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SUBROUTINE ZPBT05( UPLO, N, KD, NRHS, AB, LDAB, B, LDB, X, LDX,
$ XACT, LDXACT, FERR, BERR, RESLTS ) * * -- LAPACK test routine (version 3.1) -- * Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. * November 2006 * * .. 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, * ) * .. * * Purpose * ======= * * 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 * * Arguments * ========= * * UPLO (input) CHARACTER*1 * Specifies whether the upper or lower triangular part of the * Hermitian matrix A is stored. * = 'U': Upper triangular * = 'L': Lower triangular * * N (input) INTEGER * The number of rows of the matrices X, B, and XACT, and the * order of the matrix A. N >= 0. * * KD (input) INTEGER * The number of super-diagonals of the matrix A if UPLO = 'U', * or the number of sub-diagonals if UPLO = 'L'. KD >= 0. * * NRHS (input) INTEGER * The number of columns of the matrices X, B, and XACT. * NRHS >= 0. * * AB (input) 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). * * LDAB (input) INTEGER * The leading dimension of the array AB. LDAB >= KD+1. * * B (input) COMPLEX*16 array, dimension (LDB,NRHS) * The right hand side vectors for the system of linear * equations. * * LDB (input) INTEGER * The leading dimension of the array B. LDB >= max(1,N). * * X (input) COMPLEX*16 array, dimension (LDX,NRHS) * The computed solution vectors. Each vector is stored as a * column of the matrix X. * * LDX (input) INTEGER * The leading dimension of the array X. LDX >= max(1,N). * * XACT (input) COMPLEX*16 array, dimension (LDX,NRHS) * The exact solution vectors. Each vector is stored as a * column of the matrix XACT. * * LDXACT (input) INTEGER * The leading dimension of the array XACT. LDXACT >= max(1,N). * * FERR (input) 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. * * BERR (input) 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). * * RESLTS (output) 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 + (*) ) * * ===================================================================== * * .. 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 |