509 lines
18 KiB
Fortran
509 lines
18 KiB
Fortran
*> \brief \b ZEBCHVXX
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*
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* =========== DOCUMENTATION ===========
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*
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* Online html documentation available at
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* http://www.netlib.org/lapack/explore-html/
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*
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* Definition:
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* ===========
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*
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* SUBROUTINE ZEBCHVXX( THRESH, PATH )
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*
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* .. Scalar Arguments ..
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* DOUBLE PRECISION THRESH
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* CHARACTER*3 PATH
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* ..
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*
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* Purpose
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* ======
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*
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*> \details \b Purpose:
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*> \verbatim
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*>
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*> ZEBCHVXX will run Z**SVXX on a series of Hilbert matrices and then
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*> compare the error bounds returned by Z**SVXX to see if the returned
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*> answer indeed falls within those bounds.
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*>
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*> Eight test ratios will be computed. The tests will pass if they are .LT.
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*> THRESH. There are two cases that are determined by 1 / (SQRT( N ) * EPS).
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*> If that value is .LE. to the component wise reciprocal condition number,
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*> it uses the guaranteed case, other wise it uses the unguaranteed case.
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*>
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*> Test ratios:
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*> Let Xc be X_computed and Xt be X_truth.
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*> The norm used is the infinity norm.
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*>
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*> Let A be the guaranteed case and B be the unguaranteed case.
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*>
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*> 1. Normwise guaranteed forward error bound.
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*> A: norm ( abs( Xc - Xt ) / norm ( Xt ) .LE. ERRBND( *, nwise_i, bnd_i ) and
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*> ERRBND( *, nwise_i, bnd_i ) .LE. MAX(SQRT(N),10) * EPS.
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*> If these conditions are met, the test ratio is set to be
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*> ERRBND( *, nwise_i, bnd_i ) / MAX(SQRT(N), 10). Otherwise it is 1/EPS.
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*> B: For this case, CGESVXX should just return 1. If it is less than
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*> one, treat it the same as in 1A. Otherwise it fails. (Set test
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*> ratio to ERRBND( *, nwise_i, bnd_i ) * THRESH?)
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*>
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*> 2. Componentwise guaranteed forward error bound.
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*> A: norm ( abs( Xc(j) - Xt(j) ) ) / norm (Xt(j)) .LE. ERRBND( *, cwise_i, bnd_i )
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*> for all j .AND. ERRBND( *, cwise_i, bnd_i ) .LE. MAX(SQRT(N), 10) * EPS.
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*> If these conditions are met, the test ratio is set to be
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*> ERRBND( *, cwise_i, bnd_i ) / MAX(SQRT(N), 10). Otherwise it is 1/EPS.
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*> B: Same as normwise test ratio.
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*>
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*> 3. Backwards error.
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*> A: The test ratio is set to BERR/EPS.
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*> B: Same test ratio.
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*>
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*> 4. Reciprocal condition number.
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*> A: A condition number is computed with Xt and compared with the one
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*> returned from CGESVXX. Let RCONDc be the RCOND returned by CGESVXX
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*> and RCONDt be the RCOND from the truth value. Test ratio is set to
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*> MAX(RCONDc/RCONDt, RCONDt/RCONDc).
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*> B: Test ratio is set to 1 / (EPS * RCONDc).
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*>
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*> 5. Reciprocal normwise condition number.
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*> A: The test ratio is set to
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*> MAX(ERRBND( *, nwise_i, cond_i ) / NCOND, NCOND / ERRBND( *, nwise_i, cond_i )).
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*> B: Test ratio is set to 1 / (EPS * ERRBND( *, nwise_i, cond_i )).
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*>
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*> 6. Reciprocal componentwise condition number.
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*> A: Test ratio is set to
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*> MAX(ERRBND( *, cwise_i, cond_i ) / CCOND, CCOND / ERRBND( *, cwise_i, cond_i )).
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*> B: Test ratio is set to 1 / (EPS * ERRBND( *, cwise_i, cond_i )).
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*>
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*> .. Parameters ..
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*> NMAX is determined by the largest number in the inverse of the hilbert
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*> matrix. Precision is exhausted when the largest entry in it is greater
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*> than 2 to the power of the number of bits in the fraction of the data
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*> type used plus one, which is 24 for single precision.
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*> NMAX should be 6 for single and 11 for double.
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*> \endverbatim
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*
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* Authors:
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* ========
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*
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*> \author Univ. of Tennessee
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*> \author Univ. of California Berkeley
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*> \author Univ. of Colorado Denver
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*> \author NAG Ltd.
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*
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*> \ingroup complex16_lin
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*
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* =====================================================================
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SUBROUTINE ZEBCHVXX( THRESH, PATH )
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IMPLICIT NONE
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* .. Scalar Arguments ..
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DOUBLE PRECISION THRESH
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CHARACTER*3 PATH
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INTEGER NMAX, NPARAMS, NERRBND, NTESTS, KL, KU
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PARAMETER (NMAX = 10, NPARAMS = 2, NERRBND = 3,
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$ NTESTS = 6)
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* .. Local Scalars ..
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INTEGER N, NRHS, INFO, I ,J, k, NFAIL, LDA,
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$ N_AUX_TESTS, LDAB, LDAFB
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CHARACTER FACT, TRANS, UPLO, EQUED
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CHARACTER*2 C2
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CHARACTER(3) NGUAR, CGUAR
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LOGICAL printed_guide
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DOUBLE PRECISION NCOND, CCOND, M, NORMDIF, NORMT, RCOND,
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$ RNORM, RINORM, SUMR, SUMRI, EPS,
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$ BERR(NMAX), RPVGRW, ORCOND,
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$ CWISE_ERR, NWISE_ERR, CWISE_BND, NWISE_BND,
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$ CWISE_RCOND, NWISE_RCOND,
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$ CONDTHRESH, ERRTHRESH
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COMPLEX*16 ZDUM
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* .. Local Arrays ..
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DOUBLE PRECISION TSTRAT(NTESTS), RINV(NMAX), PARAMS(NPARAMS),
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$ S(NMAX),R(NMAX),C(NMAX),RWORK(3*NMAX),
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$ DIFF(NMAX, NMAX),
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$ ERRBND_N(NMAX*3), ERRBND_C(NMAX*3)
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INTEGER IPIV(NMAX)
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COMPLEX*16 A(NMAX,NMAX),INVHILB(NMAX,NMAX),X(NMAX,NMAX),
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$ WORK(NMAX*3*5), AF(NMAX, NMAX),B(NMAX, NMAX),
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$ ACOPY(NMAX, NMAX),
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$ AB( (NMAX-1)+(NMAX-1)+1, NMAX ),
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$ ABCOPY( (NMAX-1)+(NMAX-1)+1, NMAX ),
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$ AFB( 2*(NMAX-1)+(NMAX-1)+1, NMAX )
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* .. External Functions ..
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DOUBLE PRECISION DLAMCH
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* .. External Subroutines ..
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EXTERNAL ZLAHILB, ZGESVXX, ZPOSVXX, ZSYSVXX,
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$ ZGBSVXX, ZLACPY, LSAMEN
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LOGICAL LSAMEN
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* .. Intrinsic Functions ..
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INTRINSIC SQRT, MAX, ABS, DBLE, DIMAG
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* .. Statement Functions ..
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DOUBLE PRECISION CABS1
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* .. Statement Function Definitions ..
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CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
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* .. Parameters ..
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INTEGER NWISE_I, CWISE_I
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PARAMETER (NWISE_I = 1, CWISE_I = 1)
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INTEGER BND_I, COND_I
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PARAMETER (BND_I = 2, COND_I = 3)
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* Create the loop to test out the Hilbert matrices
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FACT = 'E'
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UPLO = 'U'
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TRANS = 'N'
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EQUED = 'N'
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EPS = DLAMCH('Epsilon')
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NFAIL = 0
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N_AUX_TESTS = 0
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LDA = NMAX
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LDAB = (NMAX-1)+(NMAX-1)+1
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LDAFB = 2*(NMAX-1)+(NMAX-1)+1
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C2 = PATH( 2: 3 )
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* Main loop to test the different Hilbert Matrices.
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printed_guide = .false.
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DO N = 1 , NMAX
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PARAMS(1) = -1
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PARAMS(2) = -1
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KL = N-1
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KU = N-1
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NRHS = n
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M = MAX(SQRT(DBLE(N)), 10.0D+0)
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* Generate the Hilbert matrix, its inverse, and the
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* right hand side, all scaled by the LCM(1,..,2N-1).
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CALL ZLAHILB(N, N, A, LDA, INVHILB, LDA, B,
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$ LDA, WORK, INFO, PATH)
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* Copy A into ACOPY.
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CALL ZLACPY('ALL', N, N, A, NMAX, ACOPY, NMAX)
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* Store A in band format for GB tests
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DO J = 1, N
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DO I = 1, KL+KU+1
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AB( I, J ) = (0.0D+0,0.0D+0)
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END DO
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END DO
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DO J = 1, N
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DO I = MAX( 1, J-KU ), MIN( N, J+KL )
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AB( KU+1+I-J, J ) = A( I, J )
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END DO
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END DO
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* Copy AB into ABCOPY.
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DO J = 1, N
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DO I = 1, KL+KU+1
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ABCOPY( I, J ) = (0.0D+0,0.0D+0)
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END DO
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END DO
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CALL ZLACPY('ALL', KL+KU+1, N, AB, LDAB, ABCOPY, LDAB)
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* Call Z**SVXX with default PARAMS and N_ERR_BND = 3.
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IF ( LSAMEN( 2, C2, 'SY' ) ) THEN
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CALL ZSYSVXX(FACT, UPLO, N, NRHS, ACOPY, LDA, AF, LDA,
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$ IPIV, EQUED, S, B, LDA, X, LDA, ORCOND,
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$ RPVGRW, BERR, NERRBND, ERRBND_N, ERRBND_C, NPARAMS,
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$ PARAMS, WORK, RWORK, INFO)
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ELSE IF ( LSAMEN( 2, C2, 'PO' ) ) THEN
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CALL ZPOSVXX(FACT, UPLO, N, NRHS, ACOPY, LDA, AF, LDA,
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$ EQUED, S, B, LDA, X, LDA, ORCOND,
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$ RPVGRW, BERR, NERRBND, ERRBND_N, ERRBND_C, NPARAMS,
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$ PARAMS, WORK, RWORK, INFO)
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ELSE IF ( LSAMEN( 2, C2, 'HE' ) ) THEN
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CALL ZHESVXX(FACT, UPLO, N, NRHS, ACOPY, LDA, AF, LDA,
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$ IPIV, EQUED, S, B, LDA, X, LDA, ORCOND,
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$ RPVGRW, BERR, NERRBND, ERRBND_N, ERRBND_C, NPARAMS,
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$ PARAMS, WORK, RWORK, INFO)
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ELSE IF ( LSAMEN( 2, C2, 'GB' ) ) THEN
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CALL ZGBSVXX(FACT, TRANS, N, KL, KU, NRHS, ABCOPY,
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$ LDAB, AFB, LDAFB, IPIV, EQUED, R, C, B,
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$ LDA, X, LDA, ORCOND, RPVGRW, BERR, NERRBND,
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$ ERRBND_N, ERRBND_C, NPARAMS, PARAMS, WORK, RWORK,
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$ INFO)
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ELSE
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CALL ZGESVXX(FACT, TRANS, N, NRHS, ACOPY, LDA, AF, LDA,
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$ IPIV, EQUED, R, C, B, LDA, X, LDA, ORCOND,
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$ RPVGRW, BERR, NERRBND, ERRBND_N, ERRBND_C, NPARAMS,
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$ PARAMS, WORK, RWORK, INFO)
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END IF
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N_AUX_TESTS = N_AUX_TESTS + 1
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IF (ORCOND .LT. EPS) THEN
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! Either factorization failed or the matrix is flagged, and 1 <=
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! INFO <= N+1. We don't decide based on rcond anymore.
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! IF (INFO .EQ. 0 .OR. INFO .GT. N+1) THEN
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! NFAIL = NFAIL + 1
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! WRITE (*, FMT=8000) N, INFO, ORCOND, RCOND
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! END IF
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ELSE
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! Either everything succeeded (INFO == 0) or some solution failed
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! to converge (INFO > N+1).
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IF (INFO .GT. 0 .AND. INFO .LE. N+1) THEN
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NFAIL = NFAIL + 1
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WRITE (*, FMT=8000) C2, N, INFO, ORCOND, RCOND
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END IF
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END IF
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* Calculating the difference between Z**SVXX's X and the true X.
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DO I = 1,N
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DO J =1,NRHS
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DIFF(I,J) = X(I,J) - INVHILB(I,J)
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END DO
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END DO
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* Calculating the RCOND
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RNORM = 0
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RINORM = 0
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IF ( LSAMEN( 2, C2, 'PO' ) .OR. LSAMEN( 2, C2, 'SY' ) .OR.
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$ LSAMEN( 2, C2, 'HE' ) ) THEN
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DO I = 1, N
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SUMR = 0
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SUMRI = 0
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DO J = 1, N
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SUMR = SUMR + S(I) * CABS1(A(I,J)) * S(J)
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SUMRI = SUMRI + CABS1(INVHILB(I, J)) / (S(J) * S(I))
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END DO
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RNORM = MAX(RNORM,SUMR)
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RINORM = MAX(RINORM,SUMRI)
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END DO
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ELSE IF ( LSAMEN( 2, C2, 'GE' ) .OR. LSAMEN( 2, C2, 'GB' ) )
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$ THEN
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DO I = 1, N
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SUMR = 0
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SUMRI = 0
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DO J = 1, N
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SUMR = SUMR + R(I) * CABS1(A(I,J)) * C(J)
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SUMRI = SUMRI + CABS1(INVHILB(I, J)) / (R(J) * C(I))
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END DO
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RNORM = MAX(RNORM,SUMR)
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RINORM = MAX(RINORM,SUMRI)
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END DO
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END IF
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RNORM = RNORM / CABS1(A(1, 1))
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RCOND = 1.0D+0/(RNORM * RINORM)
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* Calculating the R for normwise rcond.
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DO I = 1, N
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RINV(I) = 0.0D+0
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END DO
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DO J = 1, N
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DO I = 1, N
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RINV(I) = RINV(I) + CABS1(A(I,J))
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END DO
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END DO
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* Calculating the Normwise rcond.
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RINORM = 0.0D+0
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DO I = 1, N
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SUMRI = 0.0D+0
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DO J = 1, N
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SUMRI = SUMRI + CABS1(INVHILB(I,J) * RINV(J))
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END DO
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RINORM = MAX(RINORM, SUMRI)
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END DO
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! invhilb is the inverse *unscaled* Hilbert matrix, so scale its norm
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! by 1/A(1,1) to make the scaling match A (the scaled Hilbert matrix)
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NCOND = CABS1(A(1,1)) / RINORM
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CONDTHRESH = M * EPS
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ERRTHRESH = M * EPS
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DO K = 1, NRHS
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NORMT = 0.0D+0
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NORMDIF = 0.0D+0
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CWISE_ERR = 0.0D+0
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DO I = 1, N
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NORMT = MAX(CABS1(INVHILB(I, K)), NORMT)
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NORMDIF = MAX(CABS1(X(I,K) - INVHILB(I,K)), NORMDIF)
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IF (INVHILB(I,K) .NE. 0.0D+0) THEN
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CWISE_ERR = MAX(CABS1(X(I,K) - INVHILB(I,K))
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$ /CABS1(INVHILB(I,K)), CWISE_ERR)
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ELSE IF (X(I, K) .NE. 0.0D+0) THEN
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CWISE_ERR = DLAMCH('OVERFLOW')
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END IF
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END DO
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IF (NORMT .NE. 0.0D+0) THEN
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NWISE_ERR = NORMDIF / NORMT
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ELSE IF (NORMDIF .NE. 0.0D+0) THEN
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NWISE_ERR = DLAMCH('OVERFLOW')
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ELSE
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NWISE_ERR = 0.0D+0
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ENDIF
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DO I = 1, N
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RINV(I) = 0.0D+0
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END DO
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DO J = 1, N
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DO I = 1, N
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RINV(I) = RINV(I) + CABS1(A(I, J) * INVHILB(J, K))
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END DO
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END DO
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RINORM = 0.0D+0
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DO I = 1, N
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SUMRI = 0.0D+0
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DO J = 1, N
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SUMRI = SUMRI
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$ + CABS1(INVHILB(I, J) * RINV(J) / INVHILB(I, K))
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END DO
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RINORM = MAX(RINORM, SUMRI)
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END DO
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! invhilb is the inverse *unscaled* Hilbert matrix, so scale its norm
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! by 1/A(1,1) to make the scaling match A (the scaled Hilbert matrix)
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CCOND = CABS1(A(1,1))/RINORM
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! Forward error bound tests
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NWISE_BND = ERRBND_N(K + (BND_I-1)*NRHS)
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CWISE_BND = ERRBND_C(K + (BND_I-1)*NRHS)
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NWISE_RCOND = ERRBND_N(K + (COND_I-1)*NRHS)
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CWISE_RCOND = ERRBND_C(K + (COND_I-1)*NRHS)
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! write (*,*) 'nwise : ', n, k, ncond, nwise_rcond,
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! $ condthresh, ncond.ge.condthresh
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! write (*,*) 'nwise2: ', k, nwise_bnd, nwise_err, errthresh
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IF (NCOND .GE. CONDTHRESH) THEN
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NGUAR = 'YES'
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IF (NWISE_BND .GT. ERRTHRESH) THEN
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TSTRAT(1) = 1/(2.0D+0*EPS)
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ELSE
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IF (NWISE_BND .NE. 0.0D+0) THEN
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TSTRAT(1) = NWISE_ERR / NWISE_BND
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ELSE IF (NWISE_ERR .NE. 0.0D+0) THEN
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TSTRAT(1) = 1/(16.0*EPS)
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ELSE
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TSTRAT(1) = 0.0D+0
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END IF
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IF (TSTRAT(1) .GT. 1.0D+0) THEN
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TSTRAT(1) = 1/(4.0D+0*EPS)
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END IF
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END IF
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ELSE
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NGUAR = 'NO'
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IF (NWISE_BND .LT. 1.0D+0) THEN
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TSTRAT(1) = 1/(8.0D+0*EPS)
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ELSE
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TSTRAT(1) = 1.0D+0
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END IF
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END IF
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! write (*,*) 'cwise : ', n, k, ccond, cwise_rcond,
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! $ condthresh, ccond.ge.condthresh
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! write (*,*) 'cwise2: ', k, cwise_bnd, cwise_err, errthresh
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IF (CCOND .GE. CONDTHRESH) THEN
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CGUAR = 'YES'
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IF (CWISE_BND .GT. ERRTHRESH) THEN
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TSTRAT(2) = 1/(2.0D+0*EPS)
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ELSE
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IF (CWISE_BND .NE. 0.0D+0) THEN
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TSTRAT(2) = CWISE_ERR / CWISE_BND
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ELSE IF (CWISE_ERR .NE. 0.0D+0) THEN
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TSTRAT(2) = 1/(16.0D+0*EPS)
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ELSE
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TSTRAT(2) = 0.0D+0
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END IF
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IF (TSTRAT(2) .GT. 1.0D+0) TSTRAT(2) = 1/(4.0D+0*EPS)
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END IF
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ELSE
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CGUAR = 'NO'
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IF (CWISE_BND .LT. 1.0D+0) THEN
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TSTRAT(2) = 1/(8.0D+0*EPS)
|
|
ELSE
|
|
TSTRAT(2) = 1.0D+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. 1.0D+0)
|
|
$ TSTRAT(4) = 1.0D+0 / TSTRAT(4)
|
|
|
|
TSTRAT(5) = NCOND / NWISE_RCOND
|
|
IF (NCOND .GE. CONDTHRESH .AND. TSTRAT(5) .LT. 1.0D+0)
|
|
$ TSTRAT(5) = 1.0D+0 / TSTRAT(5)
|
|
|
|
TSTRAT(6) = CCOND / NWISE_RCOND
|
|
IF (CCOND .GE. CONDTHRESH .AND. TSTRAT(6) .LT. 1.0D+0)
|
|
$ TSTRAT(6) = 1.0D+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( ' Z', A2, 'SVXX: N =', I2, ', RHS = ', I2,
|
|
$ ', NWISE GUAR. = ', A, ', CWISE GUAR. = ', A,
|
|
$ ' test(',I1,') =', G12.5 )
|
|
9998 FORMAT( ' Z', A2, 'SVXX: ', I6, ' out of ', I6,
|
|
$ ' tests failed to pass the threshold' )
|
|
9997 FORMAT( ' Z', 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( ' Z', A2, 'SVXX: N =', I2, ', INFO = ', I3,
|
|
$ ', ORCOND = ', G12.5, ', real RCOND = ', G12.5 )
|
|
*
|
|
* End of ZEBCHVXX
|
|
*
|
|
END
|