*> \brief \b CTRT03 * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * * Definition: * =========== * * SUBROUTINE CTRT03( 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 * REAL RESID, SCALE, TSCAL * .. * .. Array Arguments .. * REAL CNORM( * ) * COMPLEX A( LDA, * ), B( LDB, * ), WORK( * ), * \$ X( LDX, * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> CTRT03 computes the residual for the solution to a scaled triangular *> system of equations A*x = s*b, A**T *x = s*b, or A**H *x = s*b. *> Here A is a triangular matrix, A**T denotes the transpose of A, A**H *> denotes the conjugate 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, A**T, or A**H, 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**T *x = s*b (Transpose) *> = 'C': A**H *x = s*b (Conjugate 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 COMPLEX 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 REAL *> The scaling factor s used in solving the triangular system. *> \endverbatim *> *> \param[in] CNORM *> \verbatim *> CNORM is REAL array, dimension (N) *> The 1-norms of the columns of A, not counting the diagonal. *> \endverbatim *> *> \param[in] TSCAL *> \verbatim *> TSCAL is REAL *> 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 COMPLEX 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 COMPLEX 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 COMPLEX array, dimension (N) *> \endverbatim *> *> \param[out] RESID *> \verbatim *> RESID is REAL *> 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 complex_lin * * ===================================================================== SUBROUTINE CTRT03( 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 REAL RESID, SCALE, TSCAL * .. * .. Array Arguments .. REAL CNORM( * ) COMPLEX A( LDA, * ), B( LDB, * ), WORK( * ), \$ X( LDX, * ) * .. * * ===================================================================== * * .. Parameters .. REAL ONE, ZERO PARAMETER ( ONE = 1.0E+0, ZERO = 0.0E+0 ) * .. * .. Local Scalars .. INTEGER IX, J REAL EPS, ERR, SMLNUM, TNORM, XNORM, XSCAL * .. * .. External Functions .. LOGICAL LSAME INTEGER ICAMAX REAL SLAMCH EXTERNAL LSAME, ICAMAX, SLAMCH * .. * .. External Subroutines .. EXTERNAL CAXPY, CCOPY, CSSCAL, CTRMV * .. * .. Intrinsic Functions .. INTRINSIC ABS, CMPLX, MAX, REAL * .. * .. Executable Statements .. * * Quick exit if N = 0 * IF( N.LE.0 .OR. NRHS.LE.0 ) THEN RESID = ZERO RETURN END IF EPS = SLAMCH( 'Epsilon' ) SMLNUM = SLAMCH( 'Safe minimum' ) * * Compute the norm of the triangular matrix A using the column * norms already computed by CLATRS. * 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 CCOPY( N, X( 1, J ), 1, WORK, 1 ) IX = ICAMAX( N, WORK, 1 ) XNORM = MAX( ONE, ABS( X( IX, J ) ) ) XSCAL = ( ONE / XNORM ) / REAL( N ) CALL CSSCAL( N, XSCAL, WORK, 1 ) CALL CTRMV( UPLO, TRANS, DIAG, N, A, LDA, WORK, 1 ) CALL CAXPY( N, CMPLX( -SCALE*XSCAL ), B( 1, J ), 1, WORK, 1 ) IX = ICAMAX( N, WORK, 1 ) ERR = TSCAL*ABS( WORK( IX ) ) IX = ICAMAX( 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 CTRT03 * END