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SUBROUTINE SGBT05( TRANS, N, KL, KU, 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 TRANS INTEGER KL, KU, LDAB, LDB, LDX, LDXACT, N, NRHS * .. * .. Array Arguments .. REAL AB( LDAB, * ), B( LDB, * ), BERR( * ), $ FERR( * ), RESLTS( * ), X( LDX, * ), $ XACT( LDXACT, * ) * .. * * Purpose * ======= * * SGBT05 tests the error bounds from iterative refinement for the * computed solution to a system of equations op(A)*X = B, where A is a * general band matrix of order n with kl subdiagonals and ku * superdiagonals 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, B, and XACT, and the * order of the matrix A. N >= 0. * * KL (input) INTEGER * The number of subdiagonals within the band of A. KL >= 0. * * KU (input) INTEGER * The number of superdiagonals within the band of A. KU >= 0. * * NRHS (input) INTEGER * The number of columns of the matrices X, B, and XACT. * NRHS >= 0. * * AB (input) REAL array, dimension (LDAB,N) * The original band matrix A, stored in rows 1 to KL+KU+1. * The j-th column of A is stored in the j-th column of the * array AB as follows: * AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(n,j+kl). * * LDAB (input) INTEGER * The leading dimension of the array AB. LDAB >= KL+KU+1. * * 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 = MIN( KL+KU+2, N+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 = 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 70 K = 1, NRHS DO 60 I = 1, N TMP = ABS( B( I, K ) ) IF( NOTRAN ) THEN DO 40 J = MAX( I-KL, 1 ), MIN( I+KU, N ) TMP = TMP + ABS( AB( KU+1+I-J, J ) )*ABS( X( J, K ) ) 40 CONTINUE ELSE DO 50 J = MAX( I-KU, 1 ), MIN( I+KL, N ) TMP = TMP + ABS( AB( KU+1+J-I, I ) )*ABS( X( J, K ) ) 50 CONTINUE END IF IF( I.EQ.1 ) THEN AXBI = TMP ELSE AXBI = MIN( AXBI, TMP ) END IF 60 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 70 CONTINUE * RETURN * * End of SGBT05 * END |