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*> \brief \b DTRT05
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
* Definition:
* ===========
*
* SUBROUTINE DTRT05( UPLO, TRANS, DIAG, N, NRHS, A, LDA, B, LDB, X,
* LDX, XACT, LDXACT, FERR, BERR, RESLTS )
*
* .. Scalar Arguments ..
* CHARACTER DIAG, TRANS, UPLO
* INTEGER LDA, LDB, LDX, LDXACT, N, NRHS
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), B( LDB, * ), BERR( * ), FERR( * ),
* $ RESLTS( * ), X( LDX, * ), XACT( LDXACT, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DTRT05 tests the error bounds from iterative refinement for the
*> computed solution to a system of equations A*X = B, where A is a
*> triangular 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 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] 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] 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.
*
*> \date December 2016
*
*> \ingroup double_lin
*
* =====================================================================
SUBROUTINE DTRT05( UPLO, TRANS, DIAG, N, NRHS, A, LDA, B, LDB, X,
$ LDX, XACT, LDXACT, FERR, BERR, RESLTS )
*
* -- LAPACK test routine (version 3.7.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* December 2016
*
* .. Scalar Arguments ..
CHARACTER DIAG, TRANS, UPLO
INTEGER LDA, LDB, LDX, LDXACT, N, NRHS
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), 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, 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
IF( .NOT.NOTRAN ) THEN
DO 40 J = 1, I - IFU
TMP = TMP + ABS( A( J, I ) )*ABS( X( J, K ) )
40 CONTINUE
IF( UNIT )
$ TMP = TMP + ABS( X( I, K ) )
ELSE
IF( UNIT )
$ TMP = TMP + ABS( X( I, K ) )
DO 50 J = I + IFU, N
TMP = TMP + ABS( A( I, J ) )*ABS( X( J, K ) )
50 CONTINUE
END IF
ELSE
IF( NOTRAN ) THEN
DO 60 J = 1, I - IFU
TMP = TMP + ABS( A( I, J ) )*ABS( X( J, K ) )
60 CONTINUE
IF( UNIT )
$ TMP = TMP + ABS( X( I, K ) )
ELSE
IF( UNIT )
$ TMP = TMP + ABS( X( I, K ) )
DO 70 J = I + IFU, N
TMP = TMP + ABS( A( 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 DTRT05
*
END
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