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SUBROUTINE SGTT05( TRANS, N, NRHS, DL, D, DU, 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 TRANS INTEGER LDB, LDX, LDXACT, N, NRHS * .. * .. Array Arguments .. REAL B( LDB, * ), BERR( * ), D( * ), DL( * ), $ DU( * ), FERR( * ), RESLTS( * ), X( LDX, * ), $ XACT( LDXACT, * ) * .. * * Purpose * ======= * * SGTT05 tests the error bounds from iterative refinement for the * computed solution to a system of equations A*X = B, where A is a * general tridiagonal matrix of order n and op(A) = A or A**T, * depending on TRANS. * * 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(op(A))*abs(X) +abs(b))_i ) * and NZ = max. number of nonzeros in any row of A, plus 1 * * Arguments * ========= * * TRANS (input) CHARACTER*1 * Specifies the form of the system of equations. * = 'N': A * X = B (No transpose) * = 'T': A**T * X = B (Transpose) * = 'C': A**H * X = B (Conjugate transpose = Transpose) * * N (input) INTEGER * The number of rows of the matrices X and XACT. N >= 0. * * NRHS (input) INTEGER * The number of columns of the matrices X and XACT. NRHS >= 0. * * DL (input) REAL array, dimension (N-1) * The (n-1) sub-diagonal elements of A. * * D (input) REAL array, dimension (N) * The diagonal elements of A. * * DU (input) REAL array, dimension (N-1) * The (n-1) super-diagonal elements of A. * * B (input) REAL 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) REAL 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) REAL 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) 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. * * BERR (input) 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). * * RESLTS (output) 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 / ( NZ*EPS + (*) ) * * ===================================================================== * * .. Parameters .. REAL ZERO, ONE PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 ) * .. * .. Local Scalars .. LOGICAL NOTRAN INTEGER I, IMAX, J, K, NZ 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 NOTRAN = LSAME( TRANS, 'N' ) NZ = 4 * * 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 / ( NZ*EPS + (*) ), where * (*) = NZ*UNFL / (min_i (abs(op(A))*abs(X) +abs(b))_i ) * DO 60 K = 1, NRHS IF( NOTRAN ) THEN IF( N.EQ.1 ) THEN AXBI = ABS( B( 1, K ) ) + ABS( D( 1 )*X( 1, K ) ) ELSE AXBI = ABS( B( 1, K ) ) + ABS( D( 1 )*X( 1, K ) ) + $ ABS( DU( 1 )*X( 2, K ) ) DO 40 I = 2, N - 1 TMP = ABS( B( I, K ) ) + ABS( DL( I-1 )*X( I-1, K ) ) $ + ABS( D( I )*X( I, K ) ) + $ ABS( DU( I )*X( I+1, K ) ) AXBI = MIN( AXBI, TMP ) 40 CONTINUE TMP = ABS( B( N, K ) ) + ABS( DL( N-1 )*X( N-1, K ) ) + $ ABS( D( N )*X( N, K ) ) AXBI = MIN( AXBI, TMP ) END IF ELSE IF( N.EQ.1 ) THEN AXBI = ABS( B( 1, K ) ) + ABS( D( 1 )*X( 1, K ) ) ELSE AXBI = ABS( B( 1, K ) ) + ABS( D( 1 )*X( 1, K ) ) + $ ABS( DL( 1 )*X( 2, K ) ) DO 50 I = 2, N - 1 TMP = ABS( B( I, K ) ) + ABS( DU( I-1 )*X( I-1, K ) ) $ + ABS( D( I )*X( I, K ) ) + $ ABS( DL( I )*X( I+1, K ) ) AXBI = MIN( AXBI, TMP ) 50 CONTINUE TMP = ABS( B( N, K ) ) + ABS( DU( N-1 )*X( N-1, K ) ) + $ ABS( D( N )*X( N, K ) ) AXBI = MIN( AXBI, TMP ) END IF END IF 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 60 CONTINUE * RETURN * * End of SGTT05 * END |