288 lines
8.3 KiB
Fortran
288 lines
8.3 KiB
Fortran
*> \brief \b SPOT05
|
|
*
|
|
* =========== DOCUMENTATION ===========
|
|
*
|
|
* Online html documentation available at
|
|
* http://www.netlib.org/lapack/explore-html/
|
|
*
|
|
* Definition:
|
|
* ===========
|
|
*
|
|
* SUBROUTINE SPOT05( 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 ..
|
|
* REAL A( LDA, * ), B( LDB, * ), BERR( * ), FERR( * ),
|
|
* $ RESLTS( * ), X( LDX, * ), XACT( LDXACT, * )
|
|
* ..
|
|
*
|
|
*
|
|
*> \par Purpose:
|
|
* =============
|
|
*>
|
|
*> \verbatim
|
|
*>
|
|
*> SPOT05 tests the error bounds from iterative refinement for the
|
|
*> computed solution to a system of equations A*X = B, where A is a
|
|
*> symmetric 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
|
|
*> symmetric 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 REAL array, dimension (LDA,N)
|
|
*> The symmetric 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL 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 November 2011
|
|
*
|
|
*> \ingroup single_lin
|
|
*
|
|
* =====================================================================
|
|
SUBROUTINE SPOT05( UPLO, N, NRHS, A, LDA, B, LDB, X, LDX, XACT,
|
|
$ LDXACT, FERR, BERR, RESLTS )
|
|
*
|
|
* -- LAPACK test routine (version 3.4.0) --
|
|
* -- LAPACK is a software package provided by Univ. of Tennessee, --
|
|
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
|
|
* November 2011
|
|
*
|
|
* .. Scalar Arguments ..
|
|
CHARACTER UPLO
|
|
INTEGER LDA, LDB, LDX, LDXACT, N, NRHS
|
|
* ..
|
|
* .. Array Arguments ..
|
|
REAL A( LDA, * ), B( LDB, * ), BERR( * ), FERR( * ),
|
|
$ RESLTS( * ), X( LDX, * ), XACT( LDXACT, * )
|
|
* ..
|
|
*
|
|
* =====================================================================
|
|
*
|
|
* .. Parameters ..
|
|
REAL ZERO, ONE
|
|
PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 )
|
|
* ..
|
|
* .. Local Scalars ..
|
|
LOGICAL UPPER
|
|
INTEGER I, IMAX, J, K
|
|
REAL AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM
|
|
* ..
|
|
* .. External Functions ..
|
|
LOGICAL LSAME
|
|
INTEGER ISAMAX
|
|
REAL SLAMCH
|
|
EXTERNAL LSAME, ISAMAX, SLAMCH
|
|
* ..
|
|
* .. 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 = SLAMCH( 'Epsilon' )
|
|
UNFL = SLAMCH( '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 = ISAMAX( 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 )
|
|
*
|
|
DO 90 K = 1, NRHS
|
|
DO 80 I = 1, N
|
|
TMP = ABS( B( I, K ) )
|
|
IF( UPPER ) THEN
|
|
DO 40 J = 1, I
|
|
TMP = TMP + ABS( A( J, I ) )*ABS( X( J, K ) )
|
|
40 CONTINUE
|
|
DO 50 J = I + 1, N
|
|
TMP = TMP + ABS( A( I, J ) )*ABS( X( J, K ) )
|
|
50 CONTINUE
|
|
ELSE
|
|
DO 60 J = 1, I - 1
|
|
TMP = TMP + ABS( A( I, J ) )*ABS( X( J, K ) )
|
|
60 CONTINUE
|
|
DO 70 J = I, N
|
|
TMP = TMP + ABS( A( J, I ) )*ABS( 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 SPOT05
|
|
*
|
|
END
|