*> \brief \b DTPT05 * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * * Definition: * =========== * * SUBROUTINE DTPT05( UPLO, TRANS, DIAG, N, NRHS, AP, B, LDB, X, LDX, * XACT, LDXACT, FERR, BERR, RESLTS ) * * .. Scalar Arguments .. * CHARACTER DIAG, TRANS, UPLO * INTEGER LDB, LDX, LDXACT, N, NRHS * .. * .. Array Arguments .. * DOUBLE PRECISION AP( * ), B( LDB, * ), BERR( * ), FERR( * ), * $ RESLTS( * ), X( LDX, * ), XACT( LDXACT, * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> DTPT05 tests the error bounds from iterative refinement for the *> computed solution to a system of equations A*X = B, where A is a *> triangular matrix in packed storage format. *> *> 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 matrix A is upper or lower triangular. *> = 'U': Upper triangular *> = 'L': Lower triangular *> \endverbatim *> *> \param[in] TRANS *> \verbatim *> TRANS is CHARACTER*1 *> Specifies the form of the system of equations. *> = 'N': A * X = B (No transpose) *> = 'T': A'* X = B (Transpose) *> = 'C': A'* X = B (Conjugate transpose = Transpose) *> \endverbatim *> *> \param[in] DIAG *> \verbatim *> DIAG is CHARACTER*1 *> Specifies whether or not the matrix A is unit triangular. *> = 'N': Non-unit triangular *> = 'U': Unit 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] AP *> \verbatim *> AP is DOUBLE PRECISION array, dimension (N*(N+1)/2) *> The upper or lower triangular matrix A, packed columnwise in *> a linear array. The j-th column of A is stored in the array *> AP as follows: *> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; *> if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n. *> If DIAG = 'U', the diagonal elements of A are not referenced *> and are assumed to be 1. *> \endverbatim *> *> \param[in] B *> \verbatim *> B is DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 double_lin * * ===================================================================== SUBROUTINE DTPT05( UPLO, TRANS, DIAG, N, NRHS, AP, 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 DIAG, TRANS, UPLO INTEGER LDB, LDX, LDXACT, N, NRHS * .. * .. Array Arguments .. DOUBLE PRECISION AP( * ), B( LDB, * ), BERR( * ), FERR( * ), $ RESLTS( * ), X( LDX, * ), XACT( LDXACT, * ) * .. * * ===================================================================== * * .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) * .. * .. Local Scalars .. LOGICAL NOTRAN, UNIT, UPPER INTEGER I, IFU, IMAX, J, JC, K DOUBLE PRECISION AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM * .. * .. External Functions .. LOGICAL LSAME INTEGER IDAMAX DOUBLE PRECISION DLAMCH EXTERNAL LSAME, IDAMAX, DLAMCH * .. * .. Intrinsic Functions .. INTRINSIC ABS, MAX, MIN * .. * .. 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' ) NOTRAN = LSAME( TRANS, 'N' ) UNIT = LSAME( DIAG, '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 = IDAMAX( N, X( 1, J ), 1 ) XNORM = MAX( ABS( X( IMAX, J ) ), UNFL ) DIFF = ZERO DO 10 I = 1, N DIFF = MAX( DIFF, ABS( 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 ) * IFU = 0 IF( UNIT ) $ IFU = 1 DO 90 K = 1, NRHS DO 80 I = 1, N TMP = ABS( B( I, K ) ) IF( UPPER ) THEN JC = ( ( I-1 )*I ) / 2 IF( .NOT.NOTRAN ) THEN DO 40 J = 1, I - IFU TMP = TMP + ABS( AP( JC+J ) )*ABS( X( J, K ) ) 40 CONTINUE IF( UNIT ) $ TMP = TMP + ABS( X( I, K ) ) ELSE JC = JC + I IF( UNIT ) THEN TMP = TMP + ABS( X( I, K ) ) JC = JC + I END IF DO 50 J = I + IFU, N TMP = TMP + ABS( AP( JC ) )*ABS( X( J, K ) ) JC = JC + J 50 CONTINUE END IF ELSE IF( NOTRAN ) THEN JC = I DO 60 J = 1, I - IFU TMP = TMP + ABS( AP( JC ) )*ABS( X( J, K ) ) JC = JC + N - J 60 CONTINUE IF( UNIT ) $ TMP = TMP + ABS( X( I, K ) ) ELSE JC = ( I-1 )*( N-I ) + ( I*( I+1 ) ) / 2 IF( UNIT ) $ TMP = TMP + ABS( X( I, K ) ) DO 70 J = I + IFU, N TMP = TMP + ABS( AP( JC+J-I ) )*ABS( X( J, K ) ) 70 CONTINUE END IF 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 DTPT05 * END