETA/SUM1**4 * * SOLVE TENSOR MODEL AND COMPUTE TENSOR STEP * CALL STMSLV(N,D,S,GPLS,E,VECTOR,DN,BV,GAMMA,NZ,IRN,ICN,HESS,LICN, + IWA,LIWA,XD,EPS,DSCENT,IFLAG) * IF (.NOT.DSCENT) THEN CALL DCOPY(N,DN(1),1,D(1),1) END IF * 190 ITNNO = ITNNO + 1 CALL DCOPY(N,GPLS(1),1,G(1),1) * * COMPUTE TENSOR (OR NEWTON) STEP USING A LINE SEARCH * CALL STLSCH(N,X,F,G,D,XPLS,FPLS,MXTAKE,IRETCD,STEPMX,STEPTL,SCALE, + TYPX,FCN,VECTOR) * IF (METHOD.EQ.0) GO TO 260 IF (ALMBDA.EQ.ONE) GO TO 240 * IF (DSCENT) THEN * * FULL TENSOR STEP IS A DESCENT DIRECTION BUT DOES NOT * PROVIDE ENOUGH DECREASE IN THE OBJECTIVE FUNCTION. * CALCULATE NEWTON STEP ONLY IF HESSIAN IS INDEFINITE; * OTHERWISE NEWTON STEP WAS ALREADY COMPUTED AS PART * OF THE TENSOR STEP COMPUTATION * IF (IFLAG.EQ.3) THEN DO 200 I = 1,N DN(I) = -E(I) 200 CONTINUE GO TO 230 END IF CALL DCOPY(N,G(1),1,DN(1),1) CALL STMA27(N,NZ,IRN,ICN,HESS,LICN,IWA,LIWA,DN,XD,3,.TRUE., + EPS,IFLAG) IF (IFLAG.EQ.30) THEN DO 210 I = 1,N DN(I) = -E(I) 210 CONTINUE ELSE DO 220 I = 1,N DN(I) = -DN(I) 220 CONTINUE END IF 230 CONTINUE * * CALCULATE A STEP IN THE NEWTON DIRECTION * CALL STLSCH(N,X,F,G,DN,XD,FN,MXTAKE,IRETCD,STEPMX,STEPTL, + SCALE,TYPX,FCN,VECTOR) * * COMPARE TENSOR STEP TO NEWTON STEP * IF NEWTON STEP IS BETTER, SET NEXT ITERATE TO NEW NEWTON POINT * IF (FN.LT.FPLS) THEN FPLS = FN CALL DCOPY(N,DN(1),1,D(1),1) CALL DCOPY(N,XD(1),1,XPLS(1),1) END IF END IF 240 CONTINUE DO 250 I = 1,N D(I) = XPLS(I) - X(I) 250 CONTINUE * 260 CONTINUE * * UNSCALE XPLS, AND COMPUTE FPLS AND GPLS * IF (SCALE) THEN DO 270 I = 1,N VECTOR(I) = XPLS(I)*TYPX(I) 270 CONTINUE ELSE DO 280 I = 1,N VECTOR(I) = XPLS(I) 280 CONTINUE END IF IGCNT = IGCNT + 1 IF (GRDFLG.EQ.0) THEN CALL STFDGR(N,VECTOR,FCN,FPLS,RNF,GPLS) ELSE CALL UGR(N,VECTOR,GPLS) END IF * * SCALE GPLS * IF (SCALE) THEN DO 290 I = 1,N GPLS(I) = GPLS(I)*TYPX(I) 290 CONTINUE END IF * * CHECK STOPPING CONDITIONS * CALL STCHKS(N,XPLS,FPLS,GPLS,X,ITNNO,ICSCMX,TERMCD,GRADTL,STEPTL, + FSCALE,ILIM,IRETCD,MXTAKE,RGX,IPR,MSG) * * IF TERMCD = 0 THEN EXECUTION CONTINUES * IF (TERMCD.EQ.0) GO TO 330 * 300 CONTINUE * * TRANSFORM XPLS BACK TO ORIGINAL SPACE * IF (SCALE) THEN DO 310 I = 1,N XPLS(I) = XPLS(I)*TYPX(I) 310 CONTINUE END IF * * IF MSG >= 1 THEN PRINT OUT FINAL ITERATION * IF (MSG.GE.1) THEN CALL STRSLT(N,XPLS,FPLS,GPLS,RGX,ITNNO,TERMCD,IPR) END IF * * UPDATE THE HESSIAN * IHCNT = IHCNT + 1 IF (HSNFLG.EQ.0) THEN CALL STFDHS(N,NZ,VECTOR,IRN,ICN,GPLS,RNF,GRDFLG,FCN,UGR,SCALE, + TYPX,FCALL,LISTP,NGRP,IPNTR,JPNTR,IWA,LIWA,D,XD,E, + MAXGRP,MINGRP,HESS) ELSE CALL USH(N,XPLS,NZ,LICN,HESS,IRN,ICN) END IF * * SCALE HESS * IF (SCALE) THEN DO 320 I = 1,NZ HESS(I) = HESS(I)*TYPX(IRN(I))*TYPX(ICN(I)) 320 CONTINUE END IF RETURN * * UPDATE INFORMATION AT THE CURRENT POINT * 330 CONTINUE CALL DCOPY(N,XPLS(1),1,X(1),1) DO 340 I = 1,N S(I) = -D(I) 340 CONTINUE * * IF ITERATION LIMIT EXCEEDED THEN RETURN * IF (ITNNO.GT.ILIM) GO TO 300 * * IF MSG >= 2 THEN PRINT OUT EACH ITERATION * IF (MSG.GE.2) THEN CALL STRSLT(N,XPLS,FPLS,GPLS,RGX,ITNNO,TERMCD,IPR) END IF * * UPDATE F * FP = F F = FPLS * * PERFORM NEXT ITERATION * GO TO 110 * * END OF ITERATION > 1 * * 9000 FORMAT (' STDRUO GRADIENT FLAG = ',I1) 9010 FORMAT (' STDRUO HESSIAN FLAG = ',I1) 9020 FORMAT (' STDRUO METHOD = ',I1) 9030 FORMAT (' STDRUO ITERATION LIMIT =',I5) 9040 FORMAT (' STDRUO MACHINE EPSILON = ',E20.13) 9050 FORMAT (' STDRUO STEP TOLERANCE = ',E20.13) 9060 FORMAT (' STDRUO GRADIENT TOLERANCE = ',E20.13) 9070 FORMAT (' STDRUO MAXIMUM STEP SIZE = ',E20.13) END SUBROUTINE STDUGR(N,X,G) * * PURPOSE: * ------- * * THIS IS A DUMMY ROUTINE TO PREVENT UNSATISFIED EXTERNAL DIAGNOSTIC * WHEN SPECIFIC ANALYTIC GRADIENT IS NOT SUPPLIED. * INTEGER N DOUBLE PRECISION X(N),G(N) RETURN END SUBROUTINE STDUSH(N,X,NZ,LICN,HESS,IRN,ICN) * * PURPOSE: * ------- * * THIS IS A DUMMY ROUTINE TO PREVENT UNSATISFIED EXTERNAL DIAGNOSTIC * WHEN SPECIFIC ANALYTIC HESSIAN IS NOT SUPPLIED. * INTEGER N,NZ,LICN INTEGER IRN(NZ),ICN(LICN) DOUBLE PRECISION X(N),HESS(LICN) RETURN END SUBROUTINE STFDGR(N,XPLS,FCN,FPLS,RNOISE,FDG) * * PURPOSE * ------- * * THIS ROUTINE FINDS A FORWARD FINITE-DIFFERENCE APPROXIMATION "FDG" * TO THE FIRST DERIVATIVE OF THE FUNCTION DEFINED BY THE SUBPROGRAM * "FCN" EVALUATED AT THE NEW ITERATE "XPLS". * * PARAMETERS: * ---------- * N --> DIMENSION OF PROBLEM * XPLS --> NEW ITERATE * FCN --> NAME OF SUBROUTINE THAT EVALUATES THE * OPTIMIZATION FUNCTION * FPLS --> FUNCTION VALUE AT NEW ITERATE * RNOISE --> RELATIVE NOISE IN FCN * FDG <-- FINITE-DIFFERENCE APPROXIMATION OF GRADIENT AT XPLS * INTEGER N DOUBLE PRECISION XPLS(N),FPLS,RNOISE,FDG(N) * * LOCAL VARIABLES * INTEGER J DOUBLE PRECISION XTMPJ,STEPSZ,FHAT,ONE EXTERNAL FCN INTRINSIC ABS,MAX,SQRT DATA ONE/1.0D+0/ * * FIND DERIVATIVE OF FCN WITH RESPECT TO XPLS(J), J=1,N * DO 10 J = 1,N XTMPJ = XPLS(J) STEPSZ = SQRT(RNOISE)*MAX(ABS(XPLS(J)),ONE) XPLS(J) = XTMPJ + STEPSZ CALL FCN(N,XPLS,FHAT) XPLS(J) = XTMPJ FDG(J) = (FHAT-FPLS)/STEPSZ 10 CONTINUE RETURN END SUBROUTINE STFDHS(N,NZ,X,IRN,ICN,G,RNF,GRDFLG,FCN,UGR,SCALE,TYPX, + FCALL,LISTP,NGRP,IPNTR,JPNTR,IWA,LIWA,FHESD,XD, + ETA,MAXGRP,MINGRP,HESS) * * PURPOSE: * ------- * * THIS ROUTINE FINDS A FORWARD FINITE-DIFFERENCE APPROXIMATION "HESS" * TO THE SECOND DERIVATIVE (HESSIAN) OF THE FUNCTION DEFINED BY THE * SUBPROGRAM "FCN" EVALUATED AT THE NEW ITERATE "XPLS". * * PARAMETERS: * ---------- * * N --> DIMENSION OF PROBLEM * NZ --> NUMBER OF NONZEROS IN THE LOWER TRIANGULAR PART * OF THE HESSIAN MATRIX * X --> CURRENT ITERATE * IRN --> ROW INDEX OF THE NONZERO ELEMENTS IN THE LOWER * TRIANGULAR PART OF THE HESSIAN * ICN --> COLUMN INDEX OF THE NONZERO ELEMENTS IN THE LOWER * TRIANGULAR PART OF THE HESSIAN * G --> GRADIENT VALUE AT X * RNF --> RELATIVE NOISE IN OPTIMIZATION FUNCTION FCN * GRDFLG --> =1 IF ANALYTIC GRADIENT SUPPLIED * FCN --> NAME OF A USER SUPPLIED SUBROUTINE THAT EVALUATES * THE FUNCTION F AT AN ARBITRARY VECTOR X * UGR --> NAME OF A USER SUPPLIED SUBROUTINE THAT RETURNS IN * THE VALUE OF THE GRADIENT AT AN ARBITRARY VECTOR X * SCALE --> LOGICAL VARIABLE INDICATING WHETHER OR NOT SCALING * OF THE VARIABLES IS PERFORMED * TYPX --> AN ARRAY OF LENGTH N IN WHICH THE TYPICAL SIZE OF THE * COMPONENTS OF X ARE SPECIFIED * FCALL --> LOGICAL FLAG. IF THE HESSIAN MATRIX IS EVALUATED FOR * THE FIRST TIME, THEN FCALL IS SET TO TRUE; OTHERWISE * IT IS SET TO FALSE * LISTP,NGRP,IPNTR,JPNTR,IWA --> WORKSPACE * (SEE SUBROUTINE DSSM FOR DETAILS) * LIWA --> LENGTH OF IWA ARRAY * FHESD,XD,ETA --> WORKSPACE * MAXGRP,MINGRP<-- SEE SUBROUTINE DSSM FOR DETAILS * HESS <-- FINITE-DIFFERENCE HESSIAN ON EXIT * * INTEGER N,NZ,GRDFLG,LIWA,MAXGRP,MINGRP INTEGER IRN(NZ),ICN(NZ),LISTP(N),NGRP(N),IPNTR(N+1) INTEGER JPNTR(N+1),IWA(LIWA) DOUBLE PRECISION X(N),G(N),TYPX(N),RNF,FHESD(N) DOUBLE PRECISION XD(N),ETA(N),HESS(NZ) LOGICAL SCALE * * LOCAL VARIABLES * INTEGER JP,I,INFO,J,METHOD,NUMGRP DOUBLE PRECISION FD,ONE LOGICAL FCALL EXTERNAL FCN,UGR INTRINSIC ABS,MAX,SQRT DATA ONE/1.0D+0/ * METHOD = 1 * * CALL DSM ONLY IF IT IS THE FIRST TIME * IF (FCALL) THEN CALL DSSM(N,NZ,IRN,ICN,METHOD,LISTP,NGRP,MAXGRP,MINGRP,INFO, + IPNTR,JPNTR,IWA,LIWA) ELSE CALL SETR(N,N,IRN,JPNTR,ICN,IPNTR,IWA) END IF * * UNSCALE G * IF (SCALE) THEN DO 10 I = 1,N G(I) = G(I)/TYPX(I) 10 CONTINUE END IF * * APPROXIMATE THE HESSIAN MATRIX * DO 40 NUMGRP = 1,MAXGRP DO 20 J = 1,N ETA(J) = SQRT(RNF)*MAX(ABS(X(J)),ONE) XD(J) = X(J) IF (NGRP(J).EQ.NUMGRP) XD(J) = X(J) + ETA(J) 20 CONTINUE IF (GRDFLG.EQ.0) THEN CALL FCN(N,XD,FD) CALL STFDGR(N,XD,FCN,FD,RNF,FHESD) ELSE CALL UGR(N,XD,FHESD) END IF DO 30 I = 1,N FHESD(I) = FHESD(I) - G(I) 30 CONTINUE CALL FDHS(N,IRN,JPNTR,ICN,IPNTR,LISTP,NGRP,MAXGRP,NUMGRP,ETA, + FHESD,HESS,IWA) 40 CONTINUE DO 60 J = 1,N DO 50 JP = JPNTR(J),JPNTR(J+1) - 1 ICN(JP) = J 50 CONTINUE 60 CONTINUE * * SCALE G * IF (SCALE) THEN DO 70 I = 1,N G(I) = G(I)*TYPX(I) 70 CONTINUE END IF * END SUBROUTINE STFTSM(N,F,FP,G,GP,S,ALPHA,BETA,SH,A) * * PURPOSE: * ------- * * THIS ROUTINE FORMS THE TENSOR MODEL. * * PARAMETERS: * ---------- * * N --> DIMENSION OF PROBLEM * F --> CURRENT FUNCTION VALUE * FP --> PREVIOUS FUNCTION VALUE * G --> CURRENT GRADIENT * GP --> PREVIOUS GRADIENT * S --> STEP TO PREVIOUS POINT * ALPHA <-- SCALAR TO FORM 3RD ORDER TERM OF TENSOR MODEL * BETA <-- SCALAR TO FORM 4TH ORDER TERM OF TENSOR MODEL * SH <-- HESSIAN MATRIX TIMES S (SEE SUBROUTINE STHMUV) * A <-- A=2*(GP-G-SH-S*BETA/(6*(S-TRANS*S))) * * BLAS SUBROUTINES: DDOT * INTEGER N DOUBLE PRECISION F,FP,G(N),GP(N),S(N),ALPHA,BETA,SH(N),A(N) * * LOCAL VARIABLES * INTEGER I DOUBLE PRECISION GS,GPS,SHS,B1,B2,STS DOUBLE PRECISION HALF,TWO,SIX,TFOUR,STWO DOUBLE PRECISION DDOT EXTERNAL DDOT DATA HALF,TWO,SIX,TFOUR,STWO/5.0D-1,2.0D+0,6.0D+0, + 24.0D+0,72.0D+0/ * GS = DDOT(N,G(1),1,S(1),1) GPS = DDOT(N,GP(1),1,S(1),1) SHS = DDOT(N,SH(1),1,S(1),1) B1 = GPS - GS - SHS B2 = FP - F - GS - HALF*SHS ALPHA = TFOUR*B2 - SIX*B1 BETA = TFOUR*B1 - STWO*B2 * * COMPUTE A * STS = DDOT(N,S(1),1,S(1),1) DO 10 I = 1,N A(I) = TWO* (GP(I)-G(I)-SH(I)-S(I)*BETA/ (SIX*STS)) 10 CONTINUE RETURN END SUBROUTINE STHMUV(N,NZ,IRN,ICN,HESS,S,SH) * * PURPOSE: * ------- * * THIS ROUTINE COMPUTES SH = HESS * S. * * PARAMETERS: * ---------- * N --> A POSITIVE INTEGER VARIABLE SPECIFYING THE NUMBER OF * VARIABLES IN THE PROBLEM * NZ --> NUMBER OF NONZEROS STORED IN THE LOWER HALF OF * THE HESSIAN MATRIX * IRN --> AN INTEGER ARRAY OF LENGTH LIRN. ON ENTRY, IT MUST * HOLD THE ROW INDEX OF EACH NONZERO STORED IN THE LOWER * HALF OF THE HESSIAN MATRIX * ICN --> AN INTEGER ARRAY OF LENGTH LICN. ON ENTRY, IT MUST * HOLD THE COLUMN INDEX OF THE NONZEROS STORED IN LOWER * HALF OF THE HESSIAN MATRIX * HESS --> HESSIAN MATRIX AT CURRENT ITERATE * S --> STEP TO PREVIOUS POINT * SH <-- HESS * S ON EXIT * * INTEGER N,NZ INTEGER IRN(NZ),ICN(NZ) DOUBLE PRECISION HESS(NZ),S(N),SH(N) * * LOCAL VARIABLES * INTEGER I,II,JJ DOUBLE PRECISION ZERO DATA ZERO/0.0D0/ * DO 10 I = 1,N SH(I) = ZERO 10 CONTINUE DO 20 I = 1,NZ II = IRN(I) JJ = ICN(I) SH(II) = SH(II) + HESS(I)*S(JJ) IF (II.NE.JJ) THEN SH(JJ) = SH(JJ) + HESS(I)*S(II) END IF 20 CONTINUE RETURN END SUBROUTINE STLSCH(N,X,F,G,P,XPLS,FPLS,MXTAKE,IRETCD,STEPMX,STEPTL, + SCALE,TYPX,FCN,W2) * * PURPOSE: * ------- * * THIS ROUTINE FIND A NEXT ITERATE BY A LINE SEARCH. * THIS IS THE ALPHA CONDITION ONLY LINE SEARCH. * * PARAMETERS: * ---------- * N --> DIMENSION OF PROBLEM * X --> OLD ITERATE * F --> FUNCTION VALUE AT OLD ITERATE * G --> GRADIENT AT OLD ITERATE * P --> NEWTON OR TENSOR DIRECTION * XPLS <-- NEW ITERATE * FPLS <-- FUNCTION VALUE AT NEW ITERATE * MXTAKE <-- BOOLEAN FLAG INDICATING STEP OF MAXIMUM LENGTH USED * IRETCD <-- RETURN CODE * STEPMX --> MAXIMUM ALLOWABLE STEP SIZE * STEPTL --> RELATIVE STEP SIZE AT WHICH SUCCESSIVE ITERATES * CONSIDERED CLOSE ENOUGH TO TERMINATE ALGORITHM * SCALE --> LOGICAL VARIABLE INDICATING WHETHER OR NOT SCALING * OF THE VARIABLES IS PERFORMED * TYPX --> DIAGONAL SCALING MATRIX FOR X * FCN --> NAME OF SUBROUTINE TO EVALUATE OPTIMIZATION FUNCTION * W2 --> WORKING SPACE * * BLAS SUBROUTINES: DDOT,DSCAL * INTEGER N,IRETCD DOUBLE PRECISION X(N),F,G(N),P(N),XPLS(N),FPLS,STEPMX DOUBLE PRECISION STEPTL,TYPX(N),W2(N) LOGICAL SCALE * * LOCAL VARIABLES * INTEGER I,K INTEGER IFCNT,IGCNT,IHCNT DOUBLE PRECISION TMP,SLN,SCL,SLP,RLN DOUBLE PRECISION TEMP,TEMP1,TEMP2,ALMBMN,ALMBDA DOUBLE PRECISION TLMBDA,T1,T2,T3,PLMBDA DOUBLE PRECISION A,B,DISC,PFPLS,ZERO,TENTH DOUBLE PRECISION HALF,Z99,ONE,TWO,THREE,ALPHA DOUBLE PRECISION DDOT LOGICAL MXTAKE COMMON /COUNT/IFCNT,IGCNT,IHCNT COMMON /ALMB/ALMBDA EXTERNAL FCN,DSCAL,DDOT INTRINSIC ABS,MAX,SQRT DATA ZERO,TENTH,HALF,Z99,ONE,TWO,THREE,ALPHA/0.0D+0, + 1.0D-1,5.0D-1,0.99D+0,1.0D+0,2.0D+0,3.0D+0, + 1.0D-4/ * MXTAKE = .FALSE. IRETCD = 2 *$ WRITE(IPR,954) *$ WRITE(IPR,955) (P(I),I=1,N) TMP = ZERO DO 10 I = 1,N TMP = TMP + P(I)*P(I) 10 CONTINUE SLN = SQRT(TMP) IF (SLN.LE.STEPMX) GO TO 20 * * STEP LONGER THAN MAXIMUM ALLOWED * SCL = STEPMX/SLN CALL DSCAL(N,SCL,P(1),1) SLN = STEPMX *$ WRITE(IPR,954) *$ WRITE(IPR,955) (P(I),I=1,N) 20 CONTINUE SLP = DDOT(N,G(1),1,P(1),1) RLN = ZERO DO 30 I = 1,N TEMP = ONE TEMP1 = ABS(X(I)) TEMP2 = MAX(TEMP1,TEMP) TEMP1 = ABS(P(I)) RLN = MAX(RLN,TEMP1/TEMP2) 30 CONTINUE ALMBMN = STEPTL/RLN ALMBDA = ONE *$ WRITE(IPR,952) SLN,SLP,RMNLMB,STEPMX,STEPTL * * LOOP * CHECK IF NEW ITERATE SATISFACTORY. GENERATE NEW LAMBDA IF NECESSARY. * 40 CONTINUE IF (IRETCD.LT.2) THEN RETURN END IF DO 50 I = 1,N XPLS(I) = X(I) + ALMBDA*P(I) 50 CONTINUE IFCNT = IFCNT + 1 IF (SCALE) THEN DO 60 K = 1,N W2(K) = XPLS(K)*TYPX(K) 60 CONTINUE CALL FCN(N,W2,FPLS) ELSE CALL FCN(N,XPLS,FPLS) END IF *$ WRITE(IPR,956) ALMBDA *$ WRITE(IPR,951) *$ WRITE(IPR,955) (XPLS(I),I=1,N) *$ WRITE(IPR,953) FPLS IF (FPLS.GT.F+SLP*ALPHA*ALMBDA) GO TO 70 * IF(FPLS.LE. F+SLP*1.E-4*ALMBDA) * THEN * * SOLUTION FOUND * IRETCD = 0 IF (ALMBDA.EQ.TENTH .AND. SLN.GT.Z99*STEPMX) MXTAKE = .TRUE. GO TO 40 * * SOLUTION NOT (YET) FOUND * * ELSE 70 IF (ALMBDA.GE.ALMBMN) GO TO 80 * IF(ALMBDA .LT. ALMBMN) * THEN * * NO SATISFACTORY XPLS FOUND SUFFICIENTLY DISTINCT FROM X * IRETCD = 1 GO TO 40 * ELSE * * CALCULATE NEW LAMBDA * 80 IF (ALMBDA.NE.ONE) GO TO 90 * IF(ALMBDA.EQ.1.0) * THEN * * FIRST BACKTRACK: QUADRATIC FIT * TLMBDA = -SLP/ (TWO* (FPLS-F-SLP)) GO TO 120 * ELSE * * ALL SUBSEQUENT BACKTRACKS: CUBIC FIT * 90 T1 = FPLS - F - ALMBDA*SLP T2 = PFPLS - F - PLMBDA*SLP T3 = ONE/ (ALMBDA-PLMBDA) A = T3* (T1/ (ALMBDA*ALMBDA)-T2/ (PLMBDA*PLMBDA)) B = T3* (T2*ALMBDA/ (PLMBDA*PLMBDA)-T1*PLMBDA/ (ALMBDA*ALMBDA)) DISC = B*B - THREE*A*SLP IF (DISC.LE.B*B) GO TO 100 * IF(DISC.GT. B*B) * THEN * * ONLY ONE POSITIVE CRITICAL POINT, MUST BE MINIMUM * TLMBDA = (-B+SIGN(ONE,A)*SQRT(DISC))/ (THREE*A) GO TO 110 * ELSE * * BOTH CRITICAL POINTS POSITIVE, FIRST IS MINIMUM * 100 TLMBDA = (-B-SIGN(ONE,A)*SQRT(DISC))/ (THREE*A) * ENDIF 110 IF (TLMBDA.GT.HALF*ALMBDA) TLMBDA = HALF*ALMBDA * ENDIF 120 PLMBDA = ALMBDA PFPLS = FPLS IF (TLMBDA.GE.ALMBDA*TENTH) GO TO 130 * IF(TLMBDA.LT.ALMBDA/TENTH) * THEN ALMBDA = ALMBDA*TENTH GO TO 140 * ELSE 130 ALMBDA = TLMBDA * ENDIF * ENDIF * ENDIF 140 GO TO 40 *$951 FORMAT(' STLSCH NEW ITERATE (XPLS)') *$952 FORMAT(' STLSCH SLN =',E20.13/ *$ * ' STLSCH SLP =',E20.13/ *$ * ' STLSCH ALMBMN=',E20.13/ *$ * ' STLSCH STEPMX=',E20.13/ *$ * ' STLSCH STEPTL=',E20.13) *$953 FORMAT(' STLSCH F(XPLS)=',E20.13) *$954 FORMAT(' STLSCH COMPUTED STEP (P)') *$955 FORMAT(' STLSCH ',5(E20.13,3X)) *$956 FORMAT(' STLSCH ALMBDA=',E20.13) END SUBROUTINE STMA27(N,NZ,IRN,ICN,A,MAXA,IW,MAXIW,RHS,W,ISLV,NEWTON, + EPS,IFLAG) * * PURPOSE: * -------- * * THIS ROUTINE SOLVES A SPARSE SYMMETRIC SYSTEM OF LINEAR EQUATIONS: * A * X = RHS. * * PARAMETERS: * ---------- * * N --> DIMENSION OF MATRIX A * NZ --> NUMBER OF NONZEROS IN THE LOWER HALF OF MATRIX A * IRN --> ROW INDEX OF A. IRN IS UNALTERED BY MA27 * ICN --> COLUMN INDEX OF A. ICN IS UNALTERED BY MA27 * A <-> A SYMMETRIC MATRIX OF DIMENSION N. ON EXIT, IT HOLDS * THE FACTORS L AND D OF THE L D L-TRANS FACTORIZATION * MAXA --> MAXIMUM LENGTH OF MATRIX A * IW <-- WORKSPACE OF DIMENSION MAXIW. IT IS USED TO * HOLD INTEGER INFORMATION ON THE FACTORS L AND D * RHS <-> RIGHT HAND SIDE VECTOR OF THE EQUATIONS BEING SOLVED. * ON EXIT, IT HOLDS THE SOLUTION VECTOR * W --> WORKSPACE OF DIMENSION N * ISLV --> FLAG WITH THE FOLLOWING MEANINGS: * ISLV = 0 : ANALAYSE AND FACTORIZE MATRIX A * ISLV = 1 : ANALAYSE, FACTORIZE, AND SOLVE SYSTEMS OF * EQUATIONS * ISLV = 2 : PERFORM A SOLVE ONLY (A HAS ALREADY BEEN * FACTORIZED PREVIOUSLY) * ISLV = 3 : CHANGE THE NEGATIVE SIGNS OF THE EIGENVALUES * TO PLUS IF ANY * NEWTON --> LOGICAL PARAMETER. IT IS SET TO TRUE IF NEWTON'S METHOD * IS USED AND TO FALSE IF THE TENSOR METHOD IS USED * EPS --> MACHINE PRECISION * IFLAG <-- A VALUE OF ZERO INDICATES THAT THE SUBROUTINE HAS * PERFORMED SUCCESSFULLY * INTEGER N,NZ,MAXA,MAXIW,ISLV,IFLAG INTEGER IRN(NZ),ICN(MAXA),IW(MAXIW) DOUBLE PRECISION A(MAXA),RHS(N),W(N),EPS LOGICAL NEWTON * * LOCAL VARIABLES * INTEGER NEG1,NEG2,LDIAG,LP,MP,ISTIW,LIW,NSTEPS,MAXFRT DOUBLE PRECISION U,TENTH COMMON /MA27DD/U,LP,MP,LDIAG DATA TENTH/0.1D+0/ * IF (ISLV.EQ.2) GO TO 20 IF (ISLV.EQ.3) GO TO 10 U = TENTH LDIAG = 0 * * SET UP SUBDIVISION OF ARRAYS * ISTIW = 5*N + 1 LIW = MAXIW - 5*N IFLAG = 0 * * ANALYSE SPARSITY PATTERN * CALL MA27AD(N,NZ,IRN,ICN,IW(ISTIW),LIW,IW,IW(3*N+1),NSTEPS,IFLAG) * * FACTORIZE MATRIX * CALL MA27BD(N,NZ,IRN,ICN,A,MAXA,IW(ISTIW),LIW,IW,NSTEPS,MAXFRT, + IW(3*N+1),IFLAG) * * CHANGE THE NEGATIVE SIGNS OF THE EIGENVALUES TO PLUS * 10 CONTINUE IF (NEWTON .OR. IFLAG.EQ.3) THEN CALL DSYPRC(MAXA,LIW,A,IW(ISTIW),EPS,NEG1,NEG2) END IF * IF (ISLV.EQ.3 .AND. (NEG1.EQ.0.AND.NEG2.EQ.0)) THEN IFLAG = 30 GO TO 30 END IF * 20 CONTINUE IF (ISLV.EQ.0) GO TO 30 * * SOLVE THE EQUATIONS * CALL MA27CD(N,A,MAXA,IW(ISTIW),LIW,W,MAXFRT,RHS,IW(3*N+1),NSTEPS) 30 RETURN END SUBROUTINE STMSLV(N,DT,S,G,W1,W2,W3,BV,GAMMA,NZ,IRN,ICN,HESS,LICN, + IWA,LIWA,WK,EPS,DSCENT,IFLAG) * * PURPOSE: * ------- * * THIS ROUTINE COMPUTES THE TENSOR STEP. IT ALSO COMPUTES THE * NEWTON STEP IF THE TENSOR STEP IS NOT DESCENT. * * PARAMETERS: * ---------- * * N --> DIMENSION OF PROBLEM * DT <-- TENSOR STEP ON EXIT * S --> STEP TO PREVIOUS POINT * G --> GRADIENT AT CURRENT ITERATE * W1 --> WORKSPACE ARRAY OF DIMENSION N. IT IS USED TO HOLD * THE SOLUTION OF HESS W1 = G * W2 --> WORKSPACE ARRAY OF DIMENSION N. IT IS USED TO HOLD * THE SOLUTION OF HESS W2 = S * W3 <-- NEWTON STEP (IF REQUIRED) * BV --> (3*(S-TRANS*S)*D-2*(S-TRANS*D)*S)/(3*(S-TRANS*S)**3) * (SEE D IN SUBROUTINE STFTSM) * GAMMA --> BETA/(S-TRANS*S)**4 (SEE BETA IN SUBROUTINE STFTSM) * NZ --> NUMBER OF NONZEROS IN THE LOWER TRIANGULAR PART * OF THE HESSIAN MATRIX * IRN --> ROW INDEX OF THE NONZEROS STORED IN THE LOWER * HALF OF THE HESSIAN MATRIX * ICN --> COLUMN INDEX OF THE NONZEROS STORED IN THE LOWER * HALF OF THE HESSIAN MATRIX * HESS --> HESSIAN MATRIX * LICN --> LENGTH OF ARRAY HESS * IWA --> WORKSPACE * LIWA --> LENGTH OF ARRAY IWA * WK --> WORKSPACE OF DIMENSION N * EPS --> MACHINE PRECISION * DSCENT <-- LOGICAL VARIABLE INDICATING WHETHER OR NOT THE TENSOR * STEP IS DESCENT * IFLAG <-- OUTPUT PARAMETER FROM THE MA27 PACKAGE. A VALUE OF * ZERO INDICATES THAT THE FACTORIZATION WAS PERFORMED * SUCCESSFULLY * * BLAS SUBROUTINES: DCOPY,DDOT * INTEGER N,NZ,LICN,LIWA,IFLAG INTEGER IRN(NZ),ICN(LICN),IWA(6*N) DOUBLE PRECISION DT(N),S(N),G(N),W1(N),W2(N),W3(N) DOUBLE PRECISION BV(N),GAMMA,HESS(LICN),WK(N),EPS LOGICAL DSCENT * * LOCAL VARIABLES * INTEGER NZEROS,I DOUBLE PRECISION BETA(3),THETA(3),UU,WW,VV,YY,ZZ DOUBLE PRECISION AA,A1,A2,A3,AMINB,AMINT,GD DOUBLE PRECISION DDOT,ZERO,HALF,ONE,OHALF,TWO,SIX DATA ZERO,HALF,ONE,OHALF,TWO,SIX/0.0D0,0.5D+0,1.0D+0, + 1.5D+0,2.0D+0,6.0D+0/ * DSCENT = .TRUE. CALL STMA27(N,NZ,IRN,ICN,HESS,LICN,IWA,LIWA,W1,WK,0,.FALSE.,EPS, + IFLAG) * * SOLVE HESS W1 = G FOR W1 * CALL DCOPY(N,G(1),1,W1(1),1) CALL STMA27(N,NZ,IRN,ICN,HESS,LICN,IWA,LIWA,W1,WK,2,.FALSE.,EPS, + IFLAG) * * SOLVE HESS W2 = S FOR W2 * CALL DCOPY(N,S(1),1,W2(1),1) CALL STMA27(N,NZ,IRN,ICN,HESS,LICN,IWA,LIWA,W2,WK,2,.FALSE.,EPS, + IFLAG) * * SOLVE HESS W3 = B FOR W3 * CALL DCOPY(N,BV(1),1,W3(1),1) CALL STMA27(N,NZ,IRN,ICN,HESS,LICN,IWA,LIWA,W3,WK,2,.FALSE.,EPS, + IFLAG) * * COMPUTE COEFFICIENTS OF THE SYSTEM OF TWO EQUATIONS * OF THE 3RD ORDER IN 2 UNKNOWS * UU = DDOT(N,S(1),1,W1(1),1) WW = DDOT(N,S(1),1,W2(1),1) VV = DDOT(N,S(1),1,W3(1),1) YY = DDOT(N,BV(1),1,W1(1),1) ZZ = DDOT(N,BV(1),1,W3(1),1) * * CALCULATE COEFFICIENTS OF THIRD DEGREE POLYNOMIAL * AA = HALF*WW*ZZ - (ONE/SIX)*GAMMA*WW - HALF*VV**2 A1 = -OHALF*VV/AA A2 = (YY*WW-UU*VV-ONE)/AA A3 = -UU/AA * * COMPUTE ZEROS OF THIRD DEGREE POLYNOMIAL * CALL STCZ3P(A1,A2,A3,BETA,NZEROS) * DO 10 I = 1,NZEROS THETA(I) = - (UU+BETA(I)+HALF*VV*BETA(I)**2+ + (ONE/SIX)*GAMMA*WW*BETA(I)**3)/ (WW*BETA(I)) 10 CONTINUE * * COMPUTE SMALLEST BETA IN ABSOLUTE VALUE * AMINB = BETA(1) AMINT = THETA(1) IF (NZEROS.EQ.2) THEN IF (AMINB.GT.ABS(BETA(2))) THEN AMINB = BETA(2) AMINT = THETA(2) END IF ELSE IF (NZEROS.EQ.3) THEN IF (AMINB.GT.ABS(BETA(2))) THEN AMINB = BETA(2) AMINT = THETA(2) END IF IF (AMINB.GT.ABS(BETA(3))) THEN AMINB = BETA(3) AMINT = THETA(3) END IF END IF * * COMPUTE TENSOR STEP * DO 20 I = 1,N DT(I) = -W1(I) - (ONE/TWO)*W3(I)*AMINB**2 - + W2(I)*AMINB*AMINT - (ONE/SIX)*W2(I)*GAMMA*AMINB**3 20 CONTINUE * * IF TENSOR STEP IS NOT A DESCENT DIRECTION THEN COMPUTE NEWTON STEP * GD = DDOT(N,G(1),1,DT(1),1) * IF (GD.GT.ZERO) THEN * * COMPUTE THE NEWTON STEP * DSCENT = .FALSE. IF (IFLAG.EQ.3) THEN DO 30 I = 1,N W3(I) = -W1(I) 30 CONTINUE GO TO 60 END IF CALL DCOPY(N,G(1),1,W3(1),1) CALL STMA27(N,NZ,IRN,ICN,HESS,LICN,IWA,LIWA,W3,WK,3,.TRUE., + EPS,IFLAG) IF (IFLAG.EQ.30) THEN DO 40 I = 1,N W3(I) = -W1(I) 40 CONTINUE ELSE DO 50 I = 1,N W3(I) = -W3(I) 50 CONTINUE END IF END IF * 60 CONTINUE RETURN END SUBROUTINE STRSLT(N,XPLS,FVAL,GPLS,RGX,ITN,TERMCD,IPR) * * PURPOSE: * ------- * * THIS ROUTINE PRINTS INFORMATION. * * PARAMETERS: * ---------- * * N ---> DIMENSION OF PROBLEM * XPLS ---> ITERATE TO BE PRINTED OUT * FVAL ---> FUNCTION VALUE AT XPLS * GPLS ---> GRADIENT AT XPLS * RGX ---> MAXIMUM RELATIVE GRADIENT * ITN ---> ITERATION NUMBER * IPR ---> DEVICE TO WHICH TO SEND OUTPUT * * INTEGER N,ITN,TERMCD,IPR DOUBLE PRECISION XPLS(N),FVAL,GPLS(N),RGX * * LOCAL VARIABLES * INTEGER I,IFCNT,IGCNT,IHCNT DOUBLE PRECISION D,RELGRD,ZERO,ONE COMMON /COUNT/IFCNT,IGCNT,IHCNT DATA ZERO,ONE/0.0D0,1.0D0/ * * FIND DIRECTION IN WHICH RELATIVE GRADIENT MAXIMUM * CHECK WHETHER WITHIN TOLERANCE * IF (ITN.EQ.0) THEN D = ONE RGX = ZERO DO 10 I = 1,N RELGRD = ABS(GPLS(I))*MAX(ABS(XPLS(I)),ONE)/D RGX = MAX(RGX,RELGRD) 10 CONTINUE END IF WRITE (IPR,9000) WRITE (IPR,9010) WRITE (IPR,9020) ITN WRITE (IPR,9030) WRITE (IPR,9040) FVAL WRITE (IPR,9050) WRITE (IPR,9060) RGX WRITE (IPR,9010) WRITE (IPR,9000) IF (TERMCD.GT.0) THEN WRITE (IPR,9070) IFCNT,IGCNT,IHCNT END IF * * RETURN 9000 FORMAT (' ') 9010 FORMAT ('---------------------------------------------') 9020 FORMAT (' STRSLT ITERATION K =',I5) 9030 FORMAT (' STRSLT FUNCTION AT X(K)') 9040 FORMAT (' STRSLT ',E20.13) 9050 FORMAT (' STRSLT SCALED GRADIENT AT X(K)') 9060 FORMAT (' STRSLT ',E20.13) 9070 FORMAT (' STRSLT NUMBER OF FUNCTION EVALUATIONS',I5/, + ' STRSLT NUMBER OF GRADIENT EVALUATIONS',I5/, + ' STRSLT NUMBER OF HESSIAN EVALUATIONS',I5/) END SUBROUTINE STSORT(N,NNZ,A,INDROW,INDCOL,JPNTR,IWA) INTEGER N,NNZ INTEGER INDROW(NNZ),INDCOL(NNZ),JPNTR(N+1),IWA(N) DOUBLE PRECISION A(NNZ) * ********** * * SUBROUTINE STSORT * * GIVEN THE NONZERO ELEMENTS OF AN M BY N MATRIX A IN * ARBITRARY ORDER AS SPECIFIED BY THEIR ROW AND COLUMN * INDICES, THIS SUBROUTINE PERMUTES THESE ELEMENTS SO * THAT THEIR COLUMN INDICES ARE IN NON-DECREASING ORDER. * IT ALSO PERMUTES THE CORRESPONDING A ELEMENTS. * * ON INPUT IT IS ASSUMED THAT THE ELEMENTS ARE SPECIFIED IN * * INDROW(K),INDCOL(K), A(K), K = 1,...,NNZ. * * ON OUTPUT THE ELEMENTS ARE PERMUTED SO THAT INDCOL IS * IN NON-DECREASING ORDER. IN ADDITION, THE ARRAY JPNTR * IS SET SO THAT THE ROW INDICES FOR COLUMN J ARE * * INDROW(K), K = JPNTR(J),...,JPNTR(J+1)-1. * * THE SUBROUTINE STATEMENT IS * * SUBROUTINE STSORT(N,NNZ,A,INDROW,INDCOL,JPNTR,IWA) * * WHERE * * N IS A POSITIVE INTEGER INPUT VARIABLE SET TO THE NUMBER * OF COLUMNS OF A. * * NNZ IS A POSITIVE INTEGER INPUT VARIABLE SET TO THE NUMBER * OF NONZERO ELEMENTS OF A. * * A IS AN ARRAY OF LENGTH NNZ. ON INPUT A CONTAINS THE * THE NONZEROS ELEMENTS. ON OUTPUT A IS PERMUTED SO THAT * THE CORRESPONDING COLUMN INDICES OF INDCOL ARE IN * NON-DECREASING ORDER. * * INDROW IS AN INTEGER ARRAY OF LENGTH NNZ. ON INPUT INDROW * MUST CONTAIN THE ROW INDICES OF THE NONZERO ELEMENTS OF A. * ON OUTPUT INDROW IS PERMUTED SO THAT THE CORRESPONDING * COLUMN INDICES OF INDCOL ARE IN NON-DECREASING ORDER. * * INDCOL IS AN INTEGER ARRAY OF LENGTH NNZ. ON INPUT INDCOL * MUST CONTAIN THE COLUMN INDICES OF THE NONZERO ELEMENTS * OF A. ON OUTPUT INDCOL IS PERMUTED SO THAT THESE INDICES * ARE IN NON-DECREASING ORDER. * * JPNTR IS AN INTEGER OUTPUT ARRAY OF LENGTH N + 1 WHICH * SPECIFIES THE LOCATIONS OF THE ROW INDICES IN THE OUTPUT * INDROW. THE ROW INDICES FOR COLUMN J ARE * * INDROW(K), K = JPNTR(J),...,JPNTR(J+1)-1. * * NOTE THAT JPNTR(1) IS SET TO 1 AND THAT JPNTR(N+1)-1 * IS THEN NNZ. * * IWA IS AN INTEGER WORK ARRAY OF LENGTH N. * * SUBPROGRAMS CALLED * * FORTRAN-SUPPLIED ... MAX * * ARGONNE NATIONAL LABORATORY. MINPACK PROJECT. JULY 1983. * THOMAS F. COLEMAN, BURTON S. GARBOW, JORGE J. MORE' * MODIFIED BY ALI BOUARICHA, OCTOBER 1994. * * ********** INTEGER I,J,K,L DOUBLE PRECISION S * * STORE IN ARRAY IWA THE COUNTS OF NONZEROES IN THE COLUMNS. * DO 10 J = 1,N IWA(J) = 0 10 CONTINUE DO 20 K = 1,NNZ IWA(INDCOL(K)) = IWA(INDCOL(K)) + 1 20 CONTINUE * * SET POINTERS TO THE START OF THE COLUMNS IN INDROW. * JPNTR(1) = 1 DO 30 J = 1,N JPNTR(J+1) = JPNTR(J) + IWA(J) IWA(J) = JPNTR(J) 30 CONTINUE K = 1 * * BEGIN IN-PLACE SORT. * 40 CONTINUE J = INDCOL(K) IF (K.GE.JPNTR(J)) THEN * * CURRENT ELEMENT IS IN POSITION. NOW EXAMINE THE * NEXT ELEMENT OR THE FIRST UN-SORTED ELEMENT IN * THE J-TH GROUP. * K = MAX(K+1,IWA(J)) ELSE * * CURRENT ELEMENT IS NOT IN POSITION. PLACE ELEMENT * IN POSITION AND MAKE THE DISPLACED ELEMENT THE * CURRENT ELEMENT. * L = IWA(J) IWA(J) = IWA(J) + 1 I = INDROW(K) S = A(K) INDROW(K) = INDROW(L) INDCOL(K) = INDCOL(L) A(K) = A(L) INDROW(L) = I INDCOL(L) = J A(L) = S END IF IF (K.LE.NNZ) GO TO 40 RETURN * * LAST CARD OF SUBROUTINE SRTDAT. * END SUBROUTINE STUMCD(N,X,NPAIRS,IRN,LIRN,ICN,LICN,FCN,UGR,USH,TYPX, + FSCALE,GRADTL,STEPTL,ILIM,STEPMX,IPR,METHOD, + GRDFLG,HSNFLG,NDIGIT,MSG,XPLS,FPLS,GPLS,HESS, + WRK,LWRK,IWRK,LIWRK,TERMCD,VECTOR,INFORM) * * PURPOSE: * ------- * * THIS ROUTINE PROVIDES A COMPLETE INTERFACE TO STENMIN. THE USER * HAS FULL CONTROL OVER THE OPTIONS. * * PARAMETERS: * ---------- * * N --> A POSITIVE INTEGER VARIABLE SPECIFYING THE NUMBER * OF VARIABLES IN THE PROBLEM * X --> AN ARRAY OF LENGTH N THAT CONTAINS AN INITIAL * ESTIMATE OF THE MINIMIZER * NPAIRS --> AN INTEGER VARIABLE THAT MUST BE SET BY THE USER TO * THE NUMBER OF NONZEROS STORED IN THE LOWER OR UPPER * HALF OF THE HESSIAN MATRIX * IRN --> AN INTEGER ARRAY OF LENGTH LIRN. ON ENTRY, IT MUST * HOLD THE ROW INDEX OF EACH NONZERO STORED IN THE * LOWER OR UPPER HALF OF THE HESSIAN MATRIX * LIRN --> AN INTEGER VARIABLE THAT MUST BE SET BY THE USER TO * THE LENGTH OF ARRAY IRN. LIRN NEED NOT BE AS LARGE * AS LICN; NORMALLY IT NEED NOT BE VERY MUCH GREATER * THAN NPAIRS * ICN --> AN INTEGER ARRAY OF LENGTH LICN. ON ENTRY, IT MUST * HOLD THE COLUMN INDEX OF THE NONZEROS STORED IN THE * LOWER OR UPPER HALF OF THE HESSIAN MATRIX * LICN --> AN INTEGER VARIABLE THAT MUST BE SET BY THE USER TO * THE LENGTH OF THE HESSIAN ARRAY HESS AND ICN. LICN * SHOULD ORDINARILY BE 2 TO 4 TIMES AS LARGE AS NPAIRS * FCN --> THE NAME OF A USER SUPPLIED SUBROUTINE THAT EVALUATES * THE FUNCTION F AT AN ARBITRARY VECTOR X. THE SUBROUTINE * MUST BE DECLARED EXTERNAL IN THE USER'S CALLING PROGRAM * AND MUST CONFORM TO * CALL FCN(N, X, F) * UGR --> THE NAME OF A USER SUPPLIED SUBROUTINE THAT RETURNS IN * THE VALUE OF THE GRADIENT. UGR MUST BE DECLARED EXTERNAL * IN THE USER'S CALLING PROGRAM AND MUST CONFORM TO THE * USAGE * CALL UGR(N, X, G) * IF NO ANALYTIC GRADIENT IS SUPPLIED (GRDFLG = 0), THE * USER MUST USE THE DUMMY NAME STDUGR * USH --> THE NAME OF A USER SUPPLIED SUBROUTINE THAT RETURNS * IN HESS THE VALUE OF THE HESSIAN AT THE CURRENT POINT X. * USH MUST BE DECLARED EXTERNAL IN THE USER'S CALLING * PROGRAM AND MUST CONFORM TO THE USAGE * CALL USH(N,X,NPAIRS,LICN,HESS,IRN,ICN) * ONLY THE LOWER OR UPPER TRIANGULAR PART AND THE DIAGONAL * OF HESS SHOULD BE GIVEN, WITH THEIR CORRESPONDING ROW * AND COLUMN INDICES. IF NO ANALYTIC GRADIENT IS SUPPLIED * (HSNFLG = 0), THE USER MUST USE THE DUMMY NAME STDUSH * TYPX --> AN ARRAY OF LENGTH N IN WHICH THE TYPICAL SIZE OF THE * COMPONENTS OF X ARE SPECIFIED * FSCALE --> A POSITIVE REAL NUMBER ESTIMATING THE MAGNITUDE * OF F(X) NEAR THE MINIMIZER * GRADTL --> POSITIVE SCALAR GIVING THE TOLERANCE AT WHICH THE * SCALED GRADIENT OF F(X) IS CONSIDERED CLOSE ENOUGH TO * ZERO TO TERMINATE THE ALGORITHM * STEPTL --> A POSITIVE SCALAR PROVIDING THE MINIMUM ALLOWABLE * RELATIVE STEP LENGTH * ILIM --> POSITIVE INTEGER SPECIFYING THE MAXIMUM ITERATIONS TO BE * PERFORMED BEFORE THE PROGRAM IS TERMINATED * STEPMX --> A POSITIVE SCALAR PROVIDING THE MAXIMUM ALLOWABLE SCALED * STEP LENGTH * IPR --> THE UNIT ON WHICH THE ROUTINE OUTPUTS INFORMATION * METHOD --> AN INTEGER FLAG DESIGNATING WHICH METHOD TO USE * METHOD = 0 : USE NEWTON'S METHOD * METHOD = 1 : USE THE TENSOR METHOD * GRDFLG --> INTEGER FLAG DESIGNATING WHETHER OR NOT ANALYTIC HESSIAN * HAS BEEN SUPPLIED BY THE USER * GRDFLG = 0 : NO ANALYTIC GRADIENT SUPPLIED * GRDFLG = 1 : ANALYTIC GRADIENT SUPPLIED (WILL BE CHECKED * AGAINST FINITE DIFFERENCE GRADIENT) * GRDFLG = 2 : ANALYTIC GRADIENT SUPPLIED (WILL NOT BE * CHECKED AGAINST FINITE DIFFERENCE GRADIENT) * HSNFLG --> INTEGER FLAG DESIGNATING WHETHER OR NOT ANALYTIC HESSIAN * HAS BEEN SUPPLIED BY THE USER * HSNFLG = 0 : NO ANALYTIC HESSIAN SUPPLIED * HSNFLG = 1 : ANALYTIC HESSIAN SUPPLIED (WILL BE CHECKED * AGAINST FINITE DIFFERENCE HESSIAN) * HSNFLG = 2 : ANALYTIC HESSIAN SUPPLIED (WILL NOT BE * CHECKED AGAINST FINITE DIFFERENCE HESSIAN) * NDIGIT --> INTEGER ESTIMATING THE NUMBER OF ACCURATE DIGITS IN THE * OBJECTIVE FUNCTION F(X) * MSG <--> AN INTEGER VARIABLE THAT THE USER MAY SET ON INPUT TO * INHIBIT CERTAIN AUTOMATIC CHECKS OR OVERRIDE CERTAIN * DEFAULT CHARACTERISTICS OF THE PACKAGE * XPLS <-- AN ARRAY OF LENGTH N CONTAINING THE BEST APPROXIMATION * TO THE MINIMIZER UPON RETURN. (IF THE ALGORITHM HAS NOT * CONVERGED, THE LAST ITERATE IS RETURNED) * FPLS <-- A SCALAR VARIABLE THAT CONTAINS THE FUNCTION VALUE AT * THE FINAL ITERATE XPLS * GPLS <-- AN ARRAY OF LENGTH N CONTAINING THE GRADIENT VALUE * AT XPLS * HESS <-- AN ARRAY THAT IS USED TO STORE THE HESSIAN MATRIX AT * EACH ITERATION. IT NEEDS TO BE AT LEAST OF DIMENSION * LICN. ON EXIT, HESS CONTAINS THE HESSIAN VALUE AT THE * MINIMIZER * WRK --> AN ARRAY OF LENGTH LWRK. THIS IS USED AS WORKSPACE * BY THE PACKAGE. ITS LENGTH MUST BE AT LEAST 7*N * LWRK --> AN INTEGER VARIABLE. IT MUST BE SET BY THE USER TO THE * LENGTH OF ARRAY WRK AND IS NOT ALTERED BY THE PACKAGE * IWRK --> AN INTEGER ARRAY OF LENGTH LIWRK. THIS IS USED AS * WORKSPACE BY THE PACKAGE. ITS LENGTH MUST BE AT LEAST * 2*NPAIRS+12*N+2 * LIWRK --> AN INTEGER VARIABLE. IT MUST BE SET BY THE USER TO THE * LENGTH OF ARRAY IWRK AND IS NOT ALTERED BY THE PACKAGE * TERMCD <-- AN INTEGER THAT SPECIFIES THE REASON WHY THE ALGORITHM * HAS TERMINATED * VECTOR<--> AN ARRAY OF LENGTH N. IT NEED NOT BE SET BY THE USER * ON ENTRY. IF INFORM IS SET TO 1, A RE-ENTRY MUST BE * MADE WITH VECTOR SET TO HESS TIMES VECTOR (SEE INFORM) * INFORM<--> AN INTEGER VARIABLE. IF IT IS SET TO 1, THE USER MUST * OBTAIN HESS TIMES VECTOR AND RE-ENTER WITH INFORM * UNCHANGED. THE RESULT OF HESS TIMES VECTOR MUST BE * STORED IN VECTOR. THE DEFAULT VALUE OF INFORM IS 0, * MEANING THAT HESS TIMES VECTOR IS COMPUTED BY THE * PACKAGE * INTEGER N,NPAIRS,LIRN,LICN,ILIM,IPR,METHOD INTEGER GRDFLG,HSNFLG,NDIGIT,MSG,LWRK,LIWRK,TERMCD INTEGER INFORM INTEGER IRN(LIRN),ICN(LICN),IWRK(LIWRK) DOUBLE PRECISION X(N),TYPX(N),FSCALE,GRADTL,STEPTL,STEPMX DOUBLE PRECISION XPLS(N),FPLS,GPLS(N),HESS(LICN) DOUBLE PRECISION WRK(LWRK),VECTOR(N) * EXTERNAL FCN,UGR,USH * * LOCAL VARIABLES * * NONE * * * EQUIVALENCE FOR DOUBLE PRECISION/REAL ARRAYS * * EQUIVALENCE WRK(1) = G(N) * WRK(N+1) = S(N) * WRK(2*N+1) = D(N) * WRK(3*N+1) = DN(N) * WRK(4*N+1) = E(N) * WRK(5*N+1) = XD(N) * WRK(6*N+1) = BV(N) * * EQUIVALENCE FOR INTEGER ARRAYS * * EQUIVALENCE IWRK(1) = LISTP(N) * IWRK(N+1) = NGRP(N) * IWRK(2*N+1) = IPNTR(N+1) * IWRK(3*N+2) = JPNTR(N+1) * IWRK(4*N+3) = IWA(2*NZ+8*N) * CALL STDRUO(N,X,NPAIRS,IRN,LIRN,ICN,LICN,FCN,UGR,USH,TYPX,FSCALE, + GRADTL,STEPTL,ILIM,STEPMX,IPR,METHOD,GRDFLG,HSNFLG, + NDIGIT,MSG,XPLS,FPLS,GPLS,HESS,WRK(1),WRK(N+1), + WRK(2*N+1),WRK(3*N+1),WRK(4*N+1),WRK(5*N+1), + WRK(6*N+1),IWRK(1),IWRK(N+1),IWRK(2*N+1),IWRK(3*N+2), + IWRK(4*N+3),LIWRK,LWRK,TERMCD,VECTOR,INFORM) RETURN END SUBROUTINE STUMSD(N,X,NPAIRS,IRN,LIRN,ICN,LICN,FCN,TYPX,MSG,XPLS, + FPLS,GPLS,HESS,WRK,LWRK,IWRK,LIWRK,TERMCD) * * PURPOSE: * ------- * * THIS ROUTINE PROVIDES A SIMPLE INTERFACE TO STENMIN. THE USER * HAS NO CONTROL OVER THE OPTIONS. * * PARAMETERS: * ---------- * * N --> A POSITIVE INTEGER VARIABLE SPECIFYING THE NUMBER OF * VARIABLES IN THE PROBLEM * X --> AN ARRAY OF LENGTH N THAT CONTAINS AN INITIAL * ESTIMATE OF THE MINIMIZER * NPAIRS --> AN INTEGER VARIABLE THAT MUST BE SET BY THE USER TO * THE NUMBER OF NONZEROS STORED IN THE LOWER OR UPPER * HALF OF THE HESSIAN MATRIX * IRN --> AN INTEGER ARRAY OF LENGTH LIRN. ON ENTRY, IT MUST * HOLD THE ROW INDEX OF EACH NONZERO STORED IN THE LOWER * OR UPPER HALF OF THE HESSIAN MATRIX * LIRN --> AN INTEGER VARIABLE THAT MUST BE SET BY THE USER TO * THE LENGTH OF ARRAY IRN. LIRN NEED NOT BE AS LARGE * AS LICN; NORMALLY IT NEED NOT BE VERY MUCH GREATER * THAN NPAIRS * ICN --> AN INTEGER ARRAY OF LENGTH LICN. ON ENTRY, IT MUST * HOLD THE COLUMN INDEX OF THE NONZEROS STORED IN THE * LOWER OR UPPER HALF OF THE HESSIAN MATRIX. * LICN --> AN INTEGER VARIABLE THAT MUST BE SET BY THE USER TO THE * LENGTH OF THE HESSIAN ARRAY HESS AND ICN. LICN SHOULD * ORDINARILY BE 2 TO 4 TIMES AS LARGE AS NPAIRS * FCN --> THE NAME OF A USER SUPPLIED SUBROUTINE THAT EVALUATES * THE FUNCTION F AT AN ARBITRARY VECTOR X. THE SUBROUTINE * MUST BE DECLARED EXTERNAL IN THE USER'S CALLING PROGRAM * AND MUST CONFORM TO * CALL FCN(N, X, F) * TYPX --> AN ARRAY OF LENGTH N IN WHICH THE TYPICAL SIZE OF THE * COMPONENTS OF X ARE SPECIFIED. IT NEED NOT BE SET ON * ENTRY * MSG <--> AN INTEGER VARIABLE THAT THE USER MAY SET ON INPUT TO * INHIBIT CERTAIN AUTOMATIC CHECKS OR OVERRIDE CERTAIN * DEFAULT CHARACTERISTICS OF THE PACKAGE * XPLS <-- AN ARRAY OF LENGTH N CONTAINING THE BEST APPROXIMATION * TO THE MINIMIZER UPON RETURN. (IF THE ALGORITHM HAS NOT * CONVERGED, THE LAST ITERATE IS RETURNED) * FPLS <-- A SCALAR VARIABLE THAT CONTAINS THE FUNCTION VALUE AT * THE FINAL ITERATE XPLS * GPLS <-- AN ARRAY OF LENGTH N CONTAINING THE GRADIENT VALUE * AT XPLS * HESS <-- AN ARRAY THAT IS USED TO STORE THE HESSIAN MATRIX AT * EACH ITERATION. IT NEEDS TO BE AT LEAST OF DIMENSION * LICN. ON EXIT, HESS CONTAINS THE HESSIAN VALUE AT THE * MINIMIZER * WRK --> AN ARRAY OF LENGTH LWRK. THIS IS USED AS WORKSPACE * BY THE PACKAGE. ITS LENGTH MUST BE AT LEAST 8*N * LWRK --> AN INTEGER VARIABLE. IT MUST BE SET BY THE USER TO THE * LENGTH OF ARRAY WRK AND IS NOT ALTERED BY THE PACKAGE * IWRK --> AN INTEGER ARRAY OF LENGTH LIWRK. THIS IS USED AS * WORKSPACE BY THE PACKAGE. ITS LENGTH MUST BE AT LEAST * 2*NPAIRS+12*N+2 * LIWRK --> AN INTEGER VARIABLE. IT MUST BE SET BY THE USER TO THE * LENGTH OF ARRAY IWRK AND IS NOT ALTERED BY THE PACKAGE * TERMCD <-- AN INTEGER THAT SPECIFIES THE REASON WHY THE ALGORITHM * HAS TERMINATED * * INTEGER N,NPAIRS,LIRN,LICN INTEGER MSG,LWRK,LIWRK,TERMCD INTEGER IRN(LIRN),ICN(LICN),IWRK(LIWRK) DOUBLE PRECISION X(N),TYPX(N),XPLS(N),FPLS,GPLS(N) DOUBLE PRECISION HESS(LICN),WRK(LWRK) * * LOCAL VARIABLES * INTEGER ILIM,IPR,METHOD,GRDFLG,HSNFLG,NDIGIT,INFORM DOUBLE PRECISION FSCALE,GRADTL,STEPTL,STEPMX * EXTERNAL FCN,STDUGR,STDUSH * * EQUIVALENCE FOR DOUBLE PRECISION/REAL ARRAYS * * EQUIVALENCE WRK(1) = G(N) * WRK(N+1) = S(N) * WRK(2*N+1) = D(N) * WRK(3*N+1) = DN(N) * WRK(4*N+1) = E(N) * WRK(5*N+1) = XD(N) * WRK(6*N+1) = BV(N) * WRK(7*N+1) = VECTOR(N) * * EQUIVALENCE FOR INTEGER ARRAYS * * EQUIVALENCE IWRK(1) = LISTP(N) * IWRK(N+1) = NGRP(N) * IWRK(2*N+1) = IPNTR(N+1) * IWRK(3*N+2) = JPNTR(N+1) * IWRK(4*N+3) = IWA(2*NZ+8*N) * CALL STDFLT(N,TYPX,FSCALE,GRADTL,STEPTL,ILIM,STEPMX,IPR,METHOD, + GRDFLG,HSNFLG,NDIGIT,INFORM,MSG) * CALL STDRUO(N,X,NPAIRS,IRN,LIRN,ICN,LICN,FCN,STDUGR,STDUSH,TYPX, + FSCALE,GRADTL,STEPTL,ILIM,STEPMX,IPR,METHOD,GRDFLG, + HSNFLG,NDIGIT,MSG,XPLS,FPLS,GPLS,HESS,WRK(1),WRK(N+1), + WRK(2*N+1),WRK(3*N+1),WRK(4*N+1),WRK(5*N+1), + WRK(6*N+1),IWRK(1),IWRK(N+1),IWRK(2*N+1),IWRK(3*N+2), + IWRK(4*N+3),LIWRK,LWRK,TERMCD,WRK(7*N+1),INFORM) RETURN END SUBROUTINE DSYPRC(LA,LIW,A,IW,EPSMCH,NEG1,NEG2) INTEGER LA,LIW,NEG1,NEG2 INTEGER IW(LIW) DOUBLE PRECISION A(LA),EPSMCH * * THE GILL-MURRAY-PONCELEON-SAUNDERS CODE FOR MODIFYING THE NEGATIVE * EIGEN-COMPONENTS OBTAINED WHEN FACTORIZING A SYMMETRIC INDEFINITE * MATRIX USING THE HARWELL CODE MA27. (SEE SOL 90-8, P.19-21) * * NICK GOULD, 20TH JULY 1990. * INTEGER ALEN,APOS,IBLK,NBLK,IPOS,NROWS,NCOLS,J,K DOUBLE PRECISION ZERO,ONE,TWO,ALPHA,BETA,GAMMA,TAU,T,C,S,E1,E2 LOGICAL SINGLE INTRINSIC ABS,SQRT * * SET DATA. * PARAMETER (ZERO=0.0D+0,ONE=1.0D+0,TWO=2.0D+0) * * NEG1 AND NEG2 ARE THE NUMBER OF NEGATIVE EIGENVALUES WHICH ARISE * FROM NEGATIVE 1x1 AND 2x2 BLOCK PIVOTS. * NEG1 = 0 NEG2 = 0 NBLK = ABS(IW(1)) IPOS = 2 APOS = 1 * * LOOP OVER ALL THE BLOCK PIVOTS. * DO 20 IBLK = 1,NBLK NCOLS = IW(IPOS) IF (NCOLS.LT.0) THEN NROWS = 1 NCOLS = -NCOLS ELSE IPOS = IPOS + 1 NROWS = IW(IPOS) END IF * * PROCESS THE DIAGONALS IN THIS BLOCK. * ALEN = NCOLS SINGLE = .TRUE. DO 10 K = IPOS + 1,IPOS + NROWS IF (SINGLE) THEN ALPHA = A(APOS) J = IW(K) SINGLE = J .GT. 0 IF (SINGLE) THEN * * NEGATIVE 1x1 BLOCK. * IF (ALPHA.LT.ZERO) THEN NEG1 = NEG1 + 1 A(APOS) = -ALPHA ELSE IF (ALPHA.GT.ONE/EPSMCH) THEN NEG1 = NEG1 + 1 A(APOS) = ONE/EPSMCH END IF END IF ELSE BETA = A(APOS+1) GAMMA = A(APOS+ALEN) * * 2x2 BLOCK: ( ALPHA BETA ) = ( C S ) ( E1 ) ( C S ) * ( BETA GAMMA ) ( S -C ) ( E2 ) ( S -C ) * IF (ALPHA*GAMMA.LT.BETA**2) THEN TAU = (GAMMA-ALPHA)/ (TWO*BETA) T = -ONE/ (ABS(TAU)+SQRT(ONE+TAU**2)) IF (TAU.LT.ZERO) T = -T C = ONE/ (ONE+T**2) S = T*C E1 = ALPHA + BETA*T E2 = GAMMA - BETA*T * * CHANGE E1 AND E2 TO THEIR ABSOLUTE VALUES AND THEN MULTIPLY THE * THREE 2 * 2 MATRICES TO GET THE MODIFIED ALPHA, BETA AND GAMMA. * IF (E1.LT.ZERO) THEN NEG2 = NEG2 + 1 E1 = -E1 END IF IF (E2.LT.ZERO) THEN NEG2 = NEG2 + 1 E2 = -E2 END IF A(APOS) = C**2*E1 + S**2*E2 A(APOS+1) = C*S* (E1-E2) A(APOS+ALEN) = S**2*E1 + C**2*E2 END IF END IF ELSE SINGLE = .TRUE. END IF APOS = APOS + ALEN ALEN = ALEN - 1 10 CONTINUE IPOS = IPOS + NCOLS + 1 20 CONTINUE RETURN END SHAR_EOF fi # end of overwriting check if test -f 'd1mach.f' then echo shar: will not over-write existing file "'d1mach.f'" else cat << \SHAR_EOF > 'd1mach.f' DOUBLE PRECISION FUNCTION D1MACH(I) C C DOUBLE-PRECISION MACHINE CONSTANTS C C D1MACH( 1) = B**(EMIN-1), THE SMALLEST POSITIVE MAGNITUDE. C C D1MACH( 2) = B**EMAX*(1 - B**(-T)), THE LARGEST MAGNITUDE. C C D1MACH( 3) = B**(-T), THE SMALLEST RELATIVE SPACING. C C D1MACH( 4) = B**(1-T), THE LARGEST RELATIVE SPACING. C C D1MACH( 5) = LOG10(B) C C TO ALTER THIS FUNCTION FOR A PARTICULAR ENVIRONMENT, C THE DESIRED SET OF DATA STATEMENTS SHOULD BE ACTIVATED BY C REMOVING THE C FROM COLUMN 1. C ON RARE MACHINES A STATIC STATEMENT MAY NEED TO BE ADDED. C (BUT PROBABLY MORE SYSTEMS PROHIBIT IT THAN REQUIRE IT.) C C FOR IEEE-ARITHMETIC MACHINES (BINARY STANDARD), ONE OF THE FIRST C TWO SETS OF CONSTANTS BELOW SHOULD BE APPROPRIATE. C C WHERE POSSIBLE, OCTAL OR HEXADECIMAL CONSTANTS HAVE BEEN USED C TO SPECIFY THE CONSTANTS EXACTLY, WHICH HAS IN SOME CASES C REQUIRED THE USE OF EQUIVALENT INTEGER ARRAYS. C INTEGER SMALL(4) INTEGER LARGE(4) INTEGER RIGHT(4) INTEGER DIVER(4) INTEGER LOG10(4) C DOUBLE PRECISION DMACH(5) C EQUIVALENCE (DMACH(1),SMALL(1)) EQUIVALENCE (DMACH(2),LARGE(1)) EQUIVALENCE (DMACH(3),RIGHT(1)) EQUIVALENCE (DMACH(4),DIVER(1)) EQUIVALENCE (DMACH(5),LOG10(1)) C C MACHINE CONSTANTS FOR IEEE ARITHMETIC MACHINES, SUCH AS THE AT&T C 3B SERIES AND MOTOROLA 68000 BASED MACHINES (E.G. SUN 3 AND AT&T C PC 7300), IN WHICH THE MOST SIGNIFICANT BYTE IS STORED FIRST. C C DATA SMALL(1),SMALL(2) / 1048576, 0 / C DATA LARGE(1),LARGE(2) / 2146435071, -1 / C DATA RIGHT(1),RIGHT(2) / 1017118720, 0 / C DATA DIVER(1),DIVER(2) / 1018167296, 0 / C DATA LOG10(1),LOG10(2) / 1070810131, 1352628735 / C C MACHINE CONSTANTS FOR IEEE ARITHMETIC MACHINES AND 8087-BASED C MICROS, SUCH AS THE IBM PC AND AT&T 6300, IN WHICH THE LEAST C SIGNIFICANT BYTE IS STORED FIRST. C DATA SMALL(1),SMALL(2) / 0, 1048576 / DATA LARGE(1),LARGE(2) / -1, 2146435071 / DATA RIGHT(1),RIGHT(2) / 0, 1017118720 / DATA DIVER(1),DIVER(2) / 0, 1018167296 / DATA LOG10(1),LOG10(2) / 1352628735, 1070810131 / C C MACHINE CONSTANTS FOR AMDAHL MACHINES. C C DATA SMALL(1),SMALL(2) / 1048576, 0 / C DATA LARGE(1),LARGE(2) / 2147483647, -1 / C DATA RIGHT(1),RIGHT(2) / 856686592, 0 / C DATA DIVER(1),DIVER(2) / 873463808, 0 / C DATA LOG10(1),LOG10(2) / 1091781651, 1352628735 / C C MACHINE CONSTANTS FOR THE BURROUGHS 1700 SYSTEM. C C DATA SMALL(1) / ZC00800000 / C DATA SMALL(2) / Z000000000 / C C DATA LARGE(1) / ZDFFFFFFFF / C DATA LARGE(2) / ZFFFFFFFFF / C C DATA RIGHT(1) / ZCC5800000 / C DATA RIGHT(2) / Z000000000 / C C DATA DIVER(1) / ZCC6800000 / C DATA DIVER(2) / Z000000000 / C C DATA LOG10(1) / ZD00E730E7 / C DATA LOG10(2) / ZC77800DC0 / C C MACHINE CONSTANTS FOR THE BURROUGHS 5700 SYSTEM. C C DATA SMALL(1) / O1771000000000000 / C DATA SMALL(2) / O0000000000000000 / C C DATA LARGE(1) / O0777777777777777 / C DATA LARGE(2) / O0007777777777777 / C C DATA RIGHT(1) / O1461000000000000 / C DATA RIGHT(2) / O0000000000000000 / C C DATA DIVER(1) / O1451000000000000 / C DATA DIVER(2) / O0000000000000000 / C C DATA LOG10(1) / O1157163034761674 / C DATA LOG10(2) / O0006677466732724 / C C MACHINE CONSTANTS FOR THE BURROUGHS 6700/7700 SYSTEMS. C C DATA SMALL(1) / O1771000000000000 / C DATA SMALL(2) / O7770000000000000 / C C DATA LARGE(1) / O0777777777777777 / C DATA LARGE(2) / O7777777777777777 / C C DATA RIGHT(1) / O1461000000000000 / C DATA RIGHT(2) / O0000000000000000 / C C DATA DIVER(1) / O1451000000000000 / C DATA DIVER(2) / O0000000000000000 / C C DATA LOG10(1) / O1157163034761674 / C DATA LOG10(2) / O0006677466732724 / C C MACHINE CONSTANTS FOR THE CDC 6000/7000 SERIES. C C DATA SMALL(1) / 00604000000000000000B / C DATA SMALL(2) / 00000000000000000000B / C C DATA LARGE(1) / 37767777777777777777B / C DATA LARGE(2) / 37167777777777777777B / C C DATA RIGHT(1) / 15604000000000000000B / C DATA RIGHT(2) / 15000000000000000000B / C C DATA DIVER(1) / 15614000000000000000B / C DATA DIVER(2) / 15010000000000000000B / C C DATA LOG10(1) / 17164642023241175717B / C DATA LOG10(2) / 16367571421742254654B / C C MACHINE CONSTANTS FOR CONVEX C-1 C C DATA SMALL(1),SMALL(2) / '00100000'X, '00000000'X / C DATA LARGE(1),LARGE(2) / '7FFFFFFF'X, 'FFFFFFFF'X / C DATA RIGHT(1),RIGHT(2) / '3CC00000'X, '00000000'X / C DATA DIVER(1),DIVER(2) / '3CD00000'X, '00000000'X / C DATA LOG10(1),LOG10(2) / '3FF34413'X, '509F79FF'X / C C MACHINE CONSTANTS FOR THE CRAY 1, XMP, 2, AND 3. C C DATA SMALL(1) / 201354000000000000000B / C DATA SMALL(2) / 000000000000000000000B / C C DATA LARGE(1) / 577767777777777777777B / C DATA LARGE(2) / 000007777777777777776B / C C DATA RIGHT(1) / 376434000000000000000B / C DATA RIGHT(2) / 000000000000000000000B / C C DATA DIVER(1) / 376444000000000000000B / C DATA DIVER(2) / 000000000000000000000B / C C DATA LOG10(1) / 377774642023241175717B / C DATA LOG10(2) / 000007571421742254654B / C C MACHINE CONSTANTS FOR THE DATA GENERAL ECLIPSE S/200 C C NOTE - IT MAY BE APPROPRIATE TO INCLUDE THE FOLLOWING LINE - C STATIC DMACH(5) C C DATA SMALL/20K,3*0/,LARGE/77777K,3*177777K/ C DATA RIGHT/31420K,3*0/,DIVER/32020K,3*0/ C DATA LOG10/40423K,42023K,50237K,74776K/ C C MACHINE CONSTANTS FOR THE HARRIS SLASH 6 AND SLASH 7 C C DATA SMALL(1),SMALL(2) / '20000000, '00000201 / C DATA LARGE(1),LARGE(2) / '37777777, '37777577 / C DATA RIGHT(1),RIGHT(2) / '20000000, '00000333 / C DATA DIVER(1),DIVER(2) / '20000000, '00000334 / C DATA LOG10(1),LOG10(2) / '23210115, '10237777 / C C MACHINE CONSTANTS FOR THE HONEYWELL DPS 8/70 SERIES. C C DATA SMALL(1),SMALL(2) / O402400000000, O000000000000 / C DATA LARGE(1),LARGE(2) / O376777777777, O777777777777 / C DATA RIGHT(1),RIGHT(2) / O604400000000, O000000000000 / C DATA DIVER(1),DIVER(2) / O606400000000, O000000000000 / C DATA LOG10(1),LOG10(2) / O776464202324, O117571775714 / C C MACHINE CONSTANTS FOR THE IBM 360/370 SERIES, C THE XEROX SIGMA 5/7/9 AND THE SEL SYSTEMS 85/86. C C DATA SMALL(1),SMALL(2) / Z00100000, Z00000000 / C DATA LARGE(1),LARGE(2) / Z7FFFFFFF, ZFFFFFFFF / C DATA RIGHT(1),RIGHT(2) / Z33100000, Z00000000 / C DATA DIVER(1),DIVER(2) / Z34100000, Z00000000 / C DATA LOG10(1),LOG10(2) / Z41134413, Z509F79FF / C C MACHINE CONSTANTS FOR THE INTERDATA 8/32 C WITH THE UNIX SYSTEM FORTRAN 77 COMPILER. C C FOR THE INTERDATA FORTRAN VII COMPILER REPLACE C THE Z'S SPECIFYING HEX CONSTANTS WITH Y'S. C C DATA SMALL(1),SMALL(2) / Z'00100000', Z'00000000' / C DATA LARGE(1),LARGE(2) / Z'7EFFFFFF', Z'FFFFFFFF' / C DATA RIGHT(1),RIGHT(2) / Z'33100000', Z'00000000' / C DATA DIVER(1),DIVER(2) / Z'34100000', Z'00000000' / C DATA LOG10(1),LOG10(2) / Z'41134413', Z'509F79FF' / C C MACHINE CONSTANTS FOR THE PDP-10 (KA PROCESSOR). C C DATA SMALL(1),SMALL(2) / "033400000000, "000000000000 / C DATA LARGE(1),LARGE(2) / "377777777777, "344777777777 / C DATA RIGHT(1),RIGHT(2) / "113400000000, "000000000000 / C DATA DIVER(1),DIVER(2) / "114400000000, "000000000000 / C DATA LOG10(1),LOG10(2) / "177464202324, "144117571776 / C C MACHINE CONSTANTS FOR THE PDP-10 (KI PROCESSOR). C C DATA SMALL(1),SMALL(2) / "000400000000, "000000000000 / C DATA LARGE(1),LARGE(2) / "377777777777, "377777777777 / C DATA RIGHT(1),RIGHT(2) / "103400000000, "000000000000 / C DATA DIVER(1),DIVER(2) / "104400000000, "000000000000 / C DATA LOG10(1),LOG10(2) / "177464202324, "047674776746 / C C MACHINE CONSTANTS FOR PDP-11 FORTRANS SUPPORTING C 32-BIT INTEGERS (EXPRESSED IN INTEGER AND OCTAL). C C DATA SMALL(1),SMALL(2) / 8388608, 0 / C DATA LARGE(1),LARGE(2) / 2147483647, -1 / C DATA RIGHT(1),RIGHT(2) / 612368384, 0 / C DATA DIVER(1),DIVER(2) / 620756992, 0 / C DATA LOG10(1),LOG10(2) / 1067065498, -2063872008 / C C DATA SMALL(1),SMALL(2) / O00040000000, O00000000000 / C DATA LARGE(1),LARGE(2) / O17777777777, O37777777777 / C DATA RIGHT(1),RIGHT(2) / O04440000000, O00000000000 / C DATA DIVER(1),DIVER(2) / O04500000000, O00000000000 / C DATA LOG10(1),LOG10(2) / O07746420232, O20476747770 / C C MACHINE CONSTANTS FOR PDP-11 FORTRANS SUPPORTING C 16-BIT INTEGERS (EXPRESSED IN INTEGER AND OCTAL). C C DATA SMALL(1),SMALL(2) / 128, 0 / C DATA SMALL(3),SMALL(4) / 0, 0 / C C DATA LARGE(1),LARGE(2) / 32767, -1 / C DATA LARGE(3),LARGE(4) / -1, -1 / C C DATA RIGHT(1),RIGHT(2) / 9344, 0 / C DATA RIGHT(3),RIGHT(4) / 0, 0 / C C DATA DIVER(1),DIVER(2) / 9472, 0 / C DATA DIVER(3),DIVER(4) / 0, 0 / C C DATA LOG10(1),LOG10(2) / 16282, 8346 / C DATA LOG10(3),LOG10(4) / -31493, -12296 / C C DATA SMALL(1),SMALL(2) / O000200, O000000 / C DATA SMALL(3),SMALL(4) / O000000, O000000 / C C DATA LARGE(1),LARGE(2) / O077777, O177777 / C DATA LARGE(3),LARGE(4) / O177777, O177777 / C C DATA RIGHT(1),RIGHT(2) / O022200, O000000 / C DATA RIGHT(3),RIGHT(4) / O000000, O000000 / C C DATA DIVER(1),DIVER(2) / O022400, O000000 / C DATA DIVER(3),DIVER(4) / O000000, O000000 / C C DATA LOG10(1),LOG10(2) / O037632, O020232 / C DATA LOG10(3),LOG10(4) / O102373, O147770 / C C MACHINE CONSTANTS FOR THE PRIME 50 SERIES SYSTEMS C WTIH 32-BIT INTEGERS AND 64V MODE INSTRUCTIONS, C SUPPLIED BY IGOR BRAY. C C DATA SMALL(1),SMALL(2) / :10000000000, :00000100001 / C DATA LARGE(1),LARGE(2) / :17777777777, :37777677775 / C DATA RIGHT(1),RIGHT(2) / :10000000000, :00000000122 / C DATA DIVER(1),DIVER(2) / :10000000000, :00000000123 / C DATA LOG10(1),LOG10(2) / :11504046501, :07674600177 / C C MACHINE CONSTANTS FOR THE SEQUENT BALANCE 8000 C C DATA SMALL(1),SMALL(2) / $00000000, $00100000 / C DATA LARGE(1),LARGE(2) / $FFFFFFFF, $7FEFFFFF / C DATA RIGHT(1),RIGHT(2) / $00000000, $3CA00000 / C DATA DIVER(1),DIVER(2) / $00000000, $3CB00000 / C DATA LOG10(1),LOG10(2) / $509F79FF, $3FD34413 / C C MACHINE CONSTANTS FOR THE UNIVAC 1100 SERIES. C C DATA SMALL(1),SMALL(2) / O000040000000, O000000000000 / C DATA LARGE(1),LARGE(2) / O377777777777, O777777777777 / C DATA RIGHT(1),RIGHT(2) / O170540000000, O000000000000 / C DATA DIVER(1),DIVER(2) / O170640000000, O000000000000 / C DATA LOG10(1),LOG10(2) / O177746420232, O411757177572 / C C MACHINE CONSTANTS FOR THE VAX UNIX F77 COMPILER C C DATA SMALL(1),SMALL(2) / 128, 0 / C DATA LARGE(1),LARGE(2) / -32769, -1 / C DATA RIGHT(1),RIGHT(2) / 9344, 0 / C DATA DIVER(1),DIVER(2) / 9472, 0 / C DATA LOG10(1),LOG10(2) / 546979738, -805796613 / C C MACHINE CONSTANTS FOR THE VAX-11 WITH C FORTRAN IV-PLUS COMPILER C C DATA SMALL(1),SMALL(2) / Z00000080, Z00000000 / C DATA LARGE(1),LARGE(2) / ZFFFF7FFF, ZFFFFFFFF / C DATA RIGHT(1),RIGHT(2) / Z00002480, Z00000000 / C DATA DIVER(1),DIVER(2) / Z00002500, Z00000000 / C DATA LOG10(1),LOG10(2) / Z209A3F9A, ZCFF884FB / C C MACHINE CONSTANTS FOR VAX/VMS VERSION 2.2 C C DATA SMALL(1),SMALL(2) / '80'X, '0'X / C DATA LARGE(1),LARGE(2) / 'FFFF7FFF'X, 'FFFFFFFF'X / C DATA RIGHT(1),RIGHT(2) / '2480'X, '0'X / C DATA DIVER(1),DIVER(2) / '2500'X, '0'X / C DATA LOG10(1),LOG10(2) / '209A3F9A'X, 'CFF884FB'X / C IF (I .LT. 1 .OR. I .GT. 5) GOTO 999 D1MACH = DMACH(I) RETURN 999 WRITE(I1MACH(2),1999) I 1999 FORMAT(' D1MACH - I OUT OF BOUNDS',I10) STOP END SHAR_EOF fi # end of overwriting check if test -f 'colmor.f' then echo shar: will not over-write existing file "'colmor.f'" else cat << \SHAR_EOF > 'colmor.f' SUBROUTINE DSSM(N,NPAIRS,INDROW,INDCOL,METHOD,LISTP,NGRP, * MAXGRP,MINGRP,INFO,IPNTR,JPNTR,IWA,LIWA) INTEGER N,NPAIRS,METHOD,MAXGRP,MINGRP,INFO,LIWA INTEGER INDROW(NPAIRS),INDCOL(NPAIRS),LISTP(N),NGRP(N), * IPNTR(N+1),JPNTR(N+1),IWA(LIWA) C ********** C C SUBROUTINE DSSM C C GIVEN THE SPARSITY PATTERN OF A SYMMETRIC MATRIX A OF ORDER N, C THIS SUBROUTINE DETERMINES A SYMMETRIC PERMUTATION OF A AND A C PARTITION OF THE COLUMNS OF A CONSISTENT WITH THE DETERMINATION C OF A BY A LOWER TRIANGULAR SUBSTITUTION METHOD. C C THE SPARSITY PATTERN OF THE MATRIX A IS SPECIFIED BY THE C ARRAYS INDROW AND INDCOL. ON INPUT THE INDICES FOR THE C NON-ZERO ELEMENTS IN THE LOWER TRIANGULAR PART OF A ARE C C (INDROW(K),INDCOL(K)), K = 1,2,...,NPAIRS. C C THE (INDROW(K),INDCOL(K)) PAIRS MAY BE SPECIFIED IN ANY ORDER. C DUPLICATE INPUT PAIRS ARE PERMITTED, BUT THE SUBROUTINE C ELIMINATES THEM. THE SUBROUTINE REQUIRES THAT ALL THE DIAGONAL C ELEMENTS BE PART OF THE SPARSITY PATTERN AND REPLACES ANY PAIR C (INDROW(K),INDCOL(K)) WHERE INDROW(K) IS LESS THAN INDCOL(K) C BY THE PAIR (INDCOL(K),INDROW(K)). C C THE DIRECT METHOD (METHOD = 1) FIRST DETERMINES A PARTITION C OF THE COLUMNS OF A SUCH THAT TWO COLUMNS IN A GROUP HAVE A C NON-ZERO ELEMENT IN ROW K ONLY IF COLUMN K IS IN AN EARLIER C GROUP. USING THIS PARTITION, THE SUBROUTINE THEN COMPUTES A C SYMMETRIC PERMUTATION OF A CONSISTENT WITH THE DETERMINATION C OF A BY A LOWER TRIANGULAR SUBSTITUTION METHOD. C C THE INDIRECT METHOD FIRST COMPUTES A SYMMETRIC PERMUTATION OF A C WHICH MINIMIZES THE MAXIMUM NUMBER OF NON-ZERO ELEMENTS IN ANY C ROW OF L, WHERE L IS THE LOWER TRIANGULAR PART OF THE PERMUTED C MATRIX. THE SUBROUTINE THEN PARTITIONS THE COLUMNS OF L INTO C GROUPS SUCH THAT COLUMNS OF L IN A GROUP DO NOT HAVE A NON-ZERO C IN THE SAME ROW POSITION. C C THE SUBROUTINE STATEMENT IS C C SUBROUTINE DSSM(N,NPAIRS,INDROW,INDCOL,METHOD,LISTP,NGRP, C MAXGRP,MINGRP,INFO,IPNTR,JPNTR,IWA,LIWA) C C WHERE C C N IS A POSITIVE INTEGER INPUT VARIABLE SET TO THE ORDER OF A. C C NPAIRS IS A POSITIVE INTEGER INPUT VARIABLE SET TO THE NUMBER C OF (INDROW,INDCOL) PAIRS USED TO DESCRIBE THE SPARSITY C PATTERN OF A. C C INDROW IS AN INTEGER ARRAY OF LENGTH NPAIRS. ON INPUT INDROW C MUST CONTAIN THE ROW INDICES OF THE NON-ZERO ELEMENTS IN C THE LOWER TRIANGULAR PART OF A. ON OUTPUT INDROW IS C PERMUTED SO THAT THE CORRESPONDING COLUMN INDICES ARE IN C NON-DECREASING ORDER. THE COLUMN INDICES CAN BE RECOVERED C FROM THE ARRAY JPNTR. C C INDCOL IS AN INTEGER ARRAY OF LENGTH NPAIRS. ON INPUT INDCOL C MUST CONTAIN THE COLUMN INDICES OF THE NON-ZERO ELEMENTS C IN THE LOWER TRIANGULAR PART OF A. ON OUTPUT INDCOL IS C PERMUTED SO THAT THE CORRESPONDING ROW INDICES ARE IN C NON-DECREASING ORDER. THE ROW INDICES CAN BE RECOVERED C FROM THE ARRAY IPNTR. C C METHOD IS AN INTEGER INPUT VARIABLE. IF METHOD = 1, THE C DIRECT METHOD IS USED TO DETERMINE THE PARTITION AND C SYMMETRIC PERMUTATION. OTHERWISE, THE INDIRECT METHOD IS C USED TO DETERMINE THE SYMMETRIC PERMUTATION AND PARTITION. C C LISTP IS AN INTEGER OUTPUT ARRAY OF LENGTH N WHICH SPECIFIES C THE SYMMETRIC PERMUTATION OF THE MATRIX A. ELEMENT (I,J) C OF A IS THE (LISTP(I),LISTP(J)) ELEMENT OF THE PERMUTED C MATRIX. C C NGRP IS AN INTEGER OUTPUT ARRAY OF LENGTH N WHICH SPECIFIES C THE PARTITION OF THE COLUMNS OF A. COLUMN J BELONGS TO C GROUP NGRP(J). C C MAXGRP IS AN INTEGER OUTPUT VARIABLE WHICH SPECIFIES THE C NUMBER OF GROUPS IN THE PARTITION OF THE COLUMNS OF A. C C MINGRP IS AN INTEGER OUTPUT VARIABLE WHICH SPECIFIES A LOWER C BOUND FOR THE NUMBER OF GROUPS IN ANY PARTITION OF THE C COLUMNS OF A CONSISTENT WITH THE DETERMINATION OF A BY A C LOWER TRIANGULAR SUBSTITUTION METHOD. C C INFO IS AN INTEGER OUTPUT VARIABLE SET AS FOLLOWS. FOR C NORMAL TERMINATION INFO = 1. C C IPNTR IS AN INTEGER OUTPUT ARRAY OF LENGTH N + 1 WHICH C SPECIFIES THE LOCATIONS OF THE COLUMN INDICES IN INDCOL. C THE COLUMN INDICES FOR ROW I ARE C C INDCOL(K), K = IPNTR(I),...,IPNTR(I+1)-1. C C NOTE THAT IPNTR(N+1)-1 IS THEN THE NUMBER OF NON-ZERO C ELEMENTS IN THE LOWER TRIANGULAR PART OF THE MATRIX A. C C JPNTR IS AN INTEGER OUTPUT ARRAY OF LENGTH N + 1 WHICH C SPECIFIES THE LOCATIONS OF THE ROW INDICES IN INDROW. C THE ROW INDICES FOR COLUMN J ARE C C INDROW(K), K = JPNTR(J),...,JPNTR(J+1)-1. C C NOTE THAT JPNTR(N+1)-1 IS THEN THE NUMBER OF NON-ZERO C ELEMENTS IN THE LOWER TRIANGULAR PART OF THE MATRIX A. C C IWA IS AN INTEGER WORK ARRAY OF LENGTH LIWA. C C LIWA IS A POSITIVE INTEGER INPUT VARIABLE NOT LESS THAN 6*N. C C SUBPROGRAMS CALLED C C MINPACK-SUPPLIED ... DEGR,IDO,IDOG,NUMSRT,SDPT,SEQ,SETR, C SLO,SLOG,SRTDAT C C FORTRAN-SUPPLIED ... MAX,MIN C C ARGONNE NATIONAL LABORATORY. MINPACK PROJECT. DECEMBER 1984. C THOMAS F. COLEMAN, BURTON S. GARBOW, JORGE J. MORE' C C ********** INTEGER I,J,JP,MAXID,MAXVD,MAXCLQ,NNZ,NUMGRP C INFO = 1 C C EXTEND THE DATA STRUCTURE TO ROWS. C CALL SETR(N,N,INDROW,JPNTR,INDCOL,IPNTR,IWA) C C DETERMINE THE SMALLEST-LAST ORDERING OF THE VERTICES OF THE C ADJACENCY GRAPH OF A, AND FROM IT DETERMINE A LOWER BOUND C FOR THE NUMBER OF GROUPS. C CALL SLOG(N,INDROW,JPNTR,INDCOL,IPNTR,IWA(1),MAXCLQ, * MAXVD,IWA(N+1),IWA(2*N+1),IWA(3*N+1)) MINGRP = 1 + MAXVD C C USE THE SELECTED METHOD. C IF (METHOD .EQ. 1) THEN C C DIRECT METHOD. DETERMINE A PARTITION OF THE COLUMNS C OF A BY THE POWELL-TOINT METHOD. C CALL SDPT(N,INDROW,JPNTR,INDCOL,IPNTR,NGRP,MAXGRP, * IWA(N+1),IWA(2*N+1)) C C DEFINE A SYMMETRIC PERMUTATION OF A ACCORDING TO THE C ORDERING OF THE COLUMN GROUP NUMBERS IN THE PARTITION. C CALL NUMSRT(N,MAXGRP,NGRP,1,IWA(1),IWA(2*N+1),IWA(N+1)) DO 80 I = 1, N LISTP(IWA(I)) = I 80 CONTINUE ELSE C C INDIRECT METHOD. DETERMINE THE INCIDENCE DEGREE ORDERING C OF THE VERTICES OF THE ADJACENCY GRAPH OF A AND, TOGETHER C WITH THE SMALLEST-LAST ORDERING, DEFINE A SYMMETRIC C PERMUTATION OF A. C CALL IDOG(N,INDROW,JPNTR,INDCOL,IPNTR,LISTP,MAXCLQ, * MAXID,IWA(N+1),IWA(2*N+1),IWA(3*N+1)) IF (MAXID .GT. MAXVD) THEN DO 90 I = 1, N LISTP(I) = IWA(I) 90 CONTINUE END IF C C GENERATE THE SPARSITY PATTERN FOR THE LOWER C TRIANGULAR PART L OF THE PERMUTED MATRIX. C DO 110 J = 1, N DO 100 JP = JPNTR(J), JPNTR(J+1)-1 I = INDROW(JP) INDROW(JP) = MAX(LISTP(I),LISTP(J)) INDCOL(JP) = MIN(LISTP(I),LISTP(J)) 100 CONTINUE 110 CONTINUE C C SORT THE DATA STRUCTURE BY COLUMNS. C CALL SRTDAT(N,NNZ,INDROW,INDCOL,JPNTR,IWA) C C EXTEND THE DATA STRUCTURE TO ROWS. C CALL SETR(N,N,INDROW,JPNTR,INDCOL,IPNTR,IWA) C C DETERMINE THE DEGREE SEQUENCE FOR THE INTERSECTION C GRAPH OF THE COLUMNS OF L. C CALL DEGR(N,INDROW,JPNTR,INDCOL,IPNTR,IWA(5*N+1),IWA(N+1)) C C COLOR THE INTERSECTION GRAPH OF THE COLUMNS OF L C WITH THE SMALLEST-LAST (SL) ORDERING. C CALL SLO(N,INDROW,JPNTR,INDCOL,IPNTR,IWA(5*N+1),IWA(4*N+1), * MAXCLQ,IWA(1),IWA(N+1),IWA(2*N+1),IWA(3*N+1)) CALL SEQ(N,INDROW,JPNTR,INDCOL,IPNTR,IWA(4*N+1),IWA(1), * MAXGRP,IWA(N+1)) DO 120 J = 1, N NGRP(J) = IWA(LISTP(J)) 120 CONTINUE C C EXIT IF THE SMALLEST-LAST ORDERING IS OPTIMAL. C IF (MAXGRP .EQ. MAXCLQ) GO TO 140 C C COLOR THE INTERSECTION GRAPH OF THE COLUMNS OF L C WITH THE INCIDENCE DEGREE (ID) ORDERING. C CALL IDO(N,N,INDROW,JPNTR,INDCOL,IPNTR,IWA(5*N+1),IWA(4*N+1), * MAXCLQ,IWA(1),IWA(N+1),IWA(2*N+1),IWA(3*N+1)) CALL SEQ(N,INDROW,JPNTR,INDCOL,IPNTR,IWA(4*N+1),IWA(1), * NUMGRP,IWA(N+1)) C C RETAIN THE BETTER OF THE TWO ORDERINGS. C IF (NUMGRP .LT. MAXGRP) THEN MAXGRP = NUMGRP DO 130 J = 1, N NGRP(J) = IWA(LISTP(J)) 130 CONTINUE END IF 140 CONTINUE C C GENERATE THE SPARSITY PATTERN FOR THE LOWER C TRIANGULAR PART OF THE ORIGINAL MATRIX. C DO 150 J = 1, N IWA(LISTP(J)) = J 150 CONTINUE DO 170 J = 1, N DO 160 JP = JPNTR(J), JPNTR(J+1)-1 I = INDROW(JP) INDROW(JP) = MAX(IWA(I),IWA(J)) INDCOL(JP) = MIN(IWA(I),IWA(J)) 160 CONTINUE 170 CONTINUE C C SORT THE DATA STRUCTURE BY COLUMNS. C CALL SRTDAT(N,NNZ,INDROW,INDCOL,JPNTR,IWA) C C EXTEND THE DATA STRUCTURE TO ROWS. C CALL SETR(N,N,INDROW,JPNTR,INDCOL,IPNTR,IWA) END IF RETURN C C LAST CARD OF SUBROUTINE DSSM. C END SUBROUTINE IDOG(N,NGHBRP,NPNTRP,NGHBRS,NPNTRS,LISTP, * MAXCLQ,MAXID,IWA1,IWA2,IWA3) INTEGER N,MAXCLQ,MAXID INTEGER NGHBRP(*),NPNTRP(N+1),NGHBRS(*),NPNTRS(N+1),LISTP(N), * IWA1(0:N-1),IWA2(N),IWA3(N) C ********** C C SUBROUTINE IDOG C C GIVEN A LOOPLESS GRAPH G = (V,E), THIS SUBROUTINE DETERMINES C THE INCIDENCE DEGREE ORDERING OF THE VERTICES OF G. C C THE INCIDENCE DEGREE ORDERING IS DETERMINED RECURSIVELY BY C LETTING LIST(K), K = 1,...,N BE A VERTEX WITH MAXIMAL C INCIDENCE TO THE SUBGRAPH SPANNED BY THE ORDERED VERTICES. C AMONG ALL THE VERTICES OF MAXIMAL INCIDENCE, A VERTEX OF C MAXIMAL DEGREE IS CHOSEN. THIS SUBROUTINE DETERMINES THE C INVERSE OF THE INCIDENCE DEGREE ORDERING, THAT IS, AN ARRAY C LISTP SUCH THAT LISTP(LIST(K)) = K FOR K = 1,2,...,N. C C THE SUBROUTINE STATEMENT IS C C SUBROUTINE IDOG(N,NGHBRP,NPNTRP,NGHBRS,NPNTRS,LISTP, C MAXCLQ,MAXID,IWA1,IWA2,IWA3) C C WHERE C C N IS A POSITIVE INTEGER INPUT VARIABLE SET TO THE NUMBER C OF VERTICES OF G. C C NGHBRP IS AN INTEGER INPUT ARRAY WHICH CONTAINS THE C PREDECESSOR ADJACENCY LISTS FOR THE GRAPH G. C C NPNTRP IS AN INTEGER INPUT ARRAY OF LENGTH N + 1 WHICH C SPECIFIES THE LOCATIONS OF THE PREDECESSOR ADJACENCY C LISTS IN NGHBRP. THE VERTICES PRECEDING AND ADJACENT C TO VERTEX J ARE C C NGHBRP(K), K = NPNTRP(J),...,NPNTRP(J+1)-1. C C NOTE THAT NPNTRP(N+1)-1 IS THEN THE NUMBER OF VERTICES C PLUS EDGES OF THE GRAPH G. C C NGHBRS IS AN INTEGER INPUT ARRAY WHICH CONTAINS THE C SUCCESSOR ADJACENCY LISTS FOR THE GRAPH G. C C NPNTRS IS AN INTEGER INPUT ARRAY OF LENGTH N + 1 WHICH C SPECIFIES THE LOCATIONS OF THE SUCCESSOR ADJACENCY C LISTS IN NGHBRS. THE VERTICES SUCCEEDING AND ADJACENT C TO VERTEX J ARE C C NGHBRS(K), K = NPNTRS(J),...,NPNTRS(J+1)-1. C C NOTE THAT NPNTRS(N+1)-1 IS THEN THE NUMBER OF VERTICES C PLUS EDGES OF THE GRAPH G. C C LISTP IS AN INTEGER OUTPUT ARRAY OF LENGTH N WHICH SPECIFIES C THE INVERSE OF THE INCIDENCE DEGREE ORDERING OF THE C VERTICES. VERTEX J IS IN POSITION LISTP(J) OF THIS ORDERING. C C MAXCLQ IS AN INTEGER OUTPUT VARIABLE SET TO THE SIZE C OF THE LARGEST CLIQUE FOUND DURING THE ORDERING. C C MAXID IS AN INTEGER OUTPUT VARIABLE SET TO THE MAXIMUM C INCIDENCE DEGREE FOUND DURING THE ORDERING. C C IWA1,IWA2, AND IWA3 ARE INTEGER WORK ARRAYS OF LENGTH N. C C SUBPROGRAMS CALLED C C MINPACK-SUPPLIED ... NUMSRT C C FORTRAN-SUPPLIED ... MAX C C ARGONNE NATIONAL LABORATORY. MINPACK PROJECT. DECEMBER 1984. C THOMAS F. COLEMAN, BURTON S. GARBOW, JORGE J. MORE' C C ********** INTEGER I,J,K,MAXINC,MAXDEG,MAXLST,NCOMP,NUMDEG,NUMINC,NUMORD C C INITIALIZATION BLOCK. C DO 10 J = 1, N LISTP(J) = (NPNTRP(J+1) - NPNTRP(J) - 1) + * (NPNTRS(J+1) - NPNTRS(J) - 1) 10 CONTINUE MAXLST = (NPNTRP(N+1) + NPNTRS(N+1))/N C C SORT THE DEGREE SEQUENCE. C CALL NUMSRT(N,N-1,LISTP,1,IWA1,IWA2,IWA3) C C CREATE A DOUBLY-LINKED LIST TO ACCESS THE INCIDENCES OF THE C VERTICES. THE POINTERS FOR THE LINKED LIST ARE AS FOLLOWS. C C EACH UN-ORDERED VERTEX I IS IN A LIST (THE INCIDENCE LIST) C OF VERTICES WITH THE SAME INCIDENCE. C C IWA1(NUMINC) IS THE FIRST VERTEX IN THE NUMINC LIST C UNLESS IWA1(NUMINC) = 0. IN THIS CASE THERE ARE C NO VERTICES IN THE NUMINC LIST. C C IWA2(I) IS THE VERTEX BEFORE I IN THE INCIDENCE LIST C UNLESS IWA2(I) = 0. IN THIS CASE I IS THE FIRST C VERTEX IN THIS INCIDENCE LIST. C C IWA3(I) IS THE VERTEX AFTER I IN THE INCIDENCE LIST C UNLESS IWA3(I) = 0. IN THIS CASE I IS THE LAST C VERTEX IN THIS INCIDENCE LIST. C C IF I IS AN UN-ORDERED VERTEX, THEN -LISTP(I) IS THE C INCIDENCE OF I TO THE GRAPH INDUCED BY THE ORDERED C VERTICES. IF J IS AN ORDERED VERTEX, THEN LISTP(J) C IS THE INCIDENCE DEGREE ORDER OF VERTEX J. C MAXINC = 0 DO 20 J = 1, N I = IWA1(J-1) IWA1(J-1) = 0 IWA2(I) = 0 IWA3(I) = IWA1(0) IF (IWA1(0) .GT. 0) IWA2(IWA1(0)) = I IWA1(0) = I LISTP(J) = 0 20 CONTINUE MAXCLQ = 0 MAXID = 0 NUMORD = 1 C C BEGINNING OF ITERATION LOOP. C 30 CONTINUE C C CHOOSE A VERTEX J OF MAXIMAL DEGREE AMONG THE C VERTICES OF MAXIMAL INCIDENCE MAXINC. C 40 CONTINUE K = IWA1(MAXINC) IF (K .GT. 0) GO TO 50 MAXINC = MAXINC - 1 GO TO 40 50 CONTINUE MAXDEG = -1 DO 60 I = 1, MAXLST NUMDEG = (NPNTRP(K+1) - NPNTRP(K) - 1) + * (NPNTRS(K+1) - NPNTRS(K) - 1) IF (NUMDEG .GT. MAXDEG) THEN MAXDEG = NUMDEG J = K END IF K = IWA3(K) IF (K .LE. 0) GO TO 70 60 CONTINUE 70 CONTINUE LISTP(J) = NUMORD MAXID = MAX(MAXID,MAXINC) C C UPDATE THE SIZE OF THE LARGEST CLIQUE C FOUND DURING THE ORDERING. C IF (MAXINC .EQ. 0) NCOMP = 0 NCOMP = NCOMP + 1 IF (MAXINC+1 .EQ. NCOMP) MAXCLQ = MAX(MAXCLQ,NCOMP) C C TERMINATION TEST. C NUMORD = NUMORD + 1 IF (NUMORD .GT. N) GO TO 100 C C DELETE VERTEX J FROM THE MAXINC LIST. C IF (IWA2(J) .EQ. 0) THEN IWA1(MAXINC) = IWA3(J) ELSE IWA3(IWA2(J)) = IWA3(J) END IF IF (IWA3(J) .GT. 0) IWA2(IWA3(J)) = IWA2(J) C C DETERMINE ALL THE NEIGHBORS OF VERTEX J WHICH PRECEDE J C IN THE SUBGRAPH SPANNED BY THE UN-ORDERED VERTICES. C DO 80 K = NPNTRP(J), NPNTRP(J+1)-1 I = NGHBRP(K) C C UPDATE THE POINTERS TO THE CURRENT INCIDENCE LISTS. C NUMINC = -LISTP(I) IF (NUMINC .GE. 0) THEN LISTP(I) = LISTP(I) - 1 MAXINC = MAX(MAXINC,-LISTP(I)) C C DELETE VERTEX I FROM THE NUMINC LIST. C IF (IWA2(I) .EQ. 0) THEN IWA1(NUMINC) = IWA3(I) ELSE IWA3(IWA2(I)) = IWA3(I) END IF IF (IWA3(I) .GT. 0) IWA2(IWA3(I)) = IWA2(I) C C ADD VERTEX I TO THE NUMINC+1 LIST. C IWA2(I) = 0 IWA3(I) = IWA1(NUMINC+1) IF (IWA1(NUMINC+1) .GT. 0) IWA2(IWA1(NUMINC+1)) = I IWA1(NUMINC+1) = I END IF 80 CONTINUE C C DETERMINE ALL THE NEIGHBORS OF VERTEX J WHICH SUCCEED J C IN THE SUBGRAPH SPANNED BY THE UN-ORDERED VERTICES. C DO 90 K = NPNTRS(J), NPNTRS(J+1)-1 I = NGHBRS(K) C C UPDATE THE POINTERS TO THE CURRENT INCIDENCE LISTS. C NUMINC = -LISTP(I) IF (NUMINC .GE. 0) THEN LISTP(I) = LISTP(I) - 1 MAXINC = MAX(MAXINC,-LISTP(I)) C C DELETE VERTEX I FROM THE NUMINC LIST. C IF (IWA2(I) .EQ. 0) THEN IWA1(NUMINC) = IWA3(I) ELSE IWA3(IWA2(I)) = IWA3(I) END IF IF (IWA3(I) .GT. 0) IWA2(IWA3(I)) = IWA2(I) C C ADD VERTEX I TO THE NUMINC+1 LIST. C IWA2(I) = 0 IWA3(I) = IWA1(NUMINC+1) IF (IWA1(NUMINC+1) .GT. 0) IWA2(IWA1(NUMINC+1)) = I IWA1(NUMINC+1) = I END IF 90 CONTINUE C C END OF ITERATION LOOP. C GO TO 30 100 CONTINUE RETURN C C LAST CARD OF SUBROUTINE IDOG. C END SUBROUTINE SDPT(N,NGHBRP,NPNTRP,NGHBRS,NPNTRS,NGRP,MAXGRP, * IWA1,IWA2) INTEGER N,MAXGRP INTEGER NGHBRP(*),NPNTRP(N+1),NGHBRS(*),NPNTRS(N+1),NGRP(N), * IWA1(0:N-1),IWA2(N) C ********** C C SUBROUTINE SDPT C C GIVEN A LOOPLESS GRAPH G = (V,E), THIS SUBROUTINE DETERMINES C A SYMMETRIC COLORING OF G BY THE POWELL-TOINT DIRECT METHOD. C C THE POWELL-TOINT METHOD ASSIGNS THE K-TH COLOR BY EXAMINING C THE UN-COLORED VERTICES U(K) IN ORDER OF NON-INCREASING DEGREE C AND ASSIGNING COLOR K TO VERTEX V IF THERE ARE NO PATHS OF C LENGTH 1 OR 2 (IN THE GRAPH INDUCED BY U(K)) BETWEEN V AND C SOME K-COLORED VERTEX. C C THE SUBROUTINE STATEMENT IS C C SUBROUTINE SDPT(N,NGHBRP,NPNTRP,NGHBRS,NPNTRS,NGRP,MAXGRP, C IWA1,IWA2) C C WHERE C C N IS A POSITIVE INTEGER INPUT VARIABLE SET TO THE NUMBER C OF VERTICES OF G. C C NGHBRP IS AN INTEGER INPUT ARRAY WHICH CONTAINS THE C PREDECESSOR ADJACENCY LISTS FOR THE GRAPH G. C C NPNTRP IS AN INTEGER INPUT ARRAY OF LENGTH N + 1 WHICH C SPECIFIES THE LOCATIONS OF THE PREDECESSOR ADJACENCY C LISTS IN NGHBRP. THE VERTICES PRECEDING AND ADJACENT C TO VERTEX J ARE C C NGHBRP(K), K = NPNTRP(J),...,NPNTRP(J+1)-1. C C NOTE THAT NPNTRP(N+1)-1 IS THEN THE NUMBER OF VERTICES C PLUS EDGES OF THE GRAPH G. C C NGHBRS IS AN INTEGER INPUT ARRAY WHICH CONTAINS THE C SUCCESSOR ADJACENCY LISTS FOR THE GRAPH G. C C NPNTRS IS AN INTEGER INPUT ARRAY OF LENGTH N + 1 WHICH C SPECIFIES THE LOCATIONS OF THE SUCCESSOR ADJACENCY C LISTS IN NGHBRS. THE VERTICES SUCCEEDING AND ADJACENT C TO VERTEX J ARE C C NGHBRS(K), K = NPNTRS(J),...,NPNTRS(J+1)-1. C C NOTE THAT NPNTRS(N+1)-1 IS THEN THE NUMBER OF VERTICES C PLUS EDGES OF THE GRAPH G. C C NGRP IS AN INTEGER OUTPUT ARRAY OF LENGTH N WHICH SPECIFIES C THE SYMMETRIC COLORING OF G. VERTEX J IS COLORED WITH C COLOR NGRP(J). C C MAXGRP IS AN INTEGER OUTPUT VARIABLE WHICH SPECIFIES THE C NUMBER OF COLORS IN THE SYMMETRIC COLORING OF G. C C IWA1 AND IWA2 ARE INTEGER WORK ARRAYS OF LENGTH N. C C SUBPROGRAMS CALLED C C FORTRAN-SUPPLIED ... MAX C C ARGONNE NATIONAL LABORATORY. MINPACK PROJECT. DECEMBER 1984. C THOMAS F. COLEMAN, BURTON S. GARBOW, JORGE J. MORE' C C ********** INTEGER J,JP,K,KP,L,MAXDEG,NUMDEG,NUMV C C INITIALIZATION BLOCK. NUMV IS THE CURRENT NUMBER OF UN-COLORED C VERTICES, MAXDEG IS THE MAXIMUM INDUCED DEGREE OF THESE C VERTICES, AND MAXGRP IS THE CURRENT GROUP NUMBER (COLOR). C NUMV = N MAXDEG = 0 DO 10 J = 1, N NGRP(J) = (NPNTRP(J) - NPNTRP(J+1) + 1) + * (NPNTRS(J) - NPNTRS(J+1) + 1) MAXDEG = MAX(MAXDEG,-NGRP(J)) IWA2(J) = -J 10 CONTINUE MAXGRP = 0 C C BEGINNING OF ITERATION LOOP. C 20 CONTINUE C C SORT THE LIST OF UN-COLORED VERTICES SO THAT THEIR C INDUCED DEGREES ARE IN NON-DECREASING ORDER. C DO 30 NUMDEG = 0, MAXDEG IWA1(NUMDEG) = 0 30 CONTINUE DO 40 L = 1, NUMV NUMDEG = -NGRP(-IWA2(L)) IWA1(NUMDEG) = IWA1(NUMDEG) + 1 40 CONTINUE K = 1 DO 50 NUMDEG = MAXDEG, 0, -1 L = IWA1(NUMDEG) IWA1(NUMDEG) = K K = K + L 50 CONTINUE K = 1 60 CONTINUE J = IWA2(K) IF (J .GT. 0) THEN K = IWA1(-NGRP(J)) ELSE NUMDEG = -NGRP(-J) L = IWA1(NUMDEG) IWA2(K) = IWA2(L) IWA2(L) = -J IWA1(NUMDEG) = IWA1(NUMDEG) + 1 END IF IF (K .LE. NUMV) GO TO 60 MAXGRP = MAXGRP + 1 C C DETERMINE THE VERTICES IN GROUP MAXGRP. C DO 160 L = 1, NUMV J = IWA2(L) C C EXAMINE EACH VERTEX K PRECEDING VERTEX J AND ALL C THE NEIGHBORS OF VERTEX K TO DETERMINE IF VERTEX C J CAN BE CONSIDERED FOR GROUP MAXGRP. C DO 90 JP = NPNTRP(J), NPNTRP(J+1)-1 K = NGHBRP(JP) IF (NGRP(K) .EQ. MAXGRP) GO TO 150 IF (NGRP(K) .LE. 0) THEN DO 70 KP = NPNTRP(K), NPNTRP(K+1)-1 IF (NGRP(NGHBRP(KP)) .EQ. MAXGRP) GO TO 150 70 CONTINUE DO 80 KP = NPNTRS(K), NPNTRS(K+1)-1 IF (NGRP(NGHBRS(KP)) .EQ. MAXGRP) GO TO 150 80 CONTINUE END IF 90 CONTINUE C C EXAMINE EACH VERTEX K SUCCEEDING VERTEX J AND ALL C THE NEIGHBORS OF VERTEX K TO DETERMINE IF VERTEX C J CAN BE ADDED TO GROUP MAXGRP. C DO 120 JP = NPNTRS(J), NPNTRS(J+1)-1 K = NGHBRS(JP) IF (NGRP(K) .EQ. MAXGRP) GO TO 150 IF (NGRP(K) .LE. 0) THEN DO 100 KP = NPNTRP(K), NPNTRP(K+1)-1 IF (NGRP(NGHBRP(KP)) .EQ. MAXGRP) GO TO 150 100 CONTINUE DO 110 KP = NPNTRS(K), NPNTRS(K+1)-1 IF (NGRP(NGHBRS(KP)) .EQ. MAXGRP) GO TO 150 110 CONTINUE END IF 120 CONTINUE C C ADD VERTEX J TO GROUP MAXGRP AND REMOVE VERTEX J C FROM THE LIST OF UN-COLORED VERTICES. C NGRP(J) = MAXGRP IWA2(L) = 0 C C UPDATE THE DEGREES OF THE NEIGHBORS OF VERTEX J. C DO 130 JP = NPNTRP(J), NPNTRP(J+1)-1 K = NGHBRP(JP) IF (NGRP(K) .LT. 0) NGRP(K) = NGRP(K) + 1 130 CONTINUE DO 140 JP = NPNTRS(J), NPNTRS(J+1)-1 K = NGHBRS(JP) IF (NGRP(K) .LT. 0) NGRP(K) = NGRP(K) + 1 140 CONTINUE 150 CONTINUE 160 CONTINUE C C COMPRESS THE UPDATED LIST OF UN-COLORED VERTICES. C RESET NUMV AND RECOMPUTE MAXDEG. C K = 0 MAXDEG = 0 DO 170 L = 1, NUMV IF (IWA2(L) .NE. 0) THEN K = K + 1 IWA2(K) = -IWA2(L) MAXDEG = MAX(MAXDEG,-NGRP(IWA2(L))) END IF 170 CONTINUE NUMV = K C C END OF ITERATION LOOP. C IF (NUMV .GT. 0) GO TO 20 RETURN C C LAST CARD OF SUBROUTINE SDPT. C END SUBROUTINE SLOG(N,NGHBRP,NPNTRP,NGHBRS,NPNTRS,LISTP, * MAXCLQ,MAXVD,IWA1,IWA2,IWA3) INTEGER N,MAXCLQ,MAXVD INTEGER NGHBRP(*),NPNTRP(N+1),NGHBRS(*),NPNTRS(N+1),LISTP(N), * IWA1(0:N-1),IWA2(N),IWA3(N) C ********** C C SUBROUTINE SLOG C C GIVEN A LOOPLESS GRAPH G = (V,E), THIS SUBROUTINE DETERMINES C THE SMALLEST-LAST ORDERING OF THE VERTICES OF G. C C THE SMALLEST-LAST ORDERING IS DETERMINED RECURSIVELY BY C LETTING LIST(K), K = N,...,1 BE A VERTEX WITH LEAST DEGREE C IN THE SUBGRAPH SPANNED BY THE UN-ORDERED VERTICES. C THIS SUBROUTINE DETERMINES THE INVERSE OF THE SMALLEST-LAST C ORDERING, THAT IS, AN ARRAY LISTP SUCH THAT LISTP(LIST(K)) = K C FOR K = 1,2,...,N. C C THE SUBROUTINE STATEMENT IS C C SUBROUTINE SLOG(N,NGHBRP,NPNTRP,NGHBRS,NPNTRS,LISTP, C MAXCLQ,MAXVD,IWA1,IWA2,IWA3) C C WHERE C C N IS A POSITIVE INTEGER INPUT VARIABLE SET TO THE NUMBER C OF VERTICES OF G. C C NGHBRP IS AN INTEGER INPUT ARRAY WHICH CONTAINS THE C PREDECESSOR ADJACENCY LISTS FOR THE GRAPH G. C C NPNTRP IS AN INTEGER INPUT ARRAY OF LENGTH N + 1 WHICH C SPECIFIES THE LOCATIONS OF THE PREDECESSOR ADJACENCY C LISTS IN NGHBRP. THE VERTICES PRECEDING AND ADJACENT C TO VERTEX J ARE C C NGHBRP(K), K = NPNTRP(J),...,NPNTRP(J+1)-1. C C NOTE THAT NPNTRP(N+1)-1 IS THEN THE NUMBER OF VERTICES C PLUS EDGES OF THE GRAPH G. C C NGHBRS IS AN INTEGER INPUT ARRAY WHICH CONTAINS THE C SUCCESSOR ADJACENCY LISTS FOR THE GRAPH G. C C NPNTRS IS AN INTEGER INPUT ARRAY OF LENGTH N + 1 WHICH C SPECIFIES THE LOCATIONS OF THE SUCCESSOR ADJACENCY C LISTS IN NGHBRS. THE VERTICES SUCCEEDING AND ADJACENT C TO VERTEX J ARE C C NGHBRS(K), K = NPNTRS(J),...,NPNTRS(J+1)-1. C C NOTE THAT NPNTRS(N+1)-1 IS THEN THE NUMBER OF VERTICES C PLUS EDGES OF THE GRAPH G. C C LISTP IS AN INTEGER OUTPUT ARRAY OF LENGTH N WHICH SPECIFIES C THE INVERSE OF THE SMALLEST-LAST ORDERING OF THE VERTICES. C VERTEX J IS IN POSITION LISTP(J) OF THIS ORDERING. C C MAXCLQ IS AN INTEGER OUTPUT VARIABLE SET TO THE SIZE C OF THE LARGEST CLIQUE FOUND DURING THE ORDERING. C C MAXVD IS AN INTEGER OUTPUT VARIABLE SET TO THE MAXIMUM C VERTEX DEGREE FOUND DURING THE ORDERING. C C IWA1,IWA2, AND IWA3 ARE INTEGER WORK ARRAYS OF LENGTH N. C C SUBPROGRAMS CALLED C C FORTRAN-SUPPLIED ... MAX,MIN C C ARGONNE NATIONAL LABORATORY. MINPACK PROJECT. DECEMBER 1984. C THOMAS F. COLEMAN, BURTON S. GARBOW, JORGE J. MORE' C C ********** INTEGER I,J,K,MINDEG,NUMDEG,NUMORD C C INITIALIZATION BLOCK. C MINDEG = N DO 10 J = 1, N IWA1(J-1) = 0 LISTP(J) = (NPNTRP(J) - NPNTRP(J+1) + 1) + * (NPNTRS(J) - NPNTRS(J+1) + 1) MINDEG = MIN(MINDEG,-LISTP(J)) 10 CONTINUE C C CREATE A DOUBLY-LINKED LIST TO ACCESS THE DEGREES OF THE C VERTICES. THE POINTERS FOR THE LINKED LIST ARE AS FOLLOWS. C C EACH UN-ORDERED VERTEX I IS IN A LIST (THE DEGREE LIST) C OF VERTICES WITH THE SAME DEGREE. C C IWA1(NUMDEG) IS THE FIRST VERTEX IN THE NUMDEG LIST C UNLESS IWA1(NUMDEG) = 0. IN THIS CASE THERE ARE C NO VERTICES IN THE NUMDEG LIST. C C IWA2(I) IS THE VERTEX BEFORE I IN THE DEGREE LIST C UNLESS IWA2(I) = 0. IN THIS CASE I IS THE FIRST C VERTEX IN THIS DEGREE LIST. C C IWA3(I) IS THE VERTEX AFTER I IN THE DEGREE LIST C UNLESS IWA3(I) = 0. IN THIS CASE I IS THE LAST C VERTEX IN THIS DEGREE LIST. C C IF I IS AN UN-ORDERED VERTEX, THEN -LISTP(I) IS THE C DEGREE OF I IN THE GRAPH INDUCED BY THE UN-ORDERED C VERTICES. IF J IS AN ORDERED VERTEX, THEN LISTP(J) C IS THE SMALLEST-LAST ORDER OF VERTEX J. C DO 20 J = 1, N NUMDEG = -LISTP(J) IWA2(J) = 0 IWA3(J) = IWA1(NUMDEG) IF (IWA1(NUMDEG) .GT. 0) IWA2(IWA1(NUMDEG)) = J IWA1(NUMDEG) = J 20 CONTINUE MAXCLQ = 0 MAXVD = 0 NUMORD = N C C BEGINNING OF ITERATION LOOP. C 30 CONTINUE C C CHOOSE A VERTEX J OF MINIMAL DEGREE MINDEG. C 40 CONTINUE J = IWA1(MINDEG) IF (J .GT. 0) GO TO 50 MINDEG = MINDEG + 1 GO TO 40 50 CONTINUE LISTP(J) = NUMORD MAXVD = MAX(MAXVD,MINDEG) C C MARK THE SIZE OF THE LARGEST CLIQUE C FOUND DURING THE ORDERING. C IF (MINDEG+1 .EQ. NUMORD .AND. MAXCLQ .EQ. 0) * MAXCLQ = NUMORD C C TERMINATION TEST. C NUMORD = NUMORD - 1 IF (NUMORD .EQ. 0) GO TO 80 C C DELETE VERTEX J FROM THE MINDEG LIST. C IWA1(MINDEG) = IWA3(J) IF (IWA3(J) .GT. 0) IWA2(IWA3(J)) = 0 C C DETERMINE ALL THE NEIGHBORS OF VERTEX J WHICH PRECEDE J C IN THE SUBGRAPH SPANNED BY THE UN-ORDERED VERTICES. C DO 60 K = NPNTRP(J), NPNTRP(J+1)-1 I = NGHBRP(K) C C UPDATE THE POINTERS TO THE CURRENT DEGREE LISTS. C NUMDEG = -LISTP(I) IF (NUMDEG .GE. 0) THEN LISTP(I) = LISTP(I) + 1 MINDEG = MIN(MINDEG,-LISTP(I)) C C DELETE VERTEX I FROM THE NUMDEG LIST. C IF (IWA2(I) .EQ. 0) THEN IWA1(NUMDEG) = IWA3(I) ELSE IWA3(IWA2(I)) = IWA3(I) END IF IF (IWA3(I) .GT. 0) IWA2(IWA3(I)) = IWA2(I) C C ADD VERTEX I TO THE NUMDEG-1 LIST. C IWA2(I) = 0 IWA3(I) = IWA1(NUMDEG-1) IF (IWA1(NUMDEG-1) .GT. 0) IWA2(IWA1(NUMDEG-1)) = I IWA1(NUMDEG-1) = I END IF 60 CONTINUE C C DETERMINE ALL THE NEIGHBORS OF VERTEX J WHICH SUCCEED J C IN THE SUBGRAPH SPANNED BY THE UN-ORDERED VERTICES. C DO 70 K = NPNTRS(J), NPNTRS(J+1)-1 I = NGHBRS(K) C C UPDATE THE POINTERS TO THE CURRENT DEGREE LISTS. C NUMDEG = -LISTP(I) IF (NUMDEG .GE. 0) THEN LISTP(I) = LISTP(I) + 1 MINDEG = MIN(MINDEG,-LISTP(I)) C C DELETE VERTEX I FROM THE NUMDEG LIST. C IF (IWA2(I) .EQ. 0) THEN IWA1(NUMDEG) = IWA3(I) ELSE IWA3(IWA2(I)) = IWA3(I) END IF IF (IWA3(I) .GT. 0) IWA2(IWA3(I)) = IWA2(I) C C ADD VERTEX I TO THE NUMDEG-1 LIST. C IWA2(I) = 0 IWA3(I) = IWA1(NUMDEG-1) IF (IWA1(NUMDEG-1) .GT. 0) IWA2(IWA1(NUMDEG-1)) = I IWA1(NUMDEG-1) = I END IF 70 CONTINUE C C END OF ITERATION LOOP. C GO TO 30 80 CONTINUE RETURN C C LAST CARD OF SUBROUTINE SLOG. C END SUBROUTINE FDHS(N,INDROW,JPNTR,INDCOL,IPNTR,LISTP,NGRP, * MAXGRP,NUMGRP,ETA,FHESD,FHES,IWA) INTEGER N,MAXGRP,NUMGRP INTEGER INDROW(*),JPNTR(N+1),INDCOL(*),IPNTR(N+1), * LISTP(N),NGRP(N),IWA(N) DOUBLE PRECISION ETA(N),FHESD(N),FHES(*) CS REAL ETA(N),FHESD(N),FHES(*) C ********** C C SUBROUTINE FDHS C C THIS SUBROUTINE COMPUTES AN APPROXIMATION TO THE (SYMMETRIC) C HESSIAN MATRIX OF A FUNCTION BY A SUBSTITUTION METHOD. C THE LOWER TRIANGULAR PART OF THE APPROXIMATION IS STORED C WITH A COLUMN-ORIENTED DEFINITION OF THE SPARSITY PATTERN. C C THIS SUBROUTINE REQUIRES A SYMMETRIC PERMUTATION OF THE C HESSIAN MATRIX AND A PARTITION OF THE COLUMNS OF THE HESSIAN C MATRIX CONSISTENT WITH THE DETERMINATION OF THE HESSIAN C MATRIX BY A LOWER TRIANGULAR SUBSTITUTION METHOD. C THIS INFORMATION CAN BE PROVIDED BY SUBROUTINE DSSM. C C THE SYMMETRIC PERMUTATION OF THE HESSIAN MATRIX IS DEFINED C BY THE ARRAY LISTP. THIS ARRAY IS ONLY USED INTERNALLY. C C THE PARTITION OF THE HESSIAN MATRIX IS DEFINED BY THE ARRAY C NGRP BY SETTING NGRP(J) TO THE GROUP NUMBER OF COLUMN J. C THE USER MUST PROVIDE AN APPROXIMATION TO THE COLUMNS OF C THE HESSIAN MATRIX IN EACH GROUP BY SPECIFYING A DIFFERENCE C PARAMETER VECTOR ETA AND AN APPROXIMATION TO H*D WHERE H IS C THE HESSIAN MATRIX AND THE VECTOR D IS DEFINED BY THE C FOLLOWING SECTION OF CODE. C C DO 10 J = 1, N C D(J) = 0.0 C IF (NGRP(J) .EQ. NUMGRP) D(J) = ETA(J) C 10 CONTINUE C C IN THE ABOVE CODE NUMGRP IS A GROUP NUMBER AND ETA(J) IS THE C DIFFERENCE PARAMETER USED TO APPROXIMATE COLUMN J OF THE C HESSIAN MATRIX. SUITABLE VALUES FOR ETA(J) MUST BE PROVIDED. C C AS MENTIONED ABOVE, AN APPROXIMATION TO H*D MUST BE PROVIDED. C FOR EXAMPLE, IF GRAD(X) IS THE GRADIENT OF THE FUNCTION AT X, C THEN C C GRAD(X+D) - GRAD(X) C C CORRESPONDS TO THE FORWARD DIFFERENCE APPROXIMATION. C C THE LOWER TRIANGULAR SUBSTITUTION METHOD REQUIRES THAT THE C APPROXIMATIONS TO H*D FOR ALL THE GROUPS BE STORED IN SPECIAL C LOCATIONS OF THE ARRAY FHES. THIS IS DONE BY CALLING FDHS C SUCCESSIVELY WITH NUMGRP = 1,2,...,MAXGRP. ON THE CALL WITH C NUMGRP = MAXGRP, THE SUBROUTINE THEN PROCEEDS TO OVERWRITE C FHES WITH THE APPROXIMATION TO THE LOWER TRIANGULAR PART OF C THE HESSIAN MATRIX. C C THE SUBROUTINE STATEMENT IS C C SUBROUTINE FDHS(N,INDROW,JPNTR,INDCOL,IPNTR,LISTP,NGRP, C MAXGRP,NUMGRP,ETA,FHESD,FHES,IWA) C C WHERE C C N IS A POSITIVE INTEGER INPUT VARIABLE SET TO THE ORDER C OF THE HESSIAN MATRIX. C C INDROW IS AN INTEGER INPUT ARRAY WHICH CONTAINS THE ROW C INDICES FOR THE NON-ZEROES IN THE LOWER TRIANGULAR PART C OF THE HESSIAN MATRIX. C C JPNTR IS AN INTEGER INPUT ARRAY OF LENGTH N + 1 WHICH C SPECIFIES THE LOCATIONS OF THE ROW INDICES IN INDROW. C THE ROW INDICES FOR COLUMN J ARE C C INDROW(K), K = JPNTR(J),...,JPNTR(J+1)-1. C C NOTE THAT JPNTR(N+1)-1 IS THEN THE NUMBER OF NON-ZEROES C IN THE LOWER TRIANGULAR PART OF THE HESSIAN MATRIX. C C INDCOL IS AN INTEGER INPUT ARRAY WHICH CONTAINS THE COLUMN C INDICES FOR THE NON-ZEROES IN THE LOWER TRIANGULAR PART C OF THE HESSIAN MATRIX. C C IPNTR IS AN INTEGER INPUT ARRAY OF LENGTH N + 1 WHICH C SPECIFIES THE LOCATIONS OF THE COLUMN INDICES IN INDCOL. C THE COLUMN INDICES FOR ROW I ARE C C INDCOL(K), K = IPNTR(I),...,IPNTR(I+1)-1. C C NOTE THAT IPNTR(N+1)-1 IS THEN THE NUMBER OF NON-ZEROES C IN THE LOWER TRIANGULAR PART OF THE HESSIAN MATRIX. C C LISTP IS AN INTEGER INPUT ARRAY OF LENGTH N WHICH SPECIFIES C THE SYMMETRIC PERMUTATION OF THE HESSIAN MATRIX. ELEMENT C (I,J) OF THE HESSIAN MATRIX IS THE (LISTP(I),LISTP(J)) C ELEMENT OF THE PERMUTED HESSIAN. C C NGRP IS AN INTEGER INPUT ARRAY OF LENGTH N WHICH SPECIFIES C THE PARTITION OF THE COLUMNS OF THE HESSIAN MATRIX. C COLUMN J BELONGS TO GROUP NGRP(J). C C MAXGRP IS A POSITIVE INTEGER INPUT VARIABLE WHICH SPECIFIES C THE NUMBER OF GROUPS IN THE PARTITION OF THE COLUMNS OF C THE HESSIAN MATRIX. C C NUMGRP IS A POSITIVE INTEGER INPUT VARIABLE SET TO A GROUP C NUMBER IN THE PARTITION. C C ETA IS AN INPUT ARRAY OF LENGTH N WHICH CONTAINS THE C DIFFERENCE PARAMETER VECTOR. C C FHESD IS AN INPUT ARRAY OF LENGTH N WHICH CONTAINS AN C APPROXIMATION TO H*D, WHERE H IS THE HESSIAN MATRIX C AND D IS THE DIFFERENCE VECTOR FOR GROUP NUMGRP. C C FHES IS AN OUTPUT ARRAY OF LENGTH NNZ, WHERE NNZ IS THE C NUMBER OF NON-ZERO ELEMENTS IN THE LOWER TRIANGULAR PART C OF THE HESSIAN MATRIX. ON OUTPUT WITH NUMGRP LESS THAN C MAXGRP, THE FHESD ARRAY FOR GROUP NUMGRP HAS BEEN STORED C IN FHES. WHEN NUMGRP = MAXGRP THE SUBROUTINE OVERWRITES C FHES WITH AN APPROXIMATION TO THE LOWER TRIANGULAR PART C OF THE HESSIAN MATRIX. THE APPROXIMATION IS STORED IN C FHES WITH A COLUMN-ORIENTED DEFINITION OF THE SPARSITY C PATTERN. THUS THE ELEMENTS IN COLUMN J OF THE LOWER C TRIANGULAR PART OF THE HESSIAN MATRIX ARE C C FHES(K), K = JPNTR(J),...,JPNTR(J+1)-1, C C AND THE ROW INDICES FOR THESE ELEMENTS ARE C C INDROW(K), K = JPNTR(J),...,JPNTR(J+1)-1. C C IWA IS AN INTEGER WORK ARRAY OF LENGTH N. C C SUBPROGRAMS CALLED C C FORTRAN-SUPPLIED ... ABS C C ARGONNE NATIONAL LABORATORY. MINPACK PROJECT. DECEMBER 1984. C THOMAS F. COLEMAN, BURTON S. GARBOW, JORGE J. MORE' C C ********** INTEGER I,IP,IROW,J,JP,K,L,NUMG,NUML DOUBLE PRECISION SUM CS REAL SUM C C STORE THE I-TH ELEMENT OF GRADIENT DIFFERENCE FHESD C CORRESPONDING TO GROUP NUMGRP IF THERE IS A POSITION C (I,J) SUCH THAT NGRP(J) = NUMGRP AND (I,J) IS MAPPED C ONTO THE LOWER TRIANGULAR PART OF THE PERMUTED MATRIX. C DO 50 J = 1, N IF (NGRP(J) .EQ. NUMGRP) THEN NUML = LISTP(J) DO 30 IP = IPNTR(J), IPNTR(J+1)-1 I = INDCOL(IP) IF (LISTP(I) .GT. NUML) THEN DO 10 JP = JPNTR(I), JPNTR(I+1)-1 IF (INDROW(JP) .EQ. J) THEN FHES(JP) = FHESD(I) GO TO 20 END IF 10 CONTINUE 20 CONTINUE END IF 30 CONTINUE DO 40 JP = JPNTR(J), JPNTR(J+1)-1 I = INDROW(JP) IF (LISTP(I) .GE. NUML) FHES(JP) = FHESD(I) 40 CONTINUE END IF 50 CONTINUE C C EXIT IF THIS IS NOT THE LAST GROUP. C IF (NUMGRP .LT. MAXGRP) RETURN C C MARK ALL COLUMN INDICES J SUCH THAT (I,J) IS MAPPED ONTO C THE LOWER TRIANGULAR PART OF THE PERMUTED MATRIX. C DO 80 I = 1, N NUML = LISTP(I) DO 60 IP = IPNTR(I), IPNTR(I+1)-1 J = INDCOL(IP) IF (NUML .GE. LISTP(J)) INDCOL(IP) = -INDCOL(IP) 60 CONTINUE DO 70 JP = JPNTR(I), JPNTR(I+1)-1 J = INDROW(JP) IF (NUML .GT. LISTP(J)) INDROW(JP) = -INDROW(JP) 70 CONTINUE 80 CONTINUE C C INVERT THE ARRAY LISTP. C DO 90 J = 1, N IWA(LISTP(J)) = J 90 CONTINUE DO 100 J = 1, N LISTP(J) = IWA(J) 100 CONTINUE C C DETERMINE THE LOWER TRIANGULAR PART OF THE ORIGINAL MATRIX. C DO 220 IROW = N, 1, -1 I = LISTP(IROW) C C FIND THE POSITIONS OF THE ELEMENTS IN THE I-TH ROW OF THE C LOWER TRIANGULAR PART OF THE ORIGINAL MATRIX THAT HAVE C ALREADY BEEN DETERMINED. C DO 130 IP = IPNTR(I), IPNTR(I+1)-1 J = INDCOL(IP) IF (J .GT. 0) THEN DO 110 JP = JPNTR(J), JPNTR(J+1)-1 IF (INDROW(JP) .EQ. I) THEN IWA(J) = JP GO TO 120 END IF 110 CONTINUE 120 CONTINUE END IF 130 CONTINUE C C DETERMINE THE ELEMENTS IN THE I-TH ROW OF THE LOWER C TRIANGULAR PART OF THE ORIGINAL MATRIX WHICH GET MAPPED C ONTO THE LOWER TRIANGULAR PART OF THE PERMUTED MATRIX. C DO 180 K = IPNTR(I), IPNTR(I+1)-1 J = -INDCOL(K) IF (J .GT. 0) THEN INDCOL(K) = J C C DETERMINE THE (I,J) ELEMENT. C NUMG = NGRP(J) SUM = 0.0 DO 140 IP = IPNTR(I), IPNTR(I+1)-1 L = ABS(INDCOL(IP)) IF (NGRP(L) .EQ. NUMG .AND. L .NE. J) * SUM = SUM + FHES(IWA(L))*ETA(L) 140 CONTINUE DO 150 JP = JPNTR(I), JPNTR(I+1)-1 L = ABS(INDROW(JP)) IF (NGRP(L) .EQ. NUMG .AND. L .NE. J) * SUM = SUM + FHES(JP)*ETA(L) 150 CONTINUE C C STORE THE (I,J) ELEMENT. C DO 160 JP = JPNTR(J), JPNTR(J+1)-1 IF (INDROW(JP) .EQ. I) THEN FHES(JP) = (FHES(JP) - SUM)/ETA(J) GO TO 170 END IF 160 CONTINUE 170 CONTINUE END IF 180 CONTINUE C C DETERMINE THE ELEMENTS IN THE I-TH ROW OF THE STRICT UPPER C TRIANGULAR PART OF THE ORIGINAL MATRIX WHICH GET MAPPED C ONTO THE LOWER TRIANGULAR PART OF THE PERMUTED MATRIX. C DO 210 K = JPNTR(I), JPNTR(I+1)-1 J = -INDROW(K) IF (J .GT. 0) THEN INDROW(K) = J C C DETERMINE THE (I,J) ELEMENT. C NUMG = NGRP(J) SUM = 0.0 DO 190 IP = IPNTR(I), IPNTR(I+1)-1 L = ABS(INDCOL(IP)) IF (NGRP(L) .EQ. NUMG) * SUM = SUM + FHES(IWA(L))*ETA(L) 190 CONTINUE DO 200 JP = JPNTR(I), JPNTR(I+1)-1 L = ABS(INDROW(JP)) IF (NGRP(L) .EQ. NUMG .AND. L .NE. J) * SUM = SUM + FHES(JP)*ETA(L) 200 CONTINUE C C STORE THE (I,J) ELEMENT. C FHES(K) = (FHES(K) - SUM)/ETA(J) END IF 210 CONTINUE 220 CONTINUE C C RE-INVERT THE ARRAY LISTP. C DO 230 J = 1, N IWA(LISTP(J)) = J 230 CONTINUE DO 240 J = 1, N LISTP(J) = IWA(J) 240 CONTINUE RETURN C C LAST CARD OF SUBROUTINE FDHS. C END SUBROUTINE DEGR(N,INDROW,JPNTR,INDCOL,IPNTR,NDEG,IWA) INTEGER N INTEGER INDROW(*),JPNTR(N+1),INDCOL(*),IPNTR(*),NDEG(N),IWA(N) C ********** C C SUBROUTINE DEGR C C GIVEN THE SPARSITY PATTERN OF AN M BY N MATRIX A, C THIS SUBROUTINE DETERMINES THE DEGREE SEQUENCE FOR C THE INTERSECTION GRAPH OF THE COLUMNS OF A. C C IN GRAPH-THEORY TERMINOLOGY, THE INTERSECTION GRAPH OF C THE COLUMNS OF A IS THE LOOPLESS GRAPH G WITH VERTICES C A(J), J = 1,2,...,N WHERE A(J) IS THE J-TH COLUMN OF A C AND WITH EDGE (A(I),A(J)) IF AND ONLY IF COLUMNS I AND J C HAVE A NON-ZERO IN THE SAME ROW POSITION. C C NOTE THAT THE VALUE OF M IS NOT NEEDED BY DEGR AND IS C THEREFORE NOT PRESENT IN THE SUBROUTINE STATEMENT. C C THE SUBROUTINE STATEMENT IS C C SUBROUTINE DEGR(N,INDROW,JPNTR,INDCOL,IPNTR,NDEG,IWA) C C WHERE C C N IS A POSITIVE INTEGER INPUT VARIABLE SET TO THE NUMBER C OF COLUMNS OF A. C C INDROW IS AN INTEGER INPUT ARRAY WHICH CONTAINS THE ROW C INDICES FOR THE NON-ZEROES IN THE MATRIX A. C C JPNTR IS AN INTEGER INPUT ARRAY OF LENGTH N + 1 WHICH C SPECIFIES THE LOCATIONS OF THE ROW INDICES IN INDROW. C THE ROW INDICES FOR COLUMN J ARE C C INDROW(K), K = JPNTR(J),...,JPNTR(J+1)-1. C C NOTE THAT JPNTR(N+1)-1 IS THEN THE NUMBER OF NON-ZERO C ELEMENTS OF THE MATRIX A. C C INDCOL IS AN INTEGER INPUT ARRAY WHICH CONTAINS THE C COLUMN INDICES FOR THE NON-ZEROES IN THE MATRIX A. C C IPNTR IS AN INTEGER INPUT ARRAY OF LENGTH M + 1 WHICH C SPECIFIES THE LOCATIONS OF THE COLUMN INDICES IN INDCOL. C THE COLUMN INDICES FOR ROW I ARE C C INDCOL(K), K = IPNTR(I),...,IPNTR(I+1)-1. C C NOTE THAT IPNTR(M+1)-1 IS THEN THE NUMBER OF NON-ZERO C ELEMENTS OF THE MATRIX A. C C NDEG IS AN INTEGER OUTPUT ARRAY OF LENGTH N WHICH C SPECIFIES THE DEGREE SEQUENCE. THE DEGREE OF THE C J-TH COLUMN OF A IS NDEG(J). C C IWA IS AN INTEGER WORK ARRAY OF LENGTH N. C C ARGONNE NATIONAL LABORATORY. MINPACK PROJECT. JULY 1983. C THOMAS F. COLEMAN, BURTON S. GARBOW, JORGE J. MORE' C C ********** INTEGER IC,IP,IR,JCOL,JP C C INITIALIZATION BLOCK. C DO 10 JP = 1, N NDEG(JP) = 0 IWA(JP) = 0 10 CONTINUE C C COMPUTE THE DEGREE SEQUENCE BY DETERMINING THE CONTRIBUTIONS C TO THE DEGREES FROM THE CURRENT(JCOL) COLUMN AND FURTHER C COLUMNS WHICH HAVE NOT YET BEEN CONSIDERED. C DO 40 JCOL = 2, N IWA(JCOL) = N C C DETERMINE ALL POSITIONS (IR,JCOL) WHICH CORRESPOND C TO NON-ZEROES IN THE MATRIX. C DO 30 JP = JPNTR(JCOL), JPNTR(JCOL+1)-1 IR = INDROW(JP) C C FOR EACH ROW IR, DETERMINE ALL POSITIONS (IR,IC) C WHICH CORRESPOND TO NON-ZEROES IN THE MATRIX. C DO 20 IP = IPNTR(IR), IPNTR(IR+1)-1 IC = INDCOL(IP) C C ARRAY IWA MARKS COLUMNS WHICH HAVE CONTRIBUTED TO C THE DEGREE COUNT OF COLUMN JCOL. UPDATE THE DEGREE C COUNTS OF THESE COLUMNS AS WELL AS COLUMN JCOL. C IF (IWA(IC) .LT. JCOL) THEN IWA(IC) = JCOL NDEG(IC) = NDEG(IC) + 1 NDEG(JCOL) = NDEG(JCOL) + 1 END IF 20 CONTINUE 30 CONTINUE 40 CONTINUE RETURN C C LAST CARD OF SUBROUTINE DEGR. C END SUBROUTINE IDO(M,N,INDROW,JPNTR,INDCOL,IPNTR,NDEG,LIST, * MAXCLQ,IWA1,IWA2,IWA3,IWA4) INTEGER M,N,MAXCLQ INTEGER INDROW(*),JPNTR(N+1),INDCOL(*),IPNTR(M+1),NDEG(N), * LIST(N),IWA1(0:N-1),IWA2(N),IWA3(N),IWA4(N) C ********** C C SUBROUTINE IDO C C GIVEN THE SPARSITY PATTERN OF AN M BY N MATRIX A, THIS C SUBROUTINE DETERMINES AN INCIDENCE-DEGREE ORDERING OF THE C COLUMNS OF A. C C THE INCIDENCE-DEGREE ORDERING IS DEFINED FOR THE LOOPLESS C GRAPH G WITH VERTICES A(J), J = 1,2,...,N WHERE A(J) IS THE C J-TH COLUMN OF A AND WITH EDGE (A(I),A(J)) IF AND ONLY IF C COLUMNS I AND J HAVE A NON-ZERO IN THE SAME ROW POSITION. C C THE INCIDENCE-DEGREE ORDERING IS DETERMINED RECURSIVELY BY C LETTING LIST(K), K = 1,...,N BE A COLUMN WITH MAXIMAL C INCIDENCE TO THE SUBGRAPH SPANNED BY THE ORDERED COLUMNS. C AMONG ALL THE COLUMNS OF MAXIMAL INCIDENCE, IDO CHOOSES A C COLUMN OF MAXIMAL DEGREE. C C THE SUBROUTINE STATEMENT IS C C SUBROUTINE IDO(M,N,INDROW,JPNTR,INDCOL,IPNTR,NDEG,LIST, C MAXCLQ,IWA1,IWA2,IWA3,IWA4) C C WHERE C C M IS A POSITIVE INTEGER INPUT VARIABLE SET TO THE NUMBER C OF ROWS OF A. C C N IS A POSITIVE INTEGER INPUT VARIABLE SET TO THE NUMBER C OF COLUMNS OF A. C C INDROW IS AN INTEGER INPUT ARRAY WHICH CONTAINS THE ROW C INDICES FOR THE NON-ZEROES IN THE MATRIX A. C C JPNTR IS AN INTEGER INPUT ARRAY OF LENGTH N + 1 WHICH C SPECIFIES THE LOCATIONS OF THE ROW INDICES IN INDROW. C THE ROW INDICES FOR COLUMN J ARE C C INDROW(K), K = JPNTR(J),...,JPNTR(J+1)-1. C C NOTE THAT JPNTR(N+1)-1 IS THEN THE NUMBER OF NON-ZERO C ELEMENTS OF THE MATRIX A. C C INDCOL IS AN INTEGER INPUT ARRAY WHICH CONTAINS THE C COLUMN INDICES FOR THE NON-ZEROES IN THE MATRIX A. C C IPNTR IS AN INTEGER INPUT ARRAY OF LENGTH M + 1 WHICH C SPECIFIES THE LOCATIONS OF THE COLUMN INDICES IN INDCOL. C THE COLUMN INDICES FOR ROW I ARE C C INDCOL(K), K = IPNTR(I),...,IPNTR(I+1)-1. C C NOTE THAT IPNTR(M+1)-1 IS THEN THE NUMBER OF NON-ZERO C ELEMENTS OF THE MATRIX A. C C NDEG IS AN INTEGER INPUT ARRAY OF LENGTH N WHICH SPECIFIES C THE DEGREE SEQUENCE. THE DEGREE OF THE J-TH COLUMN C OF A IS NDEG(J). C C LIST IS AN INTEGER OUTPUT ARRAY OF LENGTH N WHICH SPECIFIES C THE INCIDENCE-DEGREE ORDERING OF THE COLUMNS OF A. THE J-TH C COLUMN IN THIS ORDER IS LIST(J). C C MAXCLQ IS AN INTEGER OUTPUT VARIABLE SET TO THE SIZE C OF THE LARGEST CLIQUE FOUND DURING THE ORDERING. C C IWA1,IWA2,IWA3, AND IWA4 ARE INTEGER WORK ARRAYS OF LENGTH N. C C SUBPROGRAMS CALLED C C MINPACK-SUPPLIED ... NUMSRT C C FORTRAN-SUPPLIED ... MAX C C ARGONNE NATIONAL LABORATORY. MINPACK PROJECT. AUGUST 1984. C THOMAS F. COLEMAN, BURTON S. GARBOW, JORGE J. MORE' C C ********** INTEGER IC,IP,IR,JCOL,JP, * MAXINC,MAXLST,NCOMP,NUMINC,NUMLST,NUMORD,NUMWGT C C SORT THE DEGREE SEQUENCE. C CALL NUMSRT(N,N-1,NDEG,-1,IWA4,IWA2,IWA3) C C INITIALIZATION BLOCK. C C CREATE A DOUBLY-LINKED LIST TO ACCESS THE INCIDENCES OF THE C COLUMNS. THE POINTERS FOR THE LINKED LIST ARE AS FOLLOWS. C C EACH UN-ORDERED COLUMN IC IS IN A LIST (THE INCIDENCE LIST) C OF COLUMNS WITH THE SAME INCIDENCE. C C IWA1(NUMINC) IS THE FIRST COLUMN IN THE NUMINC LIST C UNLESS IWA1(NUMINC) = 0. IN THIS CASE THERE ARE C NO COLUMNS IN THE NUMINC LIST. C C IWA2(IC) IS THE COLUMN BEFORE IC IN THE INCIDENCE LIST C UNLESS IWA2(IC) = 0. IN THIS CASE IC IS THE FIRST C COLUMN IN THIS INCIDENCE LIST. C C IWA3(IC) IS THE COLUMN AFTER IC IN THE INCIDENCE LIST C UNLESS IWA3(IC) = 0. IN THIS CASE IC IS THE LAST C COLUMN IN THIS INCIDENCE LIST. C C IF IC IS AN UN-ORDERED COLUMN, THEN LIST(IC) IS THE C INCIDENCE OF IC TO THE GRAPH INDUCED BY THE ORDERED C COLUMNS. IF JCOL IS AN ORDERED COLUMN, THEN LIST(JCOL) C IS THE INCIDENCE-DEGREE ORDER OF COLUMN JCOL. C MAXINC = 0 DO 10 JP = N, 1, -1 IC = IWA4(JP) IWA1(N-JP) = 0 IWA2(IC) = 0 IWA3(IC) = IWA1(0) IF (IWA1(0) .GT. 0) IWA2(IWA1(0)) = IC IWA1(0) = IC IWA4(JP) = 0 LIST(JP) = 0 10 CONTINUE C C DETERMINE THE MAXIMAL SEARCH LENGTH FOR THE LIST C OF COLUMNS OF MAXIMAL INCIDENCE. C MAXLST = 0 DO 20 IR = 1, M MAXLST = MAXLST + (IPNTR(IR+1) - IPNTR(IR))**2 20 CONTINUE MAXLST = MAXLST/N MAXCLQ = 0 NUMORD = 1 C C BEGINNING OF ITERATION LOOP. C 30 CONTINUE C C CHOOSE A COLUMN JCOL OF MAXIMAL DEGREE AMONG THE C COLUMNS OF MAXIMAL INCIDENCE MAXINC. C 40 CONTINUE JP = IWA1(MAXINC) IF (JP .GT. 0) GO TO 50 MAXINC = MAXINC - 1 GO TO 40 50 CONTINUE NUMWGT = -1 DO 60 NUMLST = 1, MAXLST IF (NDEG(JP) .GT. NUMWGT) THEN NUMWGT = NDEG(JP) JCOL = JP END IF JP = IWA3(JP) IF (JP .LE. 0) GO TO 70 60 CONTINUE 70 CONTINUE LIST(JCOL) = NUMORD C C UPDATE THE SIZE OF THE LARGEST CLIQUE C FOUND DURING THE ORDERING. C IF (MAXINC .EQ. 0) NCOMP = 0 NCOMP = NCOMP + 1 IF (MAXINC+1 .EQ. NCOMP) MAXCLQ = MAX(MAXCLQ,NCOMP) C C TERMINATION TEST. C NUMORD = NUMORD + 1 IF (NUMORD .GT. N) GO TO 100 C C DELETE COLUMN JCOL FROM THE MAXINC LIST. C IF (IWA2(JCOL) .EQ. 0) THEN IWA1(MAXINC) = IWA3(JCOL) ELSE IWA3(IWA2(JCOL)) = IWA3(JCOL) END IF IF (IWA3(JCOL) .GT. 0) IWA2(IWA3(JCOL)) = IWA2(JCOL) C C FIND ALL COLUMNS ADJACENT TO COLUMN JCOL. C IWA4(JCOL) = N C C DETERMINE ALL POSITIONS (IR,JCOL) WHICH CORRESPOND C TO NON-ZEROES IN THE MATRIX. C DO 90 JP = JPNTR(JCOL), JPNTR(JCOL+1)-1 IR = INDROW(JP) C C FOR EACH ROW IR, DETERMINE ALL POSITIONS (IR,IC) C WHICH CORRESPOND TO NON-ZEROES IN THE MATRIX. C DO 80 IP = IPNTR(IR), IPNTR(IR+1)-1 IC = INDCOL(IP) C C ARRAY IWA4 MARKS COLUMNS WHICH ARE ADJACENT TO C COLUMN JCOL. C IF (IWA4(IC) .LT. NUMORD) THEN IWA4(IC) = NUMORD C C UPDATE THE POINTERS TO THE CURRENT INCIDENCE LISTS. C NUMINC = LIST(IC) LIST(IC) = LIST(IC) + 1 MAXINC = MAX(MAXINC,LIST(IC)) C C DELETE COLUMN IC FROM THE NUMINC LIST. C IF (IWA2(IC) .EQ. 0) THEN IWA1(NUMINC) = IWA3(IC) ELSE IWA3(IWA2(IC)) = IWA3(IC) END IF IF (IWA3(IC) .GT. 0) IWA2(IWA3(IC)) = IWA2(IC) C C ADD COLUMN IC TO THE NUMINC+1 LIST. C IWA2(IC) = 0 IWA3(IC) = IWA1(NUMINC+1) IF (IWA1(NUMINC+1) .GT. 0) IWA2(IWA1(NUMINC+1)) = IC IWA1(NUMINC+1) = IC END IF 80 CONTINUE 90 CONTINUE C C END OF ITERATION LOOP. C GO TO 30 100 CONTINUE C C INVERT THE ARRAY LIST. C DO 110 JCOL = 1, N IWA2(LIST(JCOL)) = JCOL 110 CONTINUE DO 120 JP = 1, N LIST(JP) = IWA2(JP) 120 CONTINUE RETURN C C LAST CARD OF SUBROUTINE IDO. C END SUBROUTINE NUMSRT(N,NMAX,NUM,MODE,INDEX,LAST,NEXT) INTEGER N,NMAX,MODE INTEGER NUM(N),INDEX(N),LAST(0:NMAX),NEXT(N) C **********. C C SUBROUTINE NUMSRT C C GIVEN A SEQUENCE OF INTEGERS, THIS SUBROUTINE GROUPS C TOGETHER THOSE INDICES WITH THE SAME SEQUENCE VALUE C AND, OPTIONALLY, SORTS THE SEQUENCE INTO EITHER C ASCENDING OR DESCENDING ORDER. C C THE SEQUENCE OF INTEGERS IS DEFINED BY THE ARRAY NUM, C AND IT IS ASSUMED THAT THE INTEGERS ARE EACH FROM THE SET C 0,1,...,NMAX. ON OUTPUT THE INDICES K SUCH THAT NUM(K) = L C FOR ANY L = 0,1,...,NMAX CAN BE OBTAINED FROM THE ARRAYS C LAST AND NEXT AS FOLLOWS. C C K = LAST(L) C WHILE (K .NE. 0) K = NEXT(K) C C OPTIONALLY, THE SUBROUTINE PRODUCES AN ARRAY INDEX SO THAT C THE SEQUENCE NUM(INDEX(I)), I = 1,2,...,N IS SORTED. C C THE SUBROUTINE STATEMENT IS C C SUBROUTINE NUMSRT(N,NMAX,NUM,MODE,INDEX,LAST,NEXT) C C WHERE C C N IS A POSITIVE INTEGER INPUT VARIABLE. C C NMAX IS A POSITIVE INTEGER INPUT VARIABLE. C C NUM IS AN INPUT ARRAY OF LENGTH N WHICH CONTAINS THE C SEQUENCE OF INTEGERS TO BE GROUPED AND SORTED. IT C IS ASSUMED THAT THE INTEGERS ARE EACH FROM THE SET C 0,1,...,NMAX. C C MODE IS AN INTEGER INPUT VARIABLE. THE SEQUENCE NUM IS C SORTED IN ASCENDING ORDER IF MODE IS POSITIVE AND IN C DESCENDING ORDER IF MODE IS NEGATIVE. IF MODE IS 0, C NO SORTING IS DONE. C C INDEX IS AN INTEGER OUTPUT ARRAY OF LENGTH N SET SO C THAT THE SEQUENCE C C NUM(INDEX(I)), I = 1,2,...,N C C IS SORTED ACCORDING TO THE SETTING OF MODE. IF MODE C IS 0, INDEX IS NOT REFERENCED. C C LAST IS AN INTEGER OUTPUT ARRAY OF LENGTH NMAX + 1. THE C INDEX OF NUM FOR THE LAST OCCURRENCE OF L IS LAST(L) C FOR ANY L = 0,1,...,NMAX UNLESS LAST(L) = 0. IN C THIS CASE L DOES NOT APPEAR IN NUM. C C NEXT IS AN INTEGER OUTPUT ARRAY OF LENGTH N. IF C NUM(K) = L, THEN THE INDEX OF NUM FOR THE PREVIOUS C OCCURRENCE OF L IS NEXT(K) FOR ANY L = 0,1,...,NMAX C UNLESS NEXT(K) = 0. IN THIS CASE THERE IS NO PREVIOUS C OCCURRENCE OF L IN NUM. C C ARGONNE NATIONAL LABORATORY. MINPACK PROJECT. JULY 1983. C THOMAS F. COLEMAN, BURTON S. GARBOW, JORGE J. MORE' C C ********** INTEGER I,J,JINC,JL,JU,K,L C C DETERMINE THE ARRAYS NEXT AND LAST. C DO 10 I = 0, NMAX LAST(I) = 0 10 CONTINUE DO 20 K = 1, N L = NUM(K) NEXT(K) = LAST(L) LAST(L) = K 20 CONTINUE IF (MODE .EQ. 0) RETURN C C STORE THE POINTERS TO THE SORTED ARRAY IN INDEX. C I = 1 IF (MODE .GT. 0) THEN JL = 0 JU = NMAX JINC = 1 ELSE JL = NMAX JU = 0 JINC = -1 END IF DO 50 J = JL, JU, JINC K = LAST(J) 30 CONTINUE IF (K .EQ. 0) GO TO 40 INDEX(I) = K I = I + 1 K = NEXT(K) GO TO 30 40 CONTINUE 50 CONTINUE RETURN C C LAST CARD OF SUBROUTINE NUMSRT. C END SUBROUTINE SEQ(N,INDROW,JPNTR,INDCOL,IPNTR,LIST,NGRP,MAXGRP, * IWA) INTEGER N,MAXGRP INTEGER INDROW(*),JPNTR(N+1),INDCOL(*),IPNTR(*),LIST(N), * NGRP(N),IWA(N) C ********** C C SUBROUTINE SEQ C C GIVEN THE SPARSITY PATTERN OF AN M BY N MATRIX A, THIS C SUBROUTINE DETERMINES A CONSISTENT PARTITION OF THE C COLUMNS OF A BY A SEQUENTIAL ALGORITHM. C C A CONSISTENT PARTITION IS DEFINED IN TERMS OF THE LOOPLESS C GRAPH G WITH VERTICES A(J), J = 1,2,...,N WHERE A(J) IS THE C J-TH COLUMN OF A AND WITH EDGE (A(I),A(J)) IF AND ONLY IF C COLUMNS I AND J HAVE A NON-ZERO IN THE SAME ROW POSITION. C C A PARTITION OF THE COLUMNS OF A INTO GROUPS IS CONSISTENT C IF THE COLUMNS IN ANY GROUP ARE NOT ADJACENT IN THE GRAPH G. C IN GRAPH-THEORY TERMINOLOGY, A CONSISTENT PARTITION OF THE C COLUMNS OF A CORRESPONDS TO A COLORING OF THE GRAPH G. C C THE SUBROUTINE EXAMINES THE COLUMNS IN THE ORDER SPECIFIED C BY THE ARRAY LIST, AND ASSIGNS THE CURRENT COLUMN TO THE C GROUP WITH THE SMALLEST POSSIBLE NUMBER. C C NOTE THAT THE VALUE OF M IS NOT NEEDED BY SEQ AND IS C THEREFORE NOT PRESENT IN THE SUBROUTINE STATEMENT. C C THE SUBROUTINE STATEMENT IS C C SUBROUTINE SEQ(N,INDROW,JPNTR,INDCOL,IPNTR,LIST,NGRP,MAXGRP, C IWA) C C WHERE C C N IS A POSITIVE INTEGER INPUT VARIABLE SET TO THE NUMBER C OF COLUMNS OF A. C C INDROW IS AN INTEGER INPUT ARRAY WHICH CONTAINS THE ROW C INDICES FOR THE NON-ZEROES IN THE MATRIX A. C C JPNTR IS AN INTEGER INPUT ARRAY OF LENGTH N + 1 WHICH C SPECIFIES THE LOCATIONS OF THE ROW INDICES IN INDROW. C THE ROW INDICES FOR COLUMN J ARE C C INDROW(K), K = JPNTR(J),...,JPNTR(J+1)-1. C C NOTE THAT JPNTR(N+1)-1 IS THEN THE NUMBER OF NON-ZERO C ELEMENTS OF THE MATRIX A. C C INDCOL IS AN INTEGER INPUT ARRAY WHICH CONTAINS THE C COLUMN INDICES FOR THE NON-ZEROES IN THE MATRIX A. C C IPNTR IS AN INTEGER INPUT ARRAY OF LENGTH M + 1 WHICH C SPECIFIES THE LOCATIONS OF THE COLUMN INDICES IN INDCOL. C THE COLUMN INDICES FOR ROW I ARE C C INDCOL(K), K = IPNTR(I),...,IPNTR(I+1)-1. C C NOTE THAT IPNTR(M+1)-1 IS THEN THE NUMBER OF NON-ZERO C ELEMENTS OF THE MATRIX A. C C LIST IS AN INTEGER INPUT ARRAY OF LENGTH N WHICH SPECIFIES C THE ORDER TO BE USED BY THE SEQUENTIAL ALGORITHM. C THE J-TH COLUMN IN THIS ORDER IS LIST(J). C C NGRP IS AN INTEGER OUTPUT ARRAY OF LENGTH N WHICH SPECIFIES C THE PARTITION OF THE COLUMNS OF A. COLUMN JCOL BELONGS C TO GROUP NGRP(JCOL). C C MAXGRP IS AN INTEGER OUTPUT VARIABLE WHICH SPECIFIES THE C NUMBER OF GROUPS IN THE PARTITION OF THE COLUMNS OF A. C C IWA IS AN INTEGER WORK ARRAY OF LENGTH N. C C ARGONNE NATIONAL LABORATORY. MINPACK PROJECT. JULY 1983. C THOMAS F. COLEMAN, BURTON S. GARBOW, JORGE J. MORE' C C ********** INTEGER IC,IP,IR,J,JCOL,JP C C INITIALIZATION BLOCK. C MAXGRP = 0 DO 10 JP = 1, N NGRP(JP) = N IWA(JP) = 0 10 CONTINUE C C BEGINNING OF ITERATION LOOP. C DO 60 J = 1, N JCOL = LIST(J) C C FIND ALL COLUMNS ADJACENT TO COLUMN JCOL. C C DETERMINE ALL POSITIONS (IR,JCOL) WHICH CORRESPOND C TO NON-ZEROES IN THE MATRIX. C DO 30 JP = JPNTR(JCOL), JPNTR(JCOL+1)-1 IR = INDROW(JP) C C FOR EACH ROW IR, DETERMINE ALL POSITIONS (IR,IC) C WHICH CORRESPOND TO NON-ZEROES IN THE MATRIX. C DO 20 IP = IPNTR(IR), IPNTR(IR+1)-1 IC = INDCOL(IP) C C ARRAY IWA MARKS THE GROUP NUMBERS OF THE C COLUMNS WHICH ARE ADJACENT TO COLUMN JCOL. C IWA(NGRP(IC)) = J 20 CONTINUE 30 CONTINUE C C ASSIGN THE SMALLEST UN-MARKED GROUP NUMBER TO JCOL. C DO 40 JP = 1, MAXGRP IF (IWA(JP) .NE. J) GO TO 50 40 CONTINUE MAXGRP = MAXGRP + 1 50 CONTINUE NGRP(JCOL) = JP 60 CONTINUE C C END OF ITERATION LOOP. C RETURN C C LAST CARD OF SUBROUTINE SEQ. C END SUBROUTINE SETR(M,N,INDROW,JPNTR,INDCOL,IPNTR,IWA) INTEGER M,N INTEGER INDROW(*),JPNTR(N+1),INDCOL(*),IPNTR(M+1),IWA(M) C ********** C C SUBROUTINE SETR C C GIVEN A COLUMN-ORIENTED DEFINITION OF THE SPARSITY PATTERN C OF AN M BY N MATRIX A, THIS SUBROUTINE DETERMINES A C ROW-ORIENTED DEFINITION OF THE SPARSITY PATTERN OF A. C C ON INPUT THE COLUMN-ORIENTED DEFINITION IS SPECIFIED BY C THE ARRAYS INDROW AND JPNTR. ON OUTPUT THE ROW-ORIENTED C DEFINITION IS SPECIFIED BY THE ARRAYS INDCOL AND IPNTR. C C THE SUBROUTINE STATEMENT IS C C SUBROUTINE SETR(M,N,INDROW,JPNTR,INDCOL,IPNTR,IWA) C C WHERE C C M IS A POSITIVE INTEGER INPUT VARIABLE SET TO THE NUMBER C OF ROWS OF A. C C N IS A POSITIVE INTEGER INPUT VARIABLE SET TO THE NUMBER C OF COLUMNS OF A. C C INDROW IS AN INTEGER INPUT ARRAY WHICH CONTAINS THE ROW C INDICES FOR THE NON-ZEROES IN THE MATRIX A. C C JPNTR IS AN INTEGER INPUT ARRAY OF LENGTH N + 1 WHICH C SPECIFIES THE LOCATIONS OF THE ROW INDICES IN INDROW. C THE ROW INDICES FOR COLUMN J ARE C C INDROW(K), K = JPNTR(J),...,JPNTR(J+1)-1. C C NOTE THAT JPNTR(N+1)-1 IS THEN THE NUMBER OF NON-ZERO C ELEMENTS OF THE MATRIX A. C C INDCOL IS AN INTEGER OUTPUT ARRAY WHICH CONTAINS THE C COLUMN INDICES FOR THE NON-ZEROES IN THE MATRIX A. C C IPNTR IS AN INTEGER OUTPUT ARRAY OF LENGTH M + 1 WHICH C SPECIFIES THE LOCATIONS OF THE COLUMN INDICES IN INDCOL. C THE COLUMN INDICES FOR ROW I ARE C C INDCOL(K), K = IPNTR(I),...,IPNTR(I+1)-1. C C NOTE THAT IPNTR(1) IS SET TO 1 AND THAT IPNTR(M+1)-1 IS C THEN THE NUMBER OF NON-ZERO ELEMENTS OF THE MATRIX A. C C IWA IS AN INTEGER WORK ARRAY OF LENGTH M. C C ARGONNE NATIONAL LABORATORY. MINPACK PROJECT. JULY 1983. C THOMAS F. COLEMAN, BURTON S. GARBOW, JORGE J. MORE' C C ********** INTEGER IR,JCOL,JP C C STORE IN ARRAY IWA THE COUNTS OF NON-ZEROES IN THE ROWS. C DO 10 IR = 1, M IWA(IR) = 0 10 CONTINUE DO 20 JP = 1, JPNTR(N+1)-1 IWA(INDROW(JP)) = IWA(INDROW(JP)) + 1 20 CONTINUE C C SET POINTERS TO THE START OF THE ROWS IN INDCOL. C IPNTR(1) = 1 DO 30 IR = 1, M IPNTR(IR+1) = IPNTR(IR) + IWA(IR) IWA(IR) = IPNTR(IR) 30 CONTINUE C C FILL INDCOL. C DO 50 JCOL = 1, N DO 40 JP = JPNTR(JCOL), JPNTR(JCOL+1)-1 IR = INDROW(JP) INDCOL(IWA(IR)) = JCOL IWA(IR) = IWA(IR) + 1 40 CONTINUE 50 CONTINUE RETURN C C LAST CARD OF SUBROUTINE SETR. C END SUBROUTINE SLO(N,INDROW,JPNTR,INDCOL,IPNTR,NDEG,LIST, * MAXCLQ,IWA1,IWA2,IWA3,IWA4) INTEGER N,MAXCLQ INTEGER INDROW(*),JPNTR(N+1),INDCOL(*),IPNTR(*),NDEG(N), * LIST(N),IWA1(0:N-1),IWA2(N),IWA3(N),IWA4(N) C ********** C C SUBROUTINE SLO C C GIVEN THE SPARSITY PATTERN OF AN M BY N MATRIX A, THIS C SUBROUTINE DETERMINES THE SMALLEST-LAST ORDERING OF THE C COLUMNS OF A. C C THE SMALLEST-LAST ORDERING IS DEFINED FOR THE LOOPLESS C GRAPH G WITH VERTICES A(J), J = 1,2,...,N WHERE A(J) IS THE C J-TH COLUMN OF A AND WITH EDGE (A(I),A(J)) IF AND ONLY IF C COLUMNS I AND J HAVE A NON-ZERO IN THE SAME ROW POSITION. C C THE SMALLEST-LAST ORDERING IS DETERMINED RECURSIVELY BY C LETTING LIST(K), K = N,...,1 BE A COLUMN WITH LEAST DEGREE C IN THE SUBGRAPH SPANNED BY THE UN-ORDERED COLUMNS. C C NOTE THAT THE VALUE OF M IS NOT NEEDED BY SLO AND IS C THEREFORE NOT PRESENT IN THE SUBROUTINE STATEMENT. C C THE SUBROUTINE STATEMENT IS C C SUBROUTINE SLO(N,INDROW,JPNTR,INDCOL,IPNTR,NDEG,LIST, C MAXCLQ,IWA1,IWA2,IWA3,IWA4) C C WHERE C C N IS A POSITIVE INTEGER INPUT VARIABLE SET TO THE NUMBER C OF COLUMNS OF A. C C INDROW IS AN INTEGER INPUT ARRAY WHICH CONTAINS THE ROW C INDICES FOR THE NON-ZEROES IN THE MATRIX A. C C JPNTR IS AN INTEGER INPUT ARRAY OF LENGTH N + 1 WHICH C SPECIFIES THE LOCATIONS OF THE ROW INDICES IN INDROW. C THE ROW INDICES FOR COLUMN J ARE C C INDROW(K), K = JPNTR(J),...,JPNTR(J+1)-1. C C NOTE THAT JPNTR(N+1)-1 IS THEN THE NUMBER OF NON-ZERO C ELEMENTS OF THE MATRIX A. C C INDCOL IS AN INTEGER INPUT ARRAY WHICH CONTAINS THE C COLUMN INDICES FOR THE NON-ZEROES IN THE MATRIX A. C C IPNTR IS AN INTEGER INPUT ARRAY OF LENGTH M + 1 WHICH C SPECIFIES THE LOCATIONS OF THE COLUMN INDICES IN INDCOL. C THE COLUMN INDICES FOR ROW I ARE C C INDCOL(K), K = IPNTR(I),...,IPNTR(I+1)-1. C C NOTE THAT IPNTR(M+1)-1 IS THEN THE NUMBER OF NON-ZERO C ELEMENTS OF THE MATRIX A. C C NDEG IS AN INTEGER INPUT ARRAY OF LENGTH N WHICH SPECIFIES C THE DEGREE SEQUENCE. THE DEGREE OF THE J-TH COLUMN C OF A IS NDEG(J). C C LIST IS AN INTEGER OUTPUT ARRAY OF LENGTH N WHICH SPECIFIES C THE SMALLEST-LAST ORDERING OF THE COLUMNS OF A. THE J-TH C COLUMN IN THIS ORDER IS LIST(J). C C MAXCLQ IS AN INTEGER OUTPUT VARIABLE SET TO THE SIZE C OF THE LARGEST CLIQUE FOUND DURING THE ORDERING. C C IWA1,IWA2,IWA3, AND IWA4 ARE INTEGER WORK ARRAYS OF LENGTH N. C C SUBPROGRAMS CALLED C C FORTRAN-SUPPLIED ... MIN C C ARGONNE NATIONAL LABORATORY. MINPACK PROJECT. AUGUST 1984. C THOMAS F. COLEMAN, BURTON S. GARBOW, JORGE J. MORE' C C ********** INTEGER IC,IP,IR,JCOL,JP,MINDEG,NUMDEG,NUMORD C C INITIALIZATION BLOCK. C MINDEG = N DO 10 JP = 1, N IWA1(JP-1) = 0 IWA4(JP) = N LIST(JP) = NDEG(JP) MINDEG = MIN(MINDEG,NDEG(JP)) 10 CONTINUE C C CREATE A DOUBLY-LINKED LIST TO ACCESS THE DEGREES OF THE C COLUMNS. THE POINTERS FOR THE LINKED LIST ARE AS FOLLOWS. C C EACH UN-ORDERED COLUMN IC IS IN A LIST (THE DEGREE LIST) C OF COLUMNS WITH THE SAME DEGREE. C C IWA1(NUMDEG) IS THE FIRST COLUMN IN THE NUMDEG LIST C UNLESS IWA1(NUMDEG) = 0. IN THIS CASE THERE ARE C NO COLUMNS IN THE NUMDEG LIST. C C IWA2(IC) IS THE COLUMN BEFORE IC IN THE DEGREE LIST C UNLESS IWA2(IC) = 0. IN THIS CASE IC IS THE FIRST C COLUMN IN THIS DEGREE LIST. C C IWA3(IC) IS THE COLUMN AFTER IC IN THE DEGREE LIST C UNLESS IWA3(IC) = 0. IN THIS CASE IC IS THE LAST C COLUMN IN THIS DEGREE LIST. C C IF IC IS AN UN-ORDERED COLUMN, THEN LIST(IC) IS THE C DEGREE OF IC IN THE GRAPH INDUCED BY THE UN-ORDERED C COLUMNS. IF JCOL IS AN ORDERED COLUMN, THEN LIST(JCOL) C IS THE SMALLEST-LAST ORDER OF COLUMN JCOL. C DO 20 JP = 1, N NUMDEG = NDEG(JP) IWA2(JP) = 0 IWA3(JP) = IWA1(NUMDEG) IF (IWA1(NUMDEG) .GT. 0) IWA2(IWA1(NUMDEG)) = JP IWA1(NUMDEG) = JP 20 CONTINUE MAXCLQ = 0 NUMORD = N C C BEGINNING OF ITERATION LOOP. C 30 CONTINUE C C CHOOSE A COLUMN JCOL OF MINIMAL DEGREE MINDEG. C 40 CONTINUE JCOL = IWA1(MINDEG) IF (JCOL .GT. 0) GO TO 50 MINDEG = MINDEG + 1 GO TO 40 50 CONTINUE LIST(JCOL) = NUMORD C C MARK THE SIZE OF THE LARGEST CLIQUE C FOUND DURING THE ORDERING. C IF (MINDEG+1 .EQ. NUMORD .AND. MAXCLQ .EQ. 0) * MAXCLQ = NUMORD C C TERMINATION TEST. C NUMORD = NUMORD - 1 IF (NUMORD .EQ. 0) GO TO 80 C C DELETE COLUMN JCOL FROM THE MINDEG LIST. C IWA1(MINDEG) = IWA3(JCOL) IF (IWA3(JCOL) .GT. 0) IWA2(IWA3(JCOL)) = 0 C C FIND ALL COLUMNS ADJACENT TO COLUMN JCOL. C IWA4(JCOL) = 0 C C DETERMINE ALL POSITIONS (IR,JCOL) WHICH CORRESPOND C TO NON-ZEROES IN THE MATRIX. C DO 70 JP = JPNTR(JCOL), JPNTR(JCOL+1)-1 IR = INDROW(JP) C C FOR EACH ROW IR, DETERMINE ALL POSITIONS (IR,IC) C WHICH CORRESPOND TO NON-ZEROES IN THE MATRIX. C DO 60 IP = IPNTR(IR), IPNTR(IR+1)-1 IC = INDCOL(IP) C C ARRAY IWA4 MARKS COLUMNS WHICH ARE ADJACENT TO C COLUMN JCOL. C IF (IWA4(IC) .GT. NUMORD) THEN IWA4(IC) = NUMORD C C UPDATE THE POINTERS TO THE CURRENT DEGREE LISTS. C NUMDEG = LIST(IC) LIST(IC) = LIST(IC) - 1 MINDEG = MIN(MINDEG,LIST(IC)) C C DELETE COLUMN IC FROM THE NUMDEG LIST. C IF (IWA2(IC) .EQ. 0) THEN IWA1(NUMDEG) = IWA3(IC) ELSE IWA3(IWA2(IC)) = IWA3(IC) END IF IF (IWA3(IC) .GT. 0) IWA2(IWA3(IC)) = IWA2(IC) C C ADD COLUMN IC TO THE NUMDEG-1 LIST. C IWA2(IC) = 0 IWA3(IC) = IWA1(NUMDEG-1) IF (IWA1(NUMDEG-1) .GT. 0) IWA2(IWA1(NUMDEG-1)) = IC IWA1(NUMDEG-1) = IC END IF 60 CONTINUE 70 CONTINUE C C END OF ITERATION LOOP. C GO TO 30 80 CONTINUE C C INVERT THE ARRAY LIST. C DO 90 JCOL = 1, N IWA2(LIST(JCOL)) = JCOL 90 CONTINUE DO 100 JP = 1, N LIST(JP) = IWA2(JP) 100 CONTINUE RETURN C C LAST CARD OF SUBROUTINE SLO. C END SUBROUTINE SRTDAT(N,NNZ,INDROW,INDCOL,JPNTR,IWA) INTEGER N,NNZ INTEGER INDROW(NNZ),INDCOL(NNZ),JPNTR(N+1),IWA(N) C ********** C C SUBROUTINE SRTDAT C C GIVEN THE NON-ZERO ELEMENTS OF AN M BY N MATRIX A IN C ARBITRARY ORDER AS SPECIFIED BY THEIR ROW AND COLUMN C INDICES, THIS SUBROUTINE PERMUTES THESE ELEMENTS SO C THAT THEIR COLUMN INDICES ARE IN NON-DECREASING ORDER. C C ON INPUT IT IS ASSUMED THAT THE ELEMENTS ARE SPECIFIED IN C C INDROW(K),INDCOL(K), K = 1,...,NNZ. C C ON OUTPUT THE ELEMENTS ARE PERMUTED SO THAT INDCOL IS C IN NON-DECREASING ORDER. IN ADDITION, THE ARRAY JPNTR C IS SET SO THAT THE ROW INDICES FOR COLUMN J ARE C C INDROW(K), K = JPNTR(J),...,JPNTR(J+1)-1. C C NOTE THAT THE VALUE OF M IS NOT NEEDED BY SRTDAT AND IS C THEREFORE NOT PRESENT IN THE SUBROUTINE STATEMENT. C C THE SUBROUTINE STATEMENT IS C C SUBROUTINE SRTDAT(N,NNZ,INDROW,INDCOL,JPNTR,IWA) C C WHERE C C N IS A POSITIVE INTEGER INPUT VARIABLE SET TO THE NUMBER C OF COLUMNS OF A. C C NNZ IS A POSITIVE INTEGER INPUT VARIABLE SET TO THE NUMBER C OF NON-ZERO ELEMENTS OF A. C C INDROW IS AN INTEGER ARRAY OF LENGTH NNZ. ON INPUT INDROW C MUST CONTAIN THE ROW INDICES OF THE NON-ZERO ELEMENTS OF A. C ON OUTPUT INDROW IS PERMUTED SO THAT THE CORRESPONDING C COLUMN INDICES OF INDCOL ARE IN NON-DECREASING ORDER. C C INDCOL IS AN INTEGER ARRAY OF LENGTH NNZ. ON INPUT INDCOL C MUST CONTAIN THE COLUMN INDICES OF THE NON-ZERO ELEMENTS C OF A. ON OUTPUT INDCOL IS PERMUTED SO THAT THESE INDICES C ARE IN NON-DECREASING ORDER. C C JPNTR IS AN INTEGER OUTPUT ARRAY OF LENGTH N + 1 WHICH C SPECIFIES THE LOCATIONS OF THE ROW INDICES IN THE OUTPUT C INDROW. THE ROW INDICES FOR COLUMN J ARE C C INDROW(K), K = JPNTR(J),...,JPNTR(J+1)-1. C C NOTE THAT JPNTR(1) IS SET TO 1 AND THAT JPNTR(N+1)-1 C IS THEN NNZ. C C IWA IS AN INTEGER WORK ARRAY OF LENGTH N. C C SUBPROGRAMS CALLED C C FORTRAN-SUPPLIED ... MAX C C ARGONNE NATIONAL LABORATORY. MINPACK PROJECT. JULY 1983. C THOMAS F. COLEMAN, BURTON S. GARBOW, JORGE J. MORE' C C ********** INTEGER I,J,K,L C C STORE IN ARRAY IWA THE COUNTS OF NON-ZEROES IN THE COLUMNS. C DO 10 J = 1, N IWA(J) = 0 10 CONTINUE DO 20 K = 1, NNZ IWA(INDCOL(K)) = IWA(INDCOL(K)) + 1 20 CONTINUE C C SET POINTERS TO THE START OF THE COLUMNS IN INDROW. C JPNTR(1) = 1 DO 30 J = 1, N JPNTR(J+1) = JPNTR(J) + IWA(J) IWA(J) = JPNTR(J) 30 CONTINUE K = 1 C C BEGIN IN-PLACE SORT. C 40 CONTINUE J = INDCOL(K) IF (K .GE. JPNTR(J)) THEN C C CURRENT ELEMENT IS IN POSITION. NOW EXAMINE THE C NEXT ELEMENT OR THE FIRST UN-SORTED ELEMENT IN C THE J-TH GROUP. C K = MAX(K+1,IWA(J)) ELSE C C CURRENT ELEMENT IS NOT IN POSITION. PLACE ELEMENT C IN POSITION AND MAKE THE DISPLACED ELEMENT THE C CURRENT ELEMENT. C L = IWA(J) IWA(J) = IWA(J) + 1 I = INDROW(K) INDROW(K) = INDROW(L) INDCOL(K) = INDCOL(L) INDROW(L) = I INDCOL(L) = J END IF IF (K .LE. NNZ) GO TO 40 RETURN C C LAST CARD OF SUBROUTINE SRTDAT. C END SHAR_EOF fi # end of overwriting check if test -f 'blas1.f' then echo shar: will not over-write existing file "'blas1.f'" else cat << \SHAR_EOF > 'blas1.f' SUBROUTINE DCOPY(N,DX,INCX,DY,INCY) C C COPIES A VECTOR, X, TO A VECTOR, Y. C USES UNROLLED LOOPS FOR INCREMENTS EQUAL TO ONE. C JACK DONGARRA, LINPACK, 3/11/78. C INTEGER I,INCX,INCY,IX,IY,M,MP1,N DOUBLE PRECISION DX(*),DY(*) CS REAL DX(*),DY(*) C IF(N.LE.0)RETURN IF(INCX.EQ.1.AND.INCY.EQ.1)GO TO 20 C C CODE FOR UNEQUAL INCREMENTS OR EQUAL INCREMENTS C NOT EQUAL TO 1 C IX = 1 IY = 1 IF(INCX.LT.0)IX = (-N+1)*INCX + 1 IF(INCY.LT.0)IY = (-N+1)*INCY + 1 DO 10 I = 1,N DY(IY) = DX(IX) IX = IX + INCX IY = IY + INCY 10 CONTINUE RETURN C C CODE FOR BOTH INCREMENTS EQUAL TO 1 C C C CLEAN-UP LOOP C 20 M = MOD(N,7) IF( M .EQ. 0 ) GO TO 40 DO 30 I = 1,M DY(I) = DX(I) 30 CONTINUE IF( N .LT. 7 ) RETURN 40 MP1 = M + 1 DO 50 I = MP1,N,7 DY(I) = DX(I) DY(I + 1) = DX(I + 1) DY(I + 2) = DX(I + 2) DY(I + 3) = DX(I + 3) DY(I + 4) = DX(I + 4) DY(I + 5) = DX(I + 5) DY(I + 6) = DX(I + 6) 50 CONTINUE RETURN END DOUBLE PRECISION FUNCTION DDOT(N,DX,INCX,DY,INCY) C C FORMS THE DOT PRODUCT OF TWO VECTORS. C USES UNROLLED LOOPS FOR INCREMENTS EQUAL TO ONE. C JACK DONGARRA, LINPACK, 3/11/78. C INTEGER I,INCX,INCY,IX,IY,M,MP1,N DOUBLE PRECISION DX(*),DY(*),DTEMP,ZERO CS REAL DX(*),DY(*),DTEMP,ZERO CS PARAMETER ( ZERO = 0.0E+0 ) PARAMETER ( ZERO = 0.0D+0 ) C CS SDOT = ZERO DDOT = ZERO DTEMP = ZERO IF(N.LE.0)RETURN IF(INCX.EQ.1.AND.INCY.EQ.1)GO TO 20 C C CODE FOR UNEQUAL INCREMENTS OR EQUAL INCREMENTS C NOT EQUAL TO 1 C IX = 1 IY = 1 IF(INCX.LT.0)IX = (-N+1)*INCX + 1 IF(INCY.LT.0)IY = (-N+1)*INCY + 1 DO 10 I = 1,N DTEMP = DTEMP + DX(IX)*DY(IY) IX = IX + INCX IY = IY + INCY 10 CONTINUE CS SDOT = DTEMP DDOT = DTEMP RETURN C C CODE FOR BOTH INCREMENTS EQUAL TO 1 C C C CLEAN-UP LOOP C 20 M = MOD(N,5) IF( M .EQ. 0 ) GO TO 40 DO 30 I = 1,M DTEMP = DTEMP + DX(I)*DY(I) 30 CONTINUE IF( N .LT. 5 ) GO TO 60 40 MP1 = M + 1 DO 50 I = MP1,N,5 DTEMP = DTEMP + DX(I)*DY(I) + DX(I + 1)*DY(I + 1) + * DX(I + 2)*DY(I + 2) + DX(I + 3)*DY(I + 3) + DX(I + 4)*DY(I + 4) 50 CONTINUE 60 CONTINUE CS SDOT = DTEMP DDOT = DTEMP RETURN END DOUBLE PRECISION FUNCTION DNRM2 ( N, DX, INCX) INTEGER N, INCX, NEXT INTEGER I, J, NN DOUBLE PRECISION DX( * ), CUTLO, CUTHI, HITEST, SUM, * XMAX, ZERO, ONE CS REAL DX( * ), CUTLO, CUTHI, HITEST, SUM, CS * XMAX, ZERO, ONE INTRINSIC ABS, SQRT CS PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 ) PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) C C EUCLIDEAN NORM OF THE N-VECTOR STORED IN DX() WITH STORAGE C INCREMENT INCX . C IF N .LE. 0 RETURN WITH RESULT = 0. C IF N .GE. 1 THEN INCX MUST BE .GE. 1 C C C.L.LAWSON, 1978 JAN 08 C C FOUR PHASE METHOD USING TWO BUILT-IN CONSTANTS THAT ARE C HOPEFULLY APPLICABLE TO ALL MACHINES. C CUTLO = MAXIMUM OF DSQRT(U/EPS) OVER ALL KNOWN MACHINES. C CUTHI = MINIMUM OF DSQRT(V) OVER ALL KNOWN MACHINES. C WHERE C EPS = SMALLEST NO. SUCH THAT EPS + 1. .GT. 1. C U = SMALLEST POSITIVE NO. (UNDERFLOW LIMIT) C V = LARGEST NO. (OVERFLOW LIMIT) C C BRIEF OUTLINE OF ALGORITHM.. C C PHASE 1 SCANS ZERO COMPONENTS. C MOVE TO PHASE 2 WHEN A COMPONENT IS NONZERO AND .LE. CUTLO C MOVE TO PHASE 3 WHEN A COMPONENT IS .GT. CUTLO C MOVE TO PHASE 4 WHEN A COMPONENT IS .GE. CUTHI/M C WHERE M = N FOR X() REAL AND M = 2*N FOR COMPLEX. C C VALUES FOR CUTLO AND CUTHI.. C FROM THE ENVIRONMENTAL PARAMETERS LISTED IN THE IMSL CONVERTER C DOCUMENT THE LIMITING VALUES ARE AS FOLLOWS.. C CUTLO, S.P. U/EPS = 2**(-102) FOR HONEYWELL. CLOSE SECONDS ARE C UNIVAC AND DEC AT 2**(-103) C THUS CUTLO = 2**(-51) = 4.44089E-16 C CUTHI, S.P. V = 2**127 FOR UNIVAC, HONEYWELL, AND DEC. C THUS CUTHI = 2**(63.5) = 1.30438E19 C CUTLO, D.P. U/EPS = 2**(-67) FOR HONEYWELL AND DEC. C THUS CUTLO = 2**(-33.5) = 8.23181D-11 C CUTHI, D.P. SAME AS S.P. CUTHI = 1.30438D19 CS DATA CUTLO, CUTHI / 4.441E-16, 1.304E19 / DATA CUTLO, CUTHI / 8.232D-11, 1.304D19 / C IF ( N .LE. 0) THEN CS SNRM2 = ZERO DNRM2 = ZERO ELSE NEXT = 1 SUM = ZERO NN = N * INCX C C BEGIN MAIN LOOP C I = 1 20 CONTINUE GO TO ( 30, 50, 70, 110 ), NEXT 30 CONTINUE IF( ABS( DX( I ) ) .GT. CUTLO ) GO TO 85 NEXT = 2 XMAX = ZERO C C PHASE 1. SUM IS ZERO C 50 CONTINUE IF ( DX( I ) .EQ. ZERO ) GO TO 200 IF ( ABS( DX( I ) ) .GT. CUTLO ) GO TO 85 C C PREPARE FOR PHASE 2. C NEXT = 3 GO TO 105 C C PREPARE FOR PHASE 4. C 100 CONTINUE I = J NEXT = 4 SUM = ( SUM / DX( I ) ) / DX( I ) 105 CONTINUE XMAX = ABS( DX( I ) ) SUM = SUM + ( DX( I ) / XMAX ) ** 2 GO TO 200 C C PHASE 2. SUM IS SMALL. SCALE TO AVOID DESTRUCTIVE UNDERFLOW. C 70 CONTINUE IF ( ABS( DX( I ) ) .GT. CUTLO ) THEN C C PREPARE FOR PHASE 3. C SUM = ( SUM * XMAX) * XMAX GO TO 85 ENDIF C C COMMON CODE FOR PHASES 2 AND 4. C IN PHASE 4 SUM IS LARGE. SCALE TO AVOID OVERFLOW. C 110 CONTINUE IF ( ABS( DX( I ) ) .GT. XMAX ) THEN SUM = ONE + SUM * ( XMAX / DX( I ) ) ** 2 XMAX = ABS( DX( I ) ) ELSE SUM = SUM + ( DX( I ) / XMAX ) ** 2 ENDIF 200 CONTINUE I = I + INCX IF ( I .LE. NN ) GO TO 20 C C END OF MAIN LOOP. C C COMPUTE SQUARE ROOT AND ADJUST FOR SCALING. C CS SNRM2 = XMAX * SQRT(SUM) DNRM2 = XMAX * SQRT(SUM) GO TO 300 C C FOR REAL OR D.P. SET HITEST = CUTHI/N C 85 CONTINUE HITEST = CUTHI/FLOAT( N ) C C PHASE 3. SUM IS MID-RANGE. NO SCALING. C DO 95 J = I, NN, INCX IF( ABS( DX( J ) ) .GE. HITEST ) GO TO 100 SUM = SUM + DX( J ) ** 2 95 CONTINUE CS SNRM2 = SQRT( SUM ) DNRM2 = SQRT( SUM ) ENDIF 300 CONTINUE RETURN END SUBROUTINE DSCAL(N,DA,DX,INCX) C C SCALES A VECTOR BY A CONSTANT. C USES UNROLLED LOOPS FOR INCREMENT EQUAL TO ONE. C JACK DONGARRA, LINPACK, 3/11/78. C INTEGER I,INCX,M,MP1,N,NINCX DOUBLE PRECISION DA,DX(1) CS REAL DA,DX(1) C IF(N.LE.0)RETURN IF(INCX.EQ.1)GO TO 20 C C CODE FOR INCREMENT NOT EQUAL TO 1 C NINCX = N*INCX DO 10 I = 1,NINCX,INCX DX(I) = DA*DX(I) 10 CONTINUE RETURN C C CODE FOR INCREMENT EQUAL TO 1 C C C CLEAN-UP LOOP C 20 M = MOD(N,5) IF( M .EQ. 0 ) GO TO 40 DO 30 I = 1,M DX(I) = DA*DX(I) 30 CONTINUE IF( N .LT. 5 ) RETURN 40 MP1 = M + 1 DO 50 I = MP1,N,5 DX(I) = DA*DX(I) DX(I + 1) = DA*DX(I + 1) DX(I + 2) = DA*DX(I + 2) DX(I + 3) = DA*DX(I + 3) DX(I + 4) = DA*DX(I + 4) 50 CONTINUE RETURN END SHAR_EOF fi # end of overwriting check cd .. cd .. # End of shell archive exit 0