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*> \brief \b ZPBT05
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
* Definition:
* ===========
*
* SUBROUTINE ZPBT05( 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 BERR( * ), FERR( * ), RESLTS( * )
* COMPLEX*16 AB( LDAB, * ), B( LDB, * ), X( LDX, * ),
* $ XACT( LDXACT, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> ZPBT05 tests the error bounds from iterative refinement for the
*> computed solution to a system of equations A*X = B, where A is a
*> Hermitian 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
*> Hermitian 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 COMPLEX*16 array, dimension (LDAB,N)
*> The upper or lower triangle of the Hermitian 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 COMPLEX*16 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 COMPLEX*16 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 COMPLEX*16 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 December 2016
*
*> \ingroup complex16_lin
*
* =====================================================================
SUBROUTINE ZPBT05( UPLO, N, KD, NRHS, AB, LDAB, 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 UPLO
INTEGER KD, LDAB, LDB, LDX, LDXACT, N, NRHS
* ..
* .. Array Arguments ..
DOUBLE PRECISION BERR( * ), FERR( * ), RESLTS( * )
COMPLEX*16 AB( LDAB, * ), B( LDB, * ), 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
COMPLEX*16 ZDUM
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER IZAMAX
DOUBLE PRECISION DLAMCH
EXTERNAL LSAME, IZAMAX, DLAMCH
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, DBLE, DIMAG, MAX, MIN
* ..
* .. Statement Functions ..
DOUBLE PRECISION CABS1
* ..
* .. Statement Function definitions ..
CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
* ..
* .. 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 = IZAMAX( N, X( 1, J ), 1 )
XNORM = MAX( CABS1( X( IMAX, J ) ), UNFL )
DIFF = ZERO
DO 10 I = 1, N
DIFF = MAX( DIFF, CABS1( 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 = CABS1( B( I, K ) )
IF( UPPER ) THEN
DO 40 J = MAX( I-KD, 1 ), I - 1
TMP = TMP + CABS1( AB( KD+1-I+J, I ) )*
$ CABS1( X( J, K ) )
40 CONTINUE
TMP = TMP + ABS( DBLE( AB( KD+1, I ) ) )*
$ CABS1( X( I, K ) )
DO 50 J = I + 1, MIN( I+KD, N )
TMP = TMP + CABS1( AB( KD+1+I-J, J ) )*
$ CABS1( X( J, K ) )
50 CONTINUE
ELSE
DO 60 J = MAX( I-KD, 1 ), I - 1
TMP = TMP + CABS1( AB( 1+I-J, J ) )*CABS1( X( J, K ) )
60 CONTINUE
TMP = TMP + ABS( DBLE( AB( 1, I ) ) )*CABS1( X( I, K ) )
DO 70 J = I + 1, MIN( I+KD, N )
TMP = TMP + CABS1( AB( 1+J-I, I ) )*CABS1( 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 ZPBT05
*
END
|
