real aa(200,200),a(201,200),b(200),x(200) real time(8,6),cray,ops,total,norma,normx real resid,residn,eps,epslon integer ipvt(200) lda = 201 ldaa = 200 c c this program was updated on 10/12/92 to correct a c problem with the random number generator. The previous c random number generator had a short period and produced c singular matrices occasionally. c n = 100 cray = .056 write(6,1) 1 format(' Please send the results of this run to:'// \$ ' Jack J. Dongarra'/ \$ ' Computer Science Department'/ \$ ' University of Tennessee'/ \$ ' Knoxville, Tennessee 37996-1300'// \$ ' Fax: 865-974-8296'// \$ ' Internet: dongarra@cs.utk.edu'/) ops = (2.0e0*float(n)**3)/3.0e0 + 2.0e0*float(n)**2 c call matgen(a,lda,n,b,norma) t1 = second() call sgefa(a,lda,n,ipvt,info) time(1,1) = second() - t1 t1 = second() call sgesl(a,lda,n,ipvt,b,0) time(1,2) = second() - t1 total = time(1,1) + time(1,2) C C COMPUTE A RESIDUAL TO VERIFY RESULTS. C do 10 i = 1,n x(i) = b(i) 10 continue call matgen(a,lda,n,b,norma) do 20 i = 1,n b(i) = -b(i) 20 continue CALL SMXPY(n,b,n,lda,x,a) RESID = 0.0 NORMX = 0.0 DO 30 I = 1,N RESID = amax1( RESID, ABS(b(i)) ) NORMX = amax1( NORMX, ABS(X(I)) ) 30 CONTINUE eps = epslon(1.0) RESIDn = RESID/( N*NORMA*NORMX*EPS ) write(6,40) 40 format(' norm. resid resid machep', \$ ' x(1) x(n)') write(6,50) residn,resid,eps,x(1),x(n) 50 format(1p5e16.8) c write(6,60) n 60 format(//' times are reported for matrices of order ',i5) write(6,70) 70 format(6x,'sgefa',6x,'sgesl',6x,'total',5x,'mflops',7x,'unit', \$ 6x,'ratio') c time(1,3) = total time(1,4) = ops/(1.0e6*total) time(1,5) = 2.0e0/time(1,4) time(1,6) = total/cray write(6,80) lda 80 format(' times for array with leading dimension of',i4) write(6,110) (time(1,i),i=1,6) c call matgen(a,lda,n,b,norma) t1 = second() call sgefa(a,lda,n,ipvt,info) time(2,1) = second() - t1 t1 = second() call sgesl(a,lda,n,ipvt,b,0) time(2,2) = second() - t1 total = time(2,1) + time(2,2) time(2,3) = total time(2,4) = ops/(1.0e6*total) time(2,5) = 2.0e0/time(2,4) time(2,6) = total/cray c call matgen(a,lda,n,b,norma) t1 = second() call sgefa(a,lda,n,ipvt,info) time(3,1) = second() - t1 t1 = second() call sgesl(a,lda,n,ipvt,b,0) time(3,2) = second() - t1 total = time(3,1) + time(3,2) time(3,3) = total time(3,4) = ops/(1.0e6*total) time(3,5) = 2.0e0/time(3,4) time(3,6) = total/cray c ntimes = 10 tm2 = 0 t1 = second() do 90 i = 1,ntimes tm = second() call matgen(a,lda,n,b,norma) tm2 = tm2 + second() - tm call sgefa(a,lda,n,ipvt,info) 90 continue time(4,1) = (second() - t1 - tm2)/ntimes t1 = second() do 100 i = 1,ntimes call sgesl(a,lda,n,ipvt,b,0) 100 continue time(4,2) = (second() - t1)/ntimes total = time(4,1) + time(4,2) time(4,3) = total time(4,4) = ops/(1.0e6*total) time(4,5) = 2.0e0/time(4,4) time(4,6) = total/cray c write(6,110) (time(2,i),i=1,6) write(6,110) (time(3,i),i=1,6) write(6,110) (time(4,i),i=1,6) 110 format(6(1pe11.3)) c call matgen(aa,ldaa,n,b,norma) t1 = second() call sgefa(aa,ldaa,n,ipvt,info) time(5,1) = second() - t1 t1 = second() call sgesl(aa,ldaa,n,ipvt,b,0) time(5,2) = second() - t1 total = time(5,1) + time(5,2) time(5,3) = total time(5,4) = ops/(1.0e6*total) time(5,5) = 2.0e0/time(5,4) time(5,6) = total/cray c call matgen(aa,ldaa,n,b,norma) t1 = second() call sgefa(aa,ldaa,n,ipvt,info) time(6,1) = second() - t1 t1 = second() call sgesl(aa,ldaa,n,ipvt,b,0) time(6,2) = second() - t1 total = time(6,1) + time(6,2) time(6,3) = total time(6,4) = ops/(1.0e6*total) time(6,5) = 2.0e0/time(6,4) time(6,6) = total/cray c call matgen(aa,ldaa,n,b,norma) t1 = second() call sgefa(aa,ldaa,n,ipvt,info) time(7,1) = second() - t1 t1 = second() call sgesl(aa,ldaa,n,ipvt,b,0) time(7,2) = second() - t1 total = time(7,1) + time(7,2) time(7,3) = total time(7,4) = ops/(1.0e6*total) time(7,5) = 2.0e0/time(7,4) time(7,6) = total/cray c ntimes = 10 tm2 = 0 t1 = second() do 120 i = 1,ntimes tm = second() call matgen(aa,ldaa,n,b,norma) tm2 = tm2 + second() - tm call sgefa(aa,ldaa,n,ipvt,info) 120 continue time(8,1) = (second() - t1 - tm2)/ntimes t1 = second() do 130 i = 1,ntimes call sgesl(aa,ldaa,n,ipvt,b,0) 130 continue time(8,2) = (second() - t1)/ntimes total = time(8,1) + time(8,2) time(8,3) = total time(8,4) = ops/(1.0e6*total) time(8,5) = 2.0e0/time(8,4) time(8,6) = total/cray c write(6,140) ldaa 140 format(/' times for array with leading dimension of',i4) write(6,110) (time(5,i),i=1,6) write(6,110) (time(6,i),i=1,6) write(6,110) (time(7,i),i=1,6) write(6,110) (time(8,i),i=1,6) write(6,*)' end of tests -- this version dated 10/12/92' stop end subroutine matgen(a,lda,n,b,norma) integer lda,n,init(4),i,j real a(lda,1),b(1),norma,ran external ran c init(1) = 1 init(2) = 2 init(3) = 3 init(4) = 1325 norma = 0.0 do 30 j = 1,n do 20 i = 1,n a(i,j) = ran(init) - .5 norma = amax1(abs(a(i,j)), norma) 20 continue 30 continue do 35 i = 1,n b(i) = 0.0 35 continue do 50 j = 1,n do 40 i = 1,n b(i) = b(i) + a(i,j) 40 continue 50 continue return end subroutine sgefa(a,lda,n,ipvt,info) integer lda,n,ipvt(1),info real a(lda,1) c c sgefa factors a real matrix by gaussian elimination. c c sgefa is usually called by dgeco, but it can be called c directly with a saving in time if rcond is not needed. c (time for dgeco) = (1 + 9/n)*(time for sgefa) . c c on entry c c a real(lda, n) c the matrix to be factored. c c lda integer c the leading dimension of the array a . c c n integer c the order of the matrix a . c c on return c c a an upper triangular matrix and the multipliers c 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 info integer c = 0 normal value. c = k if u(k,k) .eq. 0.0 . this is not an error c condition for this subroutine, but it does c indicate that sgesl or dgedi will divide by zero c if called. use rcond in dgeco for a reliable c indication of singularity. 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 blas saxpy,sscal,isamax c c internal variables c real t integer isamax,j,k,kp1,l,nm1 c c c gaussian elimination with partial pivoting c info = 0 nm1 = n - 1 if (nm1 .lt. 1) go to 70 do 60 k = 1, nm1 kp1 = k + 1 c c find l = pivot index c l = isamax(n-k+1,a(k,k),1) + k - 1 ipvt(k) = l c c zero pivot implies this column already triangularized c if (a(l,k) .eq. 0.0e0) go to 40 c c interchange if necessary c if (l .eq. k) go to 10 t = a(l,k) a(l,k) = a(k,k) a(k,k) = t 10 continue c c compute multipliers c t = -1.0e0/a(k,k) call sscal(n-k,t,a(k+1,k),1) c c row elimination with column indexing c do 30 j = kp1, n t = a(l,j) if (l .eq. k) go to 20 a(l,j) = a(k,j) a(k,j) = t 20 continue call saxpy(n-k,t,a(k+1,k),1,a(k+1,j),1) 30 continue go to 50 40 continue info = k 50 continue 60 continue 70 continue ipvt(n) = n if (a(n,n) .eq. 0.0e0) info = n return end subroutine sgesl(a,lda,n,ipvt,b,job) integer lda,n,ipvt(1),job real a(lda,1),b(1) c c sgesl solves the real system c a * x = b or trans(a) * x = b c using the factors computed by dgeco or sgefa. c c on entry c c a real(lda, n) c the output from dgeco or sgefa. c c lda integer c the leading dimension of the array a . c c n integer c the order of the matrix a . c c ipvt integer(n) c the pivot vector from dgeco or sgefa. c c b real(n) c the right hand side vector. c c job integer c = 0 to solve a*x = b , c = nonzero to solve trans(a)*x = b where c trans(a) is the transpose. c c on return c c b the solution vector x . c c error condition c c a division by zero will occur if the input factor contains a c zero on the diagonal. technically this indicates singularity c but it is often caused by improper arguments or improper c setting of lda . it will not occur if the subroutines are c called correctly and if dgeco has set rcond .gt. 0.0 c or sgefa has set info .eq. 0 . c c to compute inverse(a) * c where c is a matrix c with p columns c call dgeco(a,lda,n,ipvt,rcond,z) c if (rcond is too small) go to ... c do 10 j = 1, p c call sgesl(a,lda,n,ipvt,c(1,j),0) c 10 continue 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 blas saxpy,sdot c c internal variables c real sdot,t integer k,kb,l,nm1 c nm1 = n - 1 if (job .ne. 0) go to 50 c c job = 0 , solve a * x = b c first solve l*y = b c if (nm1 .lt. 1) go to 30 do 20 k = 1, nm1 l = ipvt(k) t = b(l) if (l .eq. k) go to 10 b(l) = b(k) b(k) = t 10 continue call saxpy(n-k,t,a(k+1,k),1,b(k+1),1) 20 continue 30 continue c c now solve u*x = y c do 40 kb = 1, n k = n + 1 - kb b(k) = b(k)/a(k,k) t = -b(k) call saxpy(k-1,t,a(1,k),1,b(1),1) 40 continue go to 100 50 continue c c job = nonzero, solve trans(a) * x = b c first solve trans(u)*y = b c do 60 k = 1, n t = sdot(k-1,a(1,k),1,b(1),1) b(k) = (b(k) - t)/a(k,k) 60 continue c c now solve trans(l)*x = y c if (nm1 .lt. 1) go to 90 do 80 kb = 1, nm1 k = n - kb b(k) = b(k) + sdot(n-k,a(k+1,k),1,b(k+1),1) l = ipvt(k) if (l .eq. k) go to 70 t = b(l) b(l) = b(k) b(k) = t 70 continue 80 continue 90 continue 100 continue return end subroutine saxpy(n,da,dx,incx,dy,incy) c c constant times a vector plus a vector. c jack dongarra, linpack, 3/11/78. c real dx(1),dy(1),da integer i,incx,incy,ix,iy,n c if(n.le.0)return if (da .eq. 0.0e0) 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) = dy(iy) + da*dx(ix) ix = ix + incx iy = iy + incy 10 continue return c c code for both increments equal to 1 c 20 continue do 30 i = 1,n dy(i) = dy(i) + da*dx(i) 30 continue return end real function sdot(n,dx,incx,dy,incy) c c forms the dot product of two vectors. c jack dongarra, linpack, 3/11/78. c real dx(1),dy(1),dtemp integer i,incx,incy,ix,iy,n c sdot = 0.0e0 dtemp = 0.0e0 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 return c c code for both increments equal to 1 c 20 continue do 30 i = 1,n dtemp = dtemp + dx(i)*dy(i) 30 continue sdot = dtemp return end subroutine sscal(n,da,dx,incx) c c scales a vector by a constant. c jack dongarra, linpack, 3/11/78. c real da,dx(1) integer i,incx,n,nincx 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 20 continue do 30 i = 1,n dx(i) = da*dx(i) 30 continue return end integer function isamax(n,dx,incx) c c finds the index of element having max. absolute value. c jack dongarra, linpack, 3/11/78. c real dx(1),dmax integer i,incx,ix,n c isamax = 0 if( n .lt. 1 ) return isamax = 1 if(n.eq.1)return if(incx.eq.1)go to 20 c c code for increment not equal to 1 c ix = 1 dmax = abs(dx(1)) ix = ix + incx do 10 i = 2,n if(abs(dx(ix)).le.dmax) go to 5 isamax = i dmax = abs(dx(ix)) 5 ix = ix + incx 10 continue return c c code for increment equal to 1 c 20 dmax = abs(dx(1)) do 30 i = 2,n if(abs(dx(i)).le.dmax) go to 30 isamax = i dmax = abs(dx(i)) 30 continue return end REAL FUNCTION EPSLON (X) REAL X C C ESTIMATE UNIT ROUNDOFF IN QUANTITIES OF SIZE X. C REAL A,B,C,EPS C C THIS PROGRAM SHOULD FUNCTION PROPERLY ON ALL SYSTEMS C SATISFYING THE FOLLOWING TWO ASSUMPTIONS, C 1. THE BASE USED IN REPRESENTING FLOATING POINT C NUMBERS IS NOT A POWER OF THREE. C 2. THE QUANTITY A IN STATEMENT 10 IS REPRESENTED TO C THE ACCURACY USED IN FLOATING POINT VARIABLES C THAT ARE STORED IN MEMORY. C THE STATEMENT NUMBER 10 AND THE GO TO 10 ARE INTENDED TO C FORCE OPTIMIZING COMPILERS TO GENERATE CODE SATISFYING C ASSUMPTION 2. C UNDER THESE ASSUMPTIONS, IT SHOULD BE TRUE THAT, C A IS NOT EXACTLY EQUAL TO FOUR-THIRDS, C B HAS A ZERO FOR ITS LAST BIT OR DIGIT, C C IS NOT EXACTLY EQUAL TO ONE, C EPS MEASURES THE SEPARATION OF 1.0 FROM C THE NEXT LARGER FLOATING POINT NUMBER. C THE DEVELOPERS OF EISPACK WOULD APPRECIATE BEING INFORMED C ABOUT ANY SYSTEMS WHERE THESE ASSUMPTIONS DO NOT HOLD. C C ***************************************************************** C THIS ROUTINE IS ONE OF THE AUXILIARY ROUTINES USED BY EISPACK III C TO AVOID MACHINE DEPENDENCIES. C ***************************************************************** C C THIS VERSION DATED 4/6/83. C A = 4.0E0/3.0E0 10 B = A - 1.0E0 C = B + B + B EPS = ABS(C-1.0E0) IF (EPS .EQ. 0.0E0) GO TO 10 EPSLON = EPS*ABS(X) RETURN END SUBROUTINE SMXPY (N1, Y, N2, LDM, X, M) REAL Y(*), X(*), M(LDM,*) C C PURPOSE: C MULTIPLY MATRIX M TIMES VECTOR X AND ADD THE RESULT TO VECTOR Y. C C PARAMETERS: C C N1 INTEGER, NUMBER OF ELEMENTS IN VECTOR Y, AND NUMBER OF ROWS IN C MATRIX M C C Y REAL(N1), VECTOR OF LENGTH N1 TO WHICH IS ADDED THE PRODUCT M*X C C N2 INTEGER, NUMBER OF ELEMENTS IN VECTOR X, AND NUMBER OF COLUMNS C IN MATRIX M C C LDM INTEGER, LEADING DIMENSION OF ARRAY M C C X REAL(N2), VECTOR OF LENGTH N2 C C M REAL(LDM,N2), MATRIX OF N1 ROWS AND N2 COLUMNS C C ---------------------------------------------------------------------- C C CLEANUP ODD VECTOR C J = MOD(N2,2) IF (J .GE. 1) THEN DO 10 I = 1, N1 Y(I) = (Y(I)) + X(J)*M(I,J) 10 CONTINUE ENDIF C C CLEANUP ODD GROUP OF TWO VECTORS C J = MOD(N2,4) IF (J .GE. 2) THEN DO 20 I = 1, N1 Y(I) = ( (Y(I)) \$ + X(J-1)*M(I,J-1)) + X(J)*M(I,J) 20 CONTINUE ENDIF C C CLEANUP ODD GROUP OF FOUR VECTORS C J = MOD(N2,8) IF (J .GE. 4) THEN DO 30 I = 1, N1 Y(I) = ((( (Y(I)) \$ + X(J-3)*M(I,J-3)) + X(J-2)*M(I,J-2)) \$ + X(J-1)*M(I,J-1)) + X(J) *M(I,J) 30 CONTINUE ENDIF C C CLEANUP ODD GROUP OF EIGHT VECTORS C J = MOD(N2,16) IF (J .GE. 8) THEN DO 40 I = 1, N1 Y(I) = ((((((( (Y(I)) \$ + X(J-7)*M(I,J-7)) + X(J-6)*M(I,J-6)) \$ + X(J-5)*M(I,J-5)) + X(J-4)*M(I,J-4)) \$ + X(J-3)*M(I,J-3)) + X(J-2)*M(I,J-2)) \$ + X(J-1)*M(I,J-1)) + X(J) *M(I,J) 40 CONTINUE ENDIF C C MAIN LOOP - GROUPS OF SIXTEEN VECTORS C JMIN = J+16 DO 60 J = JMIN, N2, 16 DO 50 I = 1, N1 Y(I) = ((((((((((((((( (Y(I)) \$ + X(J-15)*M(I,J-15)) + X(J-14)*M(I,J-14)) \$ + X(J-13)*M(I,J-13)) + X(J-12)*M(I,J-12)) \$ + X(J-11)*M(I,J-11)) + X(J-10)*M(I,J-10)) \$ + X(J- 9)*M(I,J- 9)) + X(J- 8)*M(I,J- 8)) \$ + X(J- 7)*M(I,J- 7)) + X(J- 6)*M(I,J- 6)) \$ + X(J- 5)*M(I,J- 5)) + X(J- 4)*M(I,J- 4)) \$ + X(J- 3)*M(I,J- 3)) + X(J- 2)*M(I,J- 2)) \$ + X(J- 1)*M(I,J- 1)) + X(J) *M(I,J) 50 CONTINUE 60 CONTINUE RETURN END REAL FUNCTION RAN( ISEED ) * * modified from the LAPACK auxiliary routine 10/12/92 JD * -- LAPACK auxiliary routine (version 1.0) -- * Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., * Courant Institute, Argonne National Lab, and Rice University * February 29, 1992 * * .. Array Arguments .. INTEGER ISEED( 4 ) * .. * * Purpose * ======= * * SLARAN returns a random real number from a uniform (0,1) * distribution. * * Arguments * ========= * * ISEED (input/output) INTEGER array, dimension (4) * On entry, the seed of the random number generator; the array * elements must be between 0 and 4095, and ISEED(4) must be * odd. * On exit, the seed is updated. * * Further Details * =============== * * This routine uses a multiplicative congruential method with modulus * 2**48 and multiplier 33952834046453 (see G.S.Fishman, * 'Multiplicative congruential random number generators with modulus * 2**b: an exhaustive analysis for b = 32 and a partial analysis for * b = 48', Math. Comp. 189, pp 331-344, 1990). * * 48-bit integers are stored in 4 integer array elements with 12 bits * per element. Hence the routine is portable across machines with * integers of 32 bits or more. * * .. Parameters .. INTEGER M1, M2, M3, M4 PARAMETER ( M1 = 494, M2 = 322, M3 = 2508, M4 = 2549 ) REAL ONE PARAMETER ( ONE = 1.0E+0 ) INTEGER IPW2 REAL R PARAMETER ( IPW2 = 4096, R = ONE / IPW2 ) * .. * .. Local Scalars .. INTEGER IT1, IT2, IT3, IT4 * .. * .. Intrinsic Functions .. INTRINSIC MOD, REAL * .. * .. Executable Statements .. * * multiply the seed by the multiplier modulo 2**48 * IT4 = ISEED( 4 )*M4 IT3 = IT4 / IPW2 IT4 = IT4 - IPW2*IT3 IT3 = IT3 + ISEED( 3 )*M4 + ISEED( 4 )*M3 IT2 = IT3 / IPW2 IT3 = IT3 - IPW2*IT2 IT2 = IT2 + ISEED( 2 )*M4 + ISEED( 3 )*M3 + ISEED( 4 )*M2 IT1 = IT2 / IPW2 IT2 = IT2 - IPW2*IT1 IT1 = IT1 + ISEED( 1 )*M4 + ISEED( 2 )*M3 + ISEED( 3 )*M2 + \$ ISEED( 4 )*M1 IT1 = MOD( IT1, IPW2 ) * * return updated seed * ISEED( 1 ) = IT1 ISEED( 2 ) = IT2 ISEED( 3 ) = IT3 ISEED( 4 ) = IT4 * * convert 48-bit integer to a real number in the interval (0,1) * RAN = R*( REAL( IT1 )+R*( REAL( IT2 )+R*( REAL( IT3 )+R* \$ ( REAL( IT4 ) ) ) ) ) RETURN * * End of RAN * END