*> \brief \b ZPOT05 * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * * Definition: * =========== * * SUBROUTINE ZPOT05( UPLO, N, NRHS, A, LDA, B, LDB, X, LDX, XACT, * LDXACT, FERR, BERR, RESLTS ) * * .. Scalar Arguments .. * CHARACTER UPLO * INTEGER LDA, LDB, LDX, LDXACT, N, NRHS * .. * .. Array Arguments .. * DOUBLE PRECISION BERR( * ), FERR( * ), RESLTS( * ) * COMPLEX*16 A( LDA, * ), B( LDB, * ), X( LDX, * ), * \$ XACT( LDXACT, * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> ZPOT05 tests the error bounds from iterative refinement for the *> computed solution to a system of equations A*X = B, where A is a *> Hermitian n by n matrix. *> *> RESLTS(1) = test of the error bound *> = norm(X - XACT) / ( norm(X) * FERR ) *> *> A large value is returned if this ratio is not less than one. *> *> RESLTS(2) = residual from the iterative refinement routine *> = the maximum of BERR / ( (n+1)*EPS + (*) ), where *> (*) = (n+1)*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i ) *> \endverbatim * * Arguments: * ========== * *> \param[in] UPLO *> \verbatim *> UPLO is CHARACTER*1 *> Specifies whether the upper or lower triangular part of the *> Hermitian matrix A is stored. *> = 'U': Upper triangular *> = 'L': Lower triangular *> \endverbatim *> *> \param[in] N *> \verbatim *> N is INTEGER *> The number of rows of the matrices X, B, and XACT, and the *> order of the matrix A. N >= 0. *> \endverbatim *> *> \param[in] NRHS *> \verbatim *> NRHS is INTEGER *> The number of columns of the matrices X, B, and XACT. *> NRHS >= 0. *> \endverbatim *> *> \param[in] A *> \verbatim *> A is COMPLEX*16 array, dimension (LDA,N) *> The Hermitian matrix A. If UPLO = 'U', the leading n by n *> upper triangular part of A contains the upper triangular part *> of the matrix A, and the strictly lower triangular part of A *> is not referenced. If UPLO = 'L', the leading n by n lower *> triangular part of A contains the lower triangular part of *> the matrix A, and the strictly upper triangular part of A is *> not referenced. *> \endverbatim *> *> \param[in] LDA *> \verbatim *> LDA is INTEGER *> The leading dimension of the array A. LDA >= max(1,N). *> \endverbatim *> *> \param[in] B *> \verbatim *> B is COMPLEX*16 array, dimension (LDB,NRHS) *> The right hand side vectors for the system of linear *> equations. *> \endverbatim *> *> \param[in] LDB *> \verbatim *> LDB is INTEGER *> The leading dimension of the array B. LDB >= max(1,N). *> \endverbatim *> *> \param[in] X *> \verbatim *> X is COMPLEX*16 array, dimension (LDX,NRHS) *> The computed solution vectors. Each vector is stored as a *> column of the matrix X. *> \endverbatim *> *> \param[in] LDX *> \verbatim *> LDX is INTEGER *> The leading dimension of the array X. LDX >= max(1,N). *> \endverbatim *> *> \param[in] XACT *> \verbatim *> XACT is COMPLEX*16 array, dimension (LDX,NRHS) *> The exact solution vectors. Each vector is stored as a *> column of the matrix XACT. *> \endverbatim *> *> \param[in] LDXACT *> \verbatim *> LDXACT is INTEGER *> The leading dimension of the array XACT. LDXACT >= max(1,N). *> \endverbatim *> *> \param[in] FERR *> \verbatim *> FERR is DOUBLE PRECISION array, dimension (NRHS) *> The estimated forward error bounds for each solution vector *> X. If XTRUE is the true solution, FERR bounds the magnitude *> of the largest entry in (X - XTRUE) divided by the magnitude *> of the largest entry in X. *> \endverbatim *> *> \param[in] BERR *> \verbatim *> BERR is DOUBLE PRECISION array, dimension (NRHS) *> The componentwise relative backward error of each solution *> vector (i.e., the smallest relative change in any entry of A *> or B that makes X an exact solution). *> \endverbatim *> *> \param[out] RESLTS *> \verbatim *> RESLTS is DOUBLE PRECISION array, dimension (2) *> The maximum over the NRHS solution vectors of the ratios: *> RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR ) *> RESLTS(2) = BERR / ( (n+1)*EPS + (*) ) *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \ingroup complex16_lin * * ===================================================================== SUBROUTINE ZPOT05( UPLO, N, NRHS, A, LDA, B, LDB, X, LDX, XACT, \$ LDXACT, FERR, BERR, RESLTS ) * * -- LAPACK test routine -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * * .. Scalar Arguments .. CHARACTER UPLO INTEGER LDA, LDB, LDX, LDXACT, N, NRHS * .. * .. Array Arguments .. DOUBLE PRECISION BERR( * ), FERR( * ), RESLTS( * ) COMPLEX*16 A( LDA, * ), B( LDB, * ), X( LDX, * ), \$ XACT( LDXACT, * ) * .. * * ===================================================================== * * .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) * .. * .. Local Scalars .. LOGICAL UPPER INTEGER I, IMAX, J, K DOUBLE PRECISION AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM COMPLEX*16 ZDUM * .. * .. External Functions .. LOGICAL LSAME INTEGER IZAMAX DOUBLE PRECISION DLAMCH EXTERNAL LSAME, IZAMAX, DLAMCH * .. * .. Intrinsic Functions .. INTRINSIC ABS, DBLE, DIMAG, MAX, MIN * .. * .. Statement Functions .. DOUBLE PRECISION CABS1 * .. * .. Statement Function definitions .. CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) ) * .. * .. Executable Statements .. * * Quick exit if N = 0 or NRHS = 0. * IF( N.LE.0 .OR. NRHS.LE.0 ) THEN RESLTS( 1 ) = ZERO RESLTS( 2 ) = ZERO RETURN END IF * EPS = DLAMCH( 'Epsilon' ) UNFL = DLAMCH( 'Safe minimum' ) OVFL = ONE / UNFL UPPER = LSAME( UPLO, 'U' ) * * Test 1: Compute the maximum of * norm(X - XACT) / ( norm(X) * FERR ) * over all the vectors X and XACT using the infinity-norm. * ERRBND = ZERO DO 30 J = 1, NRHS IMAX = IZAMAX( N, X( 1, J ), 1 ) XNORM = MAX( CABS1( X( IMAX, J ) ), UNFL ) DIFF = ZERO DO 10 I = 1, N DIFF = MAX( DIFF, CABS1( X( I, J )-XACT( I, J ) ) ) 10 CONTINUE * IF( XNORM.GT.ONE ) THEN GO TO 20 ELSE IF( DIFF.LE.OVFL*XNORM ) THEN GO TO 20 ELSE ERRBND = ONE / EPS GO TO 30 END IF * 20 CONTINUE IF( DIFF / XNORM.LE.FERR( J ) ) THEN ERRBND = MAX( ERRBND, ( DIFF / XNORM ) / FERR( J ) ) ELSE ERRBND = ONE / EPS END IF 30 CONTINUE RESLTS( 1 ) = ERRBND * * Test 2: Compute the maximum of BERR / ( (n+1)*EPS + (*) ), where * (*) = (n+1)*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i ) * DO 90 K = 1, NRHS DO 80 I = 1, N TMP = CABS1( B( I, K ) ) IF( UPPER ) THEN DO 40 J = 1, I - 1 TMP = TMP + CABS1( A( J, I ) )*CABS1( X( J, K ) ) 40 CONTINUE TMP = TMP + ABS( DBLE( A( I, I ) ) )*CABS1( X( I, K ) ) DO 50 J = I + 1, N TMP = TMP + CABS1( A( I, J ) )*CABS1( X( J, K ) ) 50 CONTINUE ELSE DO 60 J = 1, I - 1 TMP = TMP + CABS1( A( I, J ) )*CABS1( X( J, K ) ) 60 CONTINUE TMP = TMP + ABS( DBLE( A( I, I ) ) )*CABS1( X( I, K ) ) DO 70 J = I + 1, N TMP = TMP + CABS1( A( J, I ) )*CABS1( X( J, K ) ) 70 CONTINUE END IF IF( I.EQ.1 ) THEN AXBI = TMP ELSE AXBI = MIN( AXBI, TMP ) END IF 80 CONTINUE TMP = BERR( K ) / ( ( N+1 )*EPS+( N+1 )*UNFL / \$ MAX( AXBI, ( N+1 )*UNFL ) ) IF( K.EQ.1 ) THEN RESLTS( 2 ) = TMP ELSE RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP ) END IF 90 CONTINUE * RETURN * * End of ZPOT05 * END