SUBROUTINE CEBCHVXX( THRESH, PATH ) IMPLICIT NONE * .. Scalar Arguments .. REAL THRESH CHARACTER*3 PATH * * Purpose * ====== * * CEBCHVXX will run CGESVXX on a series of Hilbert matrices and then * compare the error bounds returned by CGESVXX to see if the returned * answer indeed falls within those bounds. * * Eight test ratios will be computed. The tests will pass if they are .LT. * THRESH. There are two cases that are determined by 1 / (SQRT( N ) * EPS). * If that value is .LE. to the component wise reciprocal condition number, * it uses the guaranteed case, other wise it uses the unguaranteed case. * * Test ratios: * Let Xc be X_computed and Xt be X_truth. * The norm used is the infinity norm. * Let A be the guaranteed case and B be the unguaranteed case. * * 1. Normwise guaranteed forward error bound. * A: norm ( abs( Xc - Xt ) / norm ( Xt ) .LE. ERRBND( *, nwise_i, bnd_i ) and * ERRBND( *, nwise_i, bnd_i ) .LE. MAX(SQRT(N),10) * EPS. * If these conditions are met, the test ratio is set to be * ERRBND( *, nwise_i, bnd_i ) / MAX(SQRT(N), 10). Otherwise it is 1/EPS. * B: For this case, CGESVXX should just return 1. If it is less than * one, treat it the same as in 1A. Otherwise it fails. (Set test * ratio to ERRBND( *, nwise_i, bnd_i ) * THRESH?) * * 2. Componentwise guaranteed forward error bound. * A: norm ( abs( Xc(j) - Xt(j) ) ) / norm (Xt(j)) .LE. ERRBND( *, cwise_i, bnd_i ) * for all j .AND. ERRBND( *, cwise_i, bnd_i ) .LE. MAX(SQRT(N), 10) * EPS. * If these conditions are met, the test ratio is set to be * ERRBND( *, cwise_i, bnd_i ) / MAX(SQRT(N), 10). Otherwise it is 1/EPS. * B: Same as normwise test ratio. * * 3. Backwards error. * A: The test ratio is set to BERR/EPS. * B: Same test ratio. * * 4. Reciprocal condition number. * A: A condition number is computed with Xt and compared with the one * returned from CGESVXX. Let RCONDc be the RCOND returned by CGESVXX * and RCONDt be the RCOND from the truth value. Test ratio is set to * MAX(RCONDc/RCONDt, RCONDt/RCONDc). * B: Test ratio is set to 1 / (EPS * RCONDc). * * 5. Reciprocal normwise condition number. * A: The test ratio is set to * MAX(ERRBND( *, nwise_i, cond_i ) / NCOND, NCOND / ERRBND( *, nwise_i, cond_i )). * B: Test ratio is set to 1 / (EPS * ERRBND( *, nwise_i, cond_i )). * * 6. Reciprocal componentwise condition number. * A: Test ratio is set to * MAX(ERRBND( *, cwise_i, cond_i ) / CCOND, CCOND / ERRBND( *, cwise_i, cond_i )). * B: Test ratio is set to 1 / (EPS * ERRBND( *, cwise_i, cond_i )). * * .. Parameters .. * NMAX is determined by the largest number in the inverse of the hilbert * matrix. Precision is exhausted when the largest entry in it is greater * than 2 to the power of the number of bits in the fraction of the data * type used plus one, which is 24 for single precision. * NMAX should be 6 for single and 11 for double. INTEGER NMAX, NPARAMS, NERRBND, NTESTS, KL, KU PARAMETER (NMAX = 6, NPARAMS = 2, NERRBND = 3, \$ NTESTS = 6) * .. Local Scalars .. INTEGER N, NRHS, INFO, I ,J, k, NFAIL, LDA, \$ N_AUX_TESTS, LDAB, LDAFB CHARACTER FACT, TRANS, UPLO, EQUED CHARACTER*2 C2 CHARACTER(3) NGUAR, CGUAR LOGICAL printed_guide REAL NCOND, CCOND, M, NORMDIF, NORMT, RCOND, \$ RNORM, RINORM, SUMR, SUMRI, EPS, \$ BERR(NMAX), RPVGRW, ORCOND, \$ CWISE_ERR, NWISE_ERR, CWISE_BND, NWISE_BND, \$ CWISE_RCOND, NWISE_RCOND, \$ CONDTHRESH, ERRTHRESH COMPLEX ZDUM * .. Local Arrays .. REAL TSTRAT(NTESTS), RINV(NMAX), PARAMS(NPARAMS), \$ S(NMAX), R(NMAX),C(NMAX),RWORK(3*NMAX), \$ DIFF(NMAX, NMAX), \$ ERRBND_N(NMAX*3), ERRBND_C(NMAX*3) INTEGER IPIV(NMAX) COMPLEX A(NMAX,NMAX),INVHILB(NMAX,NMAX),X(NMAX,NMAX), \$ WORK(NMAX*3*5), AF(NMAX, NMAX),B(NMAX, NMAX), \$ ACOPY(NMAX, NMAX), \$ AB( (NMAX-1)+(NMAX-1)+1, NMAX ), \$ ABCOPY( (NMAX-1)+(NMAX-1)+1, NMAX ), \$ AFB( 2*(NMAX-1)+(NMAX-1)+1, NMAX ) * .. External Functions .. REAL SLAMCH * .. External Subroutines .. EXTERNAL CLAHILB, CGESVXX, CSYSVXX, CPOSVXX, \$ CGBSVXX, CLACPY, LSAMEN LOGICAL LSAMEN * .. Intrinsic Functions .. INTRINSIC SQRT, MAX, ABS, REAL, AIMAG * .. Statement Functions .. REAL CABS1 * .. * .. Statement Function Definitions .. CABS1( ZDUM ) = ABS( REAL( ZDUM ) ) + ABS( AIMAG( ZDUM ) ) * .. Parameters .. INTEGER NWISE_I, CWISE_I PARAMETER (NWISE_I = 1, CWISE_I = 1) INTEGER BND_I, COND_I PARAMETER (BND_I = 2, COND_I = 3) * Create the loop to test out the Hilbert matrices FACT = 'E' UPLO = 'U' TRANS = 'N' EQUED = 'N' EPS = SLAMCH('Epsilon') NFAIL = 0 N_AUX_TESTS = 0 LDA = NMAX LDAB = (NMAX-1)+(NMAX-1)+1 LDAFB = 2*(NMAX-1)+(NMAX-1)+1 C2 = PATH( 2: 3 ) * Main loop to test the different Hilbert Matrices. printed_guide = .false. DO N = 1 , NMAX PARAMS(1) = -1 PARAMS(2) = -1 KL = N-1 KU = N-1 NRHS = n M = MAX(SQRT(REAL(N)), 10.0) * Generate the Hilbert matrix, its inverse, and the * right hand side, all scaled by the LCM(1,..,2N-1). CALL CLAHILB(N, N, A, LDA, INVHILB, LDA, B, \$ LDA, WORK, INFO, PATH) * Copy A into ACOPY. CALL CLACPY('ALL', N, N, A, NMAX, ACOPY, NMAX) * Store A in band format for GB tests DO J = 1, N DO I = 1, KL+KU+1 AB( I, J ) = (0.0E+0,0.0E+0) END DO END DO DO J = 1, N DO I = MAX( 1, J-KU ), MIN( N, J+KL ) AB( KU+1+I-J, J ) = A( I, J ) END DO END DO * Copy AB into ABCOPY. DO J = 1, N DO I = 1, KL+KU+1 ABCOPY( I, J ) = (0.0E+0,0.0E+0) END DO END DO CALL CLACPY('ALL', KL+KU+1, N, AB, LDAB, ABCOPY, LDAB) * Call C**SVXX with default PARAMS and N_ERR_BND = 3. IF ( LSAMEN( 2, C2, 'SY' ) ) THEN CALL CSYSVXX(FACT, UPLO, N, NRHS, ACOPY, LDA, AF, LDA, \$ IPIV, EQUED, S, B, LDA, X, LDA, ORCOND, \$ RPVGRW, BERR, NERRBND, ERRBND_N, ERRBND_C, NPARAMS, \$ PARAMS, WORK, RWORK, INFO) ELSE IF ( LSAMEN( 2, C2, 'PO' ) ) THEN CALL CPOSVXX(FACT, UPLO, N, NRHS, ACOPY, LDA, AF, LDA, \$ EQUED, S, B, LDA, X, LDA, ORCOND, \$ RPVGRW, BERR, NERRBND, ERRBND_N, ERRBND_C, NPARAMS, \$ PARAMS, WORK, RWORK, INFO) ELSE IF ( LSAMEN( 2, C2, 'HE' ) ) THEN CALL CHESVXX(FACT, UPLO, N, NRHS, ACOPY, LDA, AF, LDA, \$ IPIV, EQUED, S, B, LDA, X, LDA, ORCOND, \$ RPVGRW, BERR, NERRBND, ERRBND_N, ERRBND_C, NPARAMS, \$ PARAMS, WORK, RWORK, INFO) ELSE IF ( LSAMEN( 2, C2, 'GB' ) ) THEN CALL CGBSVXX(FACT, TRANS, N, KL, KU, NRHS, ABCOPY, \$ LDAB, AFB, LDAFB, IPIV, EQUED, R, C, B, \$ LDA, X, LDA, ORCOND, RPVGRW, BERR, NERRBND, \$ ERRBND_N, ERRBND_C, NPARAMS, PARAMS, WORK, RWORK, \$ INFO) ELSE CALL CGESVXX(FACT, TRANS, N, NRHS, ACOPY, LDA, AF, LDA, \$ IPIV, EQUED, R, C, B, LDA, X, LDA, ORCOND, \$ RPVGRW, BERR, NERRBND, ERRBND_N, ERRBND_C, NPARAMS, \$ PARAMS, WORK, RWORK, INFO) END IF N_AUX_TESTS = N_AUX_TESTS + 1 IF (ORCOND .LT. EPS) THEN ! Either factorization failed or the matrix is flagged, and 1 <= ! INFO <= N+1. We don't decide based on rcond anymore. ! IF (INFO .EQ. 0 .OR. INFO .GT. N+1) THEN ! NFAIL = NFAIL + 1 ! WRITE (*, FMT=8000) N, INFO, ORCOND, RCOND ! END IF ELSE ! Either everything succeeded (INFO == 0) or some solution failed ! to converge (INFO > N+1). IF (INFO .GT. 0 .AND. INFO .LE. N+1) THEN NFAIL = NFAIL + 1 WRITE (*, FMT=8000) C2, N, INFO, ORCOND, RCOND END IF END IF * Calculating the difference between C**SVXX's X and the true X. DO I = 1,N DO J =1,NRHS DIFF(I,J) = X(I,J) - INVHILB(I,J) END DO END DO * Calculating the RCOND RNORM = 0 RINORM = 0 IF ( LSAMEN( 2, C2, 'PO' ) .OR. LSAMEN( 2, C2, 'SY' ) .OR. \$ LSAMEN( 2, C2, 'HE' ) ) THEN DO I = 1, N SUMR = 0 SUMRI = 0 DO J = 1, N SUMR = SUMR + S(I) * CABS1(A(I,J)) * S(J) SUMRI = SUMRI + CABS1(INVHILB(I, J)) / (S(J) * S(I)) END DO RNORM = MAX(RNORM,SUMR) RINORM = MAX(RINORM,SUMRI) END DO ELSE IF ( LSAMEN( 2, C2, 'GE' ) .OR. LSAMEN( 2, C2, 'GB' ) ) \$ THEN DO I = 1, N SUMR = 0 SUMRI = 0 DO J = 1, N SUMR = SUMR + R(I) * CABS1(A(I,J)) * C(J) SUMRI = SUMRI + CABS1(INVHILB(I, J)) / (R(J) * C(I)) END DO RNORM = MAX(RNORM,SUMR) RINORM = MAX(RINORM,SUMRI) END DO END IF RNORM = RNORM / CABS1(A(1, 1)) RCOND = 1.0/(RNORM * RINORM) * Calculating the R for normwise rcond. DO I = 1, N RINV(I) = 0.0 END DO DO J = 1, N DO I = 1, N RINV(I) = RINV(I) + CABS1(A(I,J)) END DO END DO * Calculating the Normwise rcond. RINORM = 0.0 DO I = 1, N SUMRI = 0.0 DO J = 1, N SUMRI = SUMRI + CABS1(INVHILB(I,J) * RINV(J)) END DO RINORM = MAX(RINORM, SUMRI) END DO ! invhilb is the inverse *unscaled* Hilbert matrix, so scale its norm ! by 1/A(1,1) to make the scaling match A (the scaled Hilbert matrix) NCOND = CABS1(A(1,1)) / RINORM CONDTHRESH = M * EPS ERRTHRESH = M * EPS DO K = 1, NRHS NORMT = 0.0 NORMDIF = 0.0 CWISE_ERR = 0.0 DO I = 1, N NORMT = MAX(CABS1(INVHILB(I, K)), NORMT) NORMDIF = MAX(CABS1(X(I,K) - INVHILB(I,K)), NORMDIF) IF (INVHILB(I,K) .NE. 0.0) THEN CWISE_ERR = MAX(CABS1(X(I,K) - INVHILB(I,K)) \$ /CABS1(INVHILB(I,K)), CWISE_ERR) ELSE IF (X(I, K) .NE. 0.0) THEN CWISE_ERR = SLAMCH('OVERFLOW') END IF END DO IF (NORMT .NE. 0.0) THEN NWISE_ERR = NORMDIF / NORMT ELSE IF (NORMDIF .NE. 0.0) THEN NWISE_ERR = SLAMCH('OVERFLOW') ELSE NWISE_ERR = 0.0 ENDIF DO I = 1, N RINV(I) = 0.0 END DO DO J = 1, N DO I = 1, N RINV(I) = RINV(I) + CABS1(A(I, J) * INVHILB(J, K)) END DO END DO RINORM = 0.0 DO I = 1, N SUMRI = 0.0 DO J = 1, N SUMRI = SUMRI \$ + CABS1(INVHILB(I, J) * RINV(J) / INVHILB(I, K)) END DO RINORM = MAX(RINORM, SUMRI) END DO ! invhilb is the inverse *unscaled* Hilbert matrix, so scale its norm ! by 1/A(1,1) to make the scaling match A (the scaled Hilbert matrix) CCOND = CABS1(A(1,1))/RINORM ! Forward error bound tests NWISE_BND = ERRBND_N(K + (BND_I-1)*NRHS) CWISE_BND = ERRBND_C(K + (BND_I-1)*NRHS) NWISE_RCOND = ERRBND_N(K + (COND_I-1)*NRHS) CWISE_RCOND = ERRBND_C(K + (COND_I-1)*NRHS) ! write (*,*) 'nwise : ', n, k, ncond, nwise_rcond, ! \$ condthresh, ncond.ge.condthresh ! write (*,*) 'nwise2: ', k, nwise_bnd, nwise_err, errthresh IF (NCOND .GE. CONDTHRESH) THEN NGUAR = 'YES' IF (NWISE_BND .GT. ERRTHRESH) THEN TSTRAT(1) = 1/(2.0*EPS) ELSE IF (NWISE_BND .NE. 0.0) THEN TSTRAT(1) = NWISE_ERR / NWISE_BND ELSE IF (NWISE_ERR .NE. 0.0) THEN TSTRAT(1) = 1/(16.0*EPS) ELSE TSTRAT(1) = 0.0 END IF IF (TSTRAT(1) .GT. 1.0) THEN TSTRAT(1) = 1/(4.0*EPS) END IF END IF ELSE NGUAR = 'NO' IF (NWISE_BND .LT. 1.0) THEN TSTRAT(1) = 1/(8.0*EPS) ELSE TSTRAT(1) = 1.0 END IF END IF ! write (*,*) 'cwise : ', n, k, ccond, cwise_rcond, ! \$ condthresh, ccond.ge.condthresh ! write (*,*) 'cwise2: ', k, cwise_bnd, cwise_err, errthresh IF (CCOND .GE. CONDTHRESH) THEN CGUAR = 'YES' IF (CWISE_BND .GT. ERRTHRESH) THEN TSTRAT(2) = 1/(2.0*EPS) ELSE IF (CWISE_BND .NE. 0.0) THEN TSTRAT(2) = CWISE_ERR / CWISE_BND ELSE IF (CWISE_ERR .NE. 0.0) THEN TSTRAT(2) = 1/(16.0*EPS) ELSE TSTRAT(2) = 0.0 END IF IF (TSTRAT(2) .GT. 1.0) TSTRAT(2) = 1/(4.0*EPS) END IF ELSE CGUAR = 'NO' IF (CWISE_BND .LT. 1.0) THEN TSTRAT(2) = 1/(8.0*EPS) ELSE TSTRAT(2) = 1.0 END IF END IF ! Backwards error test TSTRAT(3) = BERR(K)/EPS ! Condition number tests TSTRAT(4) = RCOND / ORCOND IF (RCOND .GE. CONDTHRESH .AND. TSTRAT(4) .LT. 1.0) \$ TSTRAT(4) = 1.0 / TSTRAT(4) TSTRAT(5) = NCOND / NWISE_RCOND IF (NCOND .GE. CONDTHRESH .AND. TSTRAT(5) .LT. 1.0) \$ TSTRAT(5) = 1.0 / TSTRAT(5) TSTRAT(6) = CCOND / NWISE_RCOND IF (CCOND .GE. CONDTHRESH .AND. TSTRAT(6) .LT. 1.0) \$ TSTRAT(6) = 1.0 / TSTRAT(6) DO I = 1, NTESTS IF (TSTRAT(I) .GT. THRESH) THEN IF (.NOT.PRINTED_GUIDE) THEN WRITE(*,*) WRITE( *, 9996) 1 WRITE( *, 9995) 2 WRITE( *, 9994) 3 WRITE( *, 9993) 4 WRITE( *, 9992) 5 WRITE( *, 9991) 6 WRITE( *, 9990) 7 WRITE( *, 9989) 8 WRITE(*,*) PRINTED_GUIDE = .TRUE. END IF WRITE( *, 9999) C2, N, K, NGUAR, CGUAR, I, TSTRAT(I) NFAIL = NFAIL + 1 END IF END DO END DO c\$\$\$ WRITE(*,*) c\$\$\$ WRITE(*,*) 'Normwise Error Bounds' c\$\$\$ WRITE(*,*) 'Guaranteed error bound: ',ERRBND(NRHS,nwise_i,bnd_i) c\$\$\$ WRITE(*,*) 'Reciprocal condition number: ',ERRBND(NRHS,nwise_i,cond_i) c\$\$\$ WRITE(*,*) 'Raw error estimate: ',ERRBND(NRHS,nwise_i,rawbnd_i) c\$\$\$ WRITE(*,*) c\$\$\$ WRITE(*,*) 'Componentwise Error Bounds' c\$\$\$ WRITE(*,*) 'Guaranteed error bound: ',ERRBND(NRHS,cwise_i,bnd_i) c\$\$\$ WRITE(*,*) 'Reciprocal condition number: ',ERRBND(NRHS,cwise_i,cond_i) c\$\$\$ WRITE(*,*) 'Raw error estimate: ',ERRBND(NRHS,cwise_i,rawbnd_i) c\$\$\$ print *, 'Info: ', info c\$\$\$ WRITE(*,*) * WRITE(*,*) 'TSTRAT: ',TSTRAT END DO WRITE(*,*) IF( NFAIL .GT. 0 ) THEN WRITE(*,9998) C2, NFAIL, NTESTS*N+N_AUX_TESTS ELSE WRITE(*,9997) C2 END IF 9999 FORMAT( ' C', A2, 'SVXX: N =', I2, ', RHS = ', I2, \$ ', NWISE GUAR. = ', A, ', CWISE GUAR. = ', A, \$ ' test(',I1,') =', G12.5 ) 9998 FORMAT( ' C', A2, 'SVXX: ', I6, ' out of ', I6, \$ ' tests failed to pass the threshold' ) 9997 FORMAT( ' C', A2, 'SVXX passed the tests of error bounds' ) * Test ratios. 9996 FORMAT( 3X, I2, ': Normwise guaranteed forward error', / 5X, \$ 'Guaranteed case: if norm ( abs( Xc - Xt )', \$ ' / norm ( Xt ) .LE. ERRBND( *, nwise_i, bnd_i ), then', \$ / 5X, \$ 'ERRBND( *, nwise_i, bnd_i ) .LE. MAX(SQRT(N), 10) * EPS') 9995 FORMAT( 3X, I2, ': Componentwise guaranteed forward error' ) 9994 FORMAT( 3X, I2, ': Backwards error' ) 9993 FORMAT( 3X, I2, ': Reciprocal condition number' ) 9992 FORMAT( 3X, I2, ': Reciprocal normwise condition number' ) 9991 FORMAT( 3X, I2, ': Raw normwise error estimate' ) 9990 FORMAT( 3X, I2, ': Reciprocal componentwise condition number' ) 9989 FORMAT( 3X, I2, ': Raw componentwise error estimate' ) 8000 FORMAT( ' C', A2, 'SVXX: N =', I2, ', INFO = ', I3, \$ ', ORCOND = ', G12.5, ', real RCOND = ', G12.5 ) END