*> \brief \b DPBT05 * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * * Definition: * =========== * * SUBROUTINE DPBT05( UPLO, N, KD, NRHS, AB, LDAB, B, LDB, X, LDX, * XACT, LDXACT, FERR, BERR, RESLTS ) * * .. Scalar Arguments .. * CHARACTER UPLO * INTEGER KD, LDAB, LDB, LDX, LDXACT, N, NRHS * .. * .. Array Arguments .. * DOUBLE PRECISION AB( LDAB, * ), B( LDB, * ), BERR( * ), * $ FERR( * ), RESLTS( * ), X( LDX, * ), * $ XACT( LDXACT, * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> DPBT05 tests the error bounds from iterative refinement for the *> computed solution to a system of equations A*X = B, where A is a *> symmetric band 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 / ( NZ*EPS + (*) ), where *> (*) = NZ*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i ) *> and NZ = max. number of nonzeros in any row of A, plus 1 *> \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] KD *> \verbatim *> KD is INTEGER *> The number of super-diagonals of the matrix A if UPLO = 'U', *> or the number of sub-diagonals if UPLO = 'L'. KD >= 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] AB *> \verbatim *> AB is DOUBLE PRECISION array, dimension (LDAB,N) *> The upper or lower triangle of the symmetric band matrix A, *> stored in the first KD+1 rows of the array. The j-th column *> of A is stored in the j-th column of the array AB as follows: *> if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; *> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). *> \endverbatim *> *> \param[in] LDAB *> \verbatim *> LDAB is INTEGER *> The leading dimension of the array AB. LDAB >= KD+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 / ( NZ*EPS + (*) ) *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \date November 2011 * *> \ingroup double_lin * * ===================================================================== SUBROUTINE DPBT05( UPLO, N, KD, NRHS, AB, LDAB, 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 KD, LDAB, LDB, LDX, LDXACT, N, NRHS * .. * .. Array Arguments .. DOUBLE PRECISION AB( LDAB, * ), 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 UPPER INTEGER I, IMAX, J, K, NZ 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' ) NZ = 2*MAX( KD, N-1 ) + 1 * * 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 / ( NZ*EPS + (*) ), where * (*) = NZ*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 = MAX( I-KD, 1 ), I TMP = TMP + ABS( AB( KD+1-I+J, I ) )*ABS( X( J, K ) ) 40 CONTINUE DO 50 J = I + 1, MIN( I+KD, N ) TMP = TMP + ABS( AB( KD+1+I-J, J ) )*ABS( X( J, K ) ) 50 CONTINUE ELSE DO 60 J = MAX( I-KD, 1 ), I - 1 TMP = TMP + ABS( AB( 1+I-J, J ) )*ABS( X( J, K ) ) 60 CONTINUE DO 70 J = I, MIN( I+KD, N ) TMP = TMP + ABS( AB( 1+J-I, 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 ) / ( NZ*EPS+NZ*UNFL / MAX( AXBI, NZ*UNFL ) ) IF( K.EQ.1 ) THEN RESLTS( 2 ) = TMP ELSE RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP ) END IF 90 CONTINUE * RETURN * * End of DPBT05 * END