SUBROUTINE ZTPT05( UPLO, TRANS, DIAG, N, NRHS, AP, 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 DIAG, TRANS, UPLO INTEGER LDB, LDX, LDXACT, N, NRHS * .. * .. Array Arguments .. DOUBLE PRECISION BERR( * ), FERR( * ), RESLTS( * ) COMPLEX*16 AP( * ), B( LDB, * ), X( LDX, * ), \$ XACT( LDXACT, * ) * .. * * Purpose * ======= * * ZTPT05 tests the error bounds from iterative refinement for the * computed solution to a system of equations A*X = B, where A is a * triangular matrix in packed storage format. * * 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 ) * * Arguments * ========= * * UPLO (input) CHARACTER*1 * Specifies whether the matrix A is upper or lower triangular. * = 'U': Upper triangular * = 'L': Lower triangular * * TRANS (input) 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) * * DIAG (input) CHARACTER*1 * Specifies whether or not the matrix A is unit triangular. * = 'N': Non-unit triangular * = 'U': Unit triangular * * N (input) INTEGER * The number of rows of the matrices X, B, and XACT, and the * order of the matrix A. N >= 0. * * NRHS (input) INTEGER * The number of columns of the matrices X, B, and XACT. * NRHS >= 0. * * AP (input) COMPLEX*16 array, dimension (N*(N+1)/2) * The upper or lower triangular matrix A, packed columnwise in * a linear array. The j-th column of A is stored in the array * AP as follows: * if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; * if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n. * If DIAG = 'U', the diagonal elements of A are not referenced * and are assumed to be 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 / ( (n+1)*EPS + (*) ) * * ===================================================================== * * .. 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, JC, K 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' ) 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 = 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 / ( (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 = CABS1( B( I, K ) ) IF( UPPER ) THEN JC = ( ( I-1 )*I ) / 2 IF( .NOT.NOTRAN ) THEN DO 40 J = 1, I - IFU TMP = TMP + CABS1( AP( JC+J ) )*CABS1( X( J, K ) ) 40 CONTINUE IF( UNIT ) \$ TMP = TMP + CABS1( X( I, K ) ) ELSE JC = JC + I IF( UNIT ) THEN TMP = TMP + CABS1( X( I, K ) ) JC = JC + I END IF DO 50 J = I + IFU, N TMP = TMP + CABS1( AP( JC ) )*CABS1( X( J, K ) ) JC = JC + J 50 CONTINUE END IF ELSE IF( NOTRAN ) THEN JC = I DO 60 J = 1, I - IFU TMP = TMP + CABS1( AP( JC ) )*CABS1( X( J, K ) ) JC = JC + N - J 60 CONTINUE IF( UNIT ) \$ TMP = TMP + CABS1( X( I, K ) ) ELSE JC = ( I-1 )*( N-I ) + ( I*( I+1 ) ) / 2 IF( UNIT ) \$ TMP = TMP + CABS1( X( I, K ) ) DO 70 J = I + IFU, N TMP = TMP + CABS1( AP( JC+J-I ) )* \$ CABS1( 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 ZTPT05 * END