subroutine cgbco(abd,lda,n,ml,mu,ipvt,rcond,z) integer lda,n,ml,mu,ipvt(1) complex abd(lda,1),z(1) real rcond c c cgbco factors a complex band matrix by gaussian c elimination and estimates the condition of the matrix. c c if rcond is not needed, cgbfa is slightly faster. c to solve a*x = b , follow cgbco by cgbsl. c to compute inverse(a)*c , follow cgbco by cgbsl. c to compute determinant(a) , follow cgbco by cgbdi. c c on entry c c abd complex(lda, n) c contains the matrix in band storage. the columns c of the matrix are stored in the columns of abd and c the diagonals of the matrix are stored in rows c ml+1 through 2*ml+mu+1 of abd . c see the comments below for details. c c lda integer c the leading dimension of the array abd . c lda must be .ge. 2*ml + mu + 1 . c c n integer c the order of the original matrix. c c ml integer c number of diagonals below the main diagonal. c 0 .le. ml .lt. n . c c mu integer c number of diagonals above the main diagonal. c 0 .le. mu .lt. n . c more efficient if ml .le. mu . c c on return c c abd an upper triangular matrix in band storage and c the multipliers which were used to obtain it. c the factorization can be written a = l*u where c l is a product of permutation and unit lower c triangular matrices and u is upper triangular. c c ipvt integer(n) c an integer vector of pivot indices. c c rcond real c an estimate of the reciprocal condition of a . c for the system a*x = b , relative perturbations c in a and b of size epsilon may cause c relative perturbations in x of size epsilon/rcond . c if rcond is so small that the logical expression c 1.0 + rcond .eq. 1.0 c is true, then a may be singular to working c precision. in particular, rcond is zero if c exact singularity is detected or the estimate c underflows. c c z complex(n) c a work vector whose contents are usually unimportant. c if a is close to a singular matrix, then z is c an approximate null vector in the sense that c norm(a*z) = rcond*norm(a)*norm(z) . c c band storage c c if a is a band matrix, the following program segment c will set up the input. c c ml = (band width below the diagonal) c mu = (band width above the diagonal) c m = ml + mu + 1 c do 20 j = 1, n c i1 = max0(1, j-mu) c i2 = min0(n, j+ml) c do 10 i = i1, i2 c k = i - j + m c abd(k,j) = a(i,j) c 10 continue c 20 continue c c this uses rows ml+1 through 2*ml+mu+1 of abd . c in addition, the first ml rows in abd are used for c elements generated during the triangularization. c the total number of rows needed in abd is 2*ml+mu+1 . c the ml+mu by ml+mu upper left triangle and the c ml by ml lower right triangle are not referenced. c c example.. if the original matrix is c c 11 12 13 0 0 0 c 21 22 23 24 0 0 c 0 32 33 34 35 0 c 0 0 43 44 45 46 c 0 0 0 54 55 56 c 0 0 0 0 65 66 c c then n = 6, ml = 1, mu = 2, lda .ge. 5 and abd should contain c c * * * + + + , * = not used c * * 13 24 35 46 , + = used for pivoting c * 12 23 34 45 56 c 11 22 33 44 55 66 c 21 32 43 54 65 * c c linpack. this version dated 08/14/78 . c cleve moler, university of new mexico, argonne national lab. c c subroutines and functions c c linpack cgbfa c blas caxpy,cdotc,csscal,scasum c fortran abs,aimag,amax1,cmplx,conjg,max0,min0,real c c internal variables c complex cdotc,ek,t,wk,wkm real anorm,s,scasum,sm,ynorm integer is,info,j,ju,k,kb,kp1,l,la,lm,lz,m,mm c complex zdum,zdum1,zdum2,csign1 real cabs1 cabs1(zdum) = abs(real(zdum)) + abs(aimag(zdum)) csign1(zdum1,zdum2) = cabs1(zdum1)*(zdum2/cabs1(zdum2)) c c compute 1-norm of a c anorm = 0.0e0 l = ml + 1 is = l + mu do 10 j = 1, n anorm = amax1(anorm,scasum(l,abd(is,j),1)) if (is .gt. ml + 1) is = is - 1 if (j .le. mu) l = l + 1 if (j .ge. n - ml) l = l - 1 10 continue c c factor c call cgbfa(abd,lda,n,ml,mu,ipvt,info) c c rcond = 1/(norm(a)*(estimate of norm(inverse(a)))) . c estimate = norm(z)/norm(y) where a*z = y and ctrans(a)*y = e . c ctrans(a) is the conjugate transpose of a . c the components of e are chosen to cause maximum local c growth in the elements of w where ctrans(u)*w = e . c the vectors are frequently rescaled to avoid overflow. c c solve ctrans(u)*w = e c ek = (1.0e0,0.0e0) do 20 j = 1, n z(j) = (0.0e0,0.0e0) 20 continue m = ml + mu + 1 ju = 0 do 100 k = 1, n if (cabs1(z(k)) .ne. 0.0e0) ek = csign1(ek,-z(k)) if (cabs1(ek-z(k)) .le. cabs1(abd(m,k))) go to 30 s = cabs1(abd(m,k))/cabs1(ek-z(k)) call csscal(n,s,z,1) ek = cmplx(s,0.0e0)*ek 30 continue wk = ek - z(k) wkm = -ek - z(k) s = cabs1(wk) sm = cabs1(wkm) if (cabs1(abd(m,k)) .eq. 0.0e0) go to 40 wk = wk/conjg(abd(m,k)) wkm = wkm/conjg(abd(m,k)) go to 50 40 continue wk = (1.0e0,0.0e0) wkm = (1.0e0,0.0e0) 50 continue kp1 = k + 1 ju = min0(max0(ju,mu+ipvt(k)),n) mm = m if (kp1 .gt. ju) go to 90 do 60 j = kp1, ju mm = mm - 1 sm = sm + cabs1(z(j)+wkm*conjg(abd(mm,j))) z(j) = z(j) + wk*conjg(abd(mm,j)) s = s + cabs1(z(j)) 60 continue if (s .ge. sm) go to 80 t = wkm - wk wk = wkm mm = m do 70 j = kp1, ju mm = mm - 1 z(j) = z(j) + t*conjg(abd(mm,j)) 70 continue 80 continue 90 continue z(k) = wk 100 continue s = 1.0e0/scasum(n,z,1) call csscal(n,s,z,1) c c solve ctrans(l)*y = w c do 120 kb = 1, n k = n + 1 - kb lm = min0(ml,n-k) if (k .lt. n) z(k) = z(k) + cdotc(lm,abd(m+1,k),1,z(k+1),1) if (cabs1(z(k)) .le. 1.0e0) go to 110 s = 1.0e0/cabs1(z(k)) call csscal(n,s,z,1) 110 continue l = ipvt(k) t = z(l) z(l) = z(k) z(k) = t 120 continue s = 1.0e0/scasum(n,z,1) call csscal(n,s,z,1) c ynorm = 1.0e0 c c solve l*v = y c do 140 k = 1, n l = ipvt(k) t = z(l) z(l) = z(k) z(k) = t lm = min0(ml,n-k) if (k .lt. n) call caxpy(lm,t,abd(m+1,k),1,z(k+1),1) if (cabs1(z(k)) .le. 1.0e0) go to 130 s = 1.0e0/cabs1(z(k)) call csscal(n,s,z,1) ynorm = s*ynorm 130 continue 140 continue s = 1.0e0/scasum(n,z,1) call csscal(n,s,z,1) ynorm = s*ynorm c c solve u*z = w c do 160 kb = 1, n k = n + 1 - kb if (cabs1(z(k)) .le. cabs1(abd(m,k))) go to 150 s = cabs1(abd(m,k))/cabs1(z(k)) call csscal(n,s,z,1) ynorm = s*ynorm 150 continue if (cabs1(abd(m,k)) .ne. 0.0e0) z(k) = z(k)/abd(m,k) if (cabs1(abd(m,k)) .eq. 0.0e0) z(k) = (1.0e0,0.0e0) lm = min0(k,m) - 1 la = m - lm lz = k - lm t = -z(k) call caxpy(lm,t,abd(la,k),1,z(lz),1) 160 continue c make znorm = 1.0 s = 1.0e0/scasum(n,z,1) call csscal(n,s,z,1) ynorm = s*ynorm c if (anorm .ne. 0.0e0) rcond = ynorm/anorm if (anorm .eq. 0.0e0) rcond = 0.0e0 return end