*> \brief \b DTRT03 * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * * Definition: * =========== * * SUBROUTINE DTRT03( UPLO, TRANS, DIAG, N, NRHS, A, LDA, SCALE, * CNORM, TSCAL, X, LDX, B, LDB, WORK, RESID ) * * .. Scalar Arguments .. * CHARACTER DIAG, TRANS, UPLO * INTEGER LDA, LDB, LDX, N, NRHS * DOUBLE PRECISION RESID, SCALE, TSCAL * .. * .. Array Arguments .. * DOUBLE PRECISION A( LDA, * ), B( LDB, * ), CNORM( * ), * $ WORK( * ), X( LDX, * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> DTRT03 computes the residual for the solution to a scaled triangular *> system of equations A*x = s*b or A'*x = s*b. *> Here A is a triangular matrix, A' is the transpose of A, s is a *> scalar, and x and b are N by NRHS matrices. The test ratio is the *> maximum over the number of right hand sides of *> norm(s*b - op(A)*x) / ( norm(op(A)) * norm(x) * EPS ), *> where op(A) denotes A or A' and EPS is the machine epsilon. *> \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 operation applied to A. *> = 'N': A *x = s*b (No transpose) *> = 'T': A'*x = s*b (Transpose) *> = 'C': A'*x = s*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 order of the matrix A. N >= 0. *> \endverbatim *> *> \param[in] NRHS *> \verbatim *> NRHS is INTEGER *> The number of right hand sides, i.e., the number of columns *> of the matrices X and B. NRHS >= 0. *> \endverbatim *> *> \param[in] A *> \verbatim *> A is DOUBLE PRECISION array, dimension (LDA,N) *> The triangular matrix A. If UPLO = 'U', the leading n by n *> upper triangular part of the array A contains the upper *> triangular matrix, and the strictly lower triangular part of *> A is not referenced. If UPLO = 'L', the leading n by n lower *> triangular part of the array A contains the lower triangular *> matrix, and the strictly upper triangular part of A is not *> referenced. If DIAG = 'U', the diagonal elements of A are *> also not referenced and are assumed to be 1. *> \endverbatim *> *> \param[in] LDA *> \verbatim *> LDA is INTEGER *> The leading dimension of the array A. LDA >= max(1,N). *> \endverbatim *> *> \param[in] SCALE *> \verbatim *> SCALE is DOUBLE PRECISION *> The scaling factor s used in solving the triangular system. *> \endverbatim *> *> \param[in] CNORM *> \verbatim *> CNORM is DOUBLE PRECISION array, dimension (N) *> The 1-norms of the columns of A, not counting the diagonal. *> \endverbatim *> *> \param[in] TSCAL *> \verbatim *> TSCAL is DOUBLE PRECISION *> The scaling factor used in computing the 1-norms in CNORM. *> CNORM actually contains the column norms of TSCAL*A. *> \endverbatim *> *> \param[in] X *> \verbatim *> X is DOUBLE PRECISION array, dimension (LDX,NRHS) *> The computed solution vectors for the system of linear *> equations. *> \endverbatim *> *> \param[in] LDX *> \verbatim *> LDX is INTEGER *> The leading dimension of the array X. LDX >= max(1,N). *> \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[out] WORK *> \verbatim *> WORK is DOUBLE PRECISION array, dimension (N) *> \endverbatim *> *> \param[out] RESID *> \verbatim *> RESID is DOUBLE PRECISION *> The maximum over the number of right hand sides of *> norm(op(A)*x - s*b) / ( norm(op(A)) * norm(x) * EPS ). *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \ingroup double_lin * * ===================================================================== SUBROUTINE DTRT03( UPLO, TRANS, DIAG, N, NRHS, A, LDA, SCALE, $ CNORM, TSCAL, X, LDX, B, LDB, WORK, RESID ) * * -- 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 LDA, LDB, LDX, N, NRHS DOUBLE PRECISION RESID, SCALE, TSCAL * .. * .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), B( LDB, * ), CNORM( * ), $ WORK( * ), X( LDX, * ) * .. * * ===================================================================== * * .. Parameters .. DOUBLE PRECISION ONE, ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) * .. * .. Local Scalars .. INTEGER IX, J DOUBLE PRECISION BIGNUM, EPS, ERR, SMLNUM, TNORM, XNORM, XSCAL * .. * .. External Functions .. LOGICAL LSAME INTEGER IDAMAX DOUBLE PRECISION DLAMCH EXTERNAL LSAME, IDAMAX, DLAMCH * .. * .. External Subroutines .. EXTERNAL DAXPY, DCOPY, DLABAD, DSCAL, DTRMV * .. * .. Intrinsic Functions .. INTRINSIC ABS, DBLE, MAX * .. * .. Executable Statements .. * * Quick exit if N = 0 * IF( N.LE.0 .OR. NRHS.LE.0 ) THEN RESID = ZERO RETURN END IF EPS = DLAMCH( 'Epsilon' ) SMLNUM = DLAMCH( 'Safe minimum' ) BIGNUM = ONE / SMLNUM CALL DLABAD( SMLNUM, BIGNUM ) * * Compute the norm of the triangular matrix A using the column * norms already computed by DLATRS. * TNORM = ZERO IF( LSAME( DIAG, 'N' ) ) THEN DO 10 J = 1, N TNORM = MAX( TNORM, TSCAL*ABS( A( J, J ) )+CNORM( J ) ) 10 CONTINUE ELSE DO 20 J = 1, N TNORM = MAX( TNORM, TSCAL+CNORM( J ) ) 20 CONTINUE END IF * * Compute the maximum over the number of right hand sides of * norm(op(A)*x - s*b) / ( norm(op(A)) * norm(x) * EPS ). * RESID = ZERO DO 30 J = 1, NRHS CALL DCOPY( N, X( 1, J ), 1, WORK, 1 ) IX = IDAMAX( N, WORK, 1 ) XNORM = MAX( ONE, ABS( X( IX, J ) ) ) XSCAL = ( ONE / XNORM ) / DBLE( N ) CALL DSCAL( N, XSCAL, WORK, 1 ) CALL DTRMV( UPLO, TRANS, DIAG, N, A, LDA, WORK, 1 ) CALL DAXPY( N, -SCALE*XSCAL, B( 1, J ), 1, WORK, 1 ) IX = IDAMAX( N, WORK, 1 ) ERR = TSCAL*ABS( WORK( IX ) ) IX = IDAMAX( N, X( 1, J ), 1 ) XNORM = ABS( X( IX, J ) ) IF( ERR*SMLNUM.LE.XNORM ) THEN IF( XNORM.GT.ZERO ) $ ERR = ERR / XNORM ELSE IF( ERR.GT.ZERO ) $ ERR = ONE / EPS END IF IF( ERR*SMLNUM.LE.TNORM ) THEN IF( TNORM.GT.ZERO ) $ ERR = ERR / TNORM ELSE IF( ERR.GT.ZERO ) $ ERR = ONE / EPS END IF RESID = MAX( RESID, ERR ) 30 CONTINUE * RETURN * * End of DTRT03 * END