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SUBROUTINE DBDT03( UPLO, N, KD, D, E, U, LDU, S, VT, LDVT, WORK,
$ RESID ) * * -- LAPACK test routine (version 3.1) -- * Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. * November 2006 * * .. Scalar Arguments .. CHARACTER UPLO INTEGER KD, LDU, LDVT, N DOUBLE PRECISION RESID * .. * .. Array Arguments .. DOUBLE PRECISION D( * ), E( * ), S( * ), U( LDU, * ), $ VT( LDVT, * ), WORK( * ) * .. * * Purpose * ======= * * DBDT03 reconstructs a bidiagonal matrix B from its SVD: * S = U' * B * V * where U and V are orthogonal matrices and S is diagonal. * * The test ratio to test the singular value decomposition is * RESID = norm( B - U * S * VT ) / ( n * norm(B) * EPS ) * where VT = V' and EPS is the machine precision. * * Arguments * ========= * * UPLO (input) CHARACTER*1 * Specifies whether the matrix B is upper or lower bidiagonal. * = 'U': Upper bidiagonal * = 'L': Lower bidiagonal * * N (input) INTEGER * The order of the matrix B. * * KD (input) INTEGER * The bandwidth of the bidiagonal matrix B. If KD = 1, the * matrix B is bidiagonal, and if KD = 0, B is diagonal and E is * not referenced. If KD is greater than 1, it is assumed to be * 1, and if KD is less than 0, it is assumed to be 0. * * D (input) DOUBLE PRECISION array, dimension (N) * The n diagonal elements of the bidiagonal matrix B. * * E (input) DOUBLE PRECISION array, dimension (N-1) * The (n-1) superdiagonal elements of the bidiagonal matrix B * if UPLO = 'U', or the (n-1) subdiagonal elements of B if * UPLO = 'L'. * * U (input) DOUBLE PRECISION array, dimension (LDU,N) * The n by n orthogonal matrix U in the reduction B = U'*A*P. * * LDU (input) INTEGER * The leading dimension of the array U. LDU >= max(1,N) * * S (input) DOUBLE PRECISION array, dimension (N) * The singular values from the SVD of B, sorted in decreasing * order. * * VT (input) DOUBLE PRECISION array, dimension (LDVT,N) * The n by n orthogonal matrix V' in the reduction * B = U * S * V'. * * LDVT (input) INTEGER * The leading dimension of the array VT. * * WORK (workspace) DOUBLE PRECISION array, dimension (2*N) * * RESID (output) DOUBLE PRECISION * The test ratio: norm(B - U * S * V') / ( n * norm(A) * EPS ) * * ====================================================================== * * .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) * .. * .. Local Scalars .. INTEGER I, J DOUBLE PRECISION BNORM, EPS * .. * .. External Functions .. LOGICAL LSAME INTEGER IDAMAX DOUBLE PRECISION DASUM, DLAMCH EXTERNAL LSAME, IDAMAX, DASUM, DLAMCH * .. * .. External Subroutines .. EXTERNAL DGEMV * .. * .. Intrinsic Functions .. INTRINSIC ABS, DBLE, MAX, MIN * .. * .. Executable Statements .. * * Quick return if possible * RESID = ZERO IF( N.LE.0 ) $ RETURN * * Compute B - U * S * V' one column at a time. * BNORM = ZERO IF( KD.GE.1 ) THEN * * B is bidiagonal. * IF( LSAME( UPLO, 'U' ) ) THEN * * B is upper bidiagonal. * DO 20 J = 1, N DO 10 I = 1, N WORK( N+I ) = S( I )*VT( I, J ) 10 CONTINUE CALL DGEMV( 'No transpose', N, N, -ONE, U, LDU, $ WORK( N+1 ), 1, ZERO, WORK, 1 ) WORK( J ) = WORK( J ) + D( J ) IF( J.GT.1 ) THEN WORK( J-1 ) = WORK( J-1 ) + E( J-1 ) BNORM = MAX( BNORM, ABS( D( J ) )+ABS( E( J-1 ) ) ) ELSE BNORM = MAX( BNORM, ABS( D( J ) ) ) END IF RESID = MAX( RESID, DASUM( N, WORK, 1 ) ) 20 CONTINUE ELSE * * B is lower bidiagonal. * DO 40 J = 1, N DO 30 I = 1, N WORK( N+I ) = S( I )*VT( I, J ) 30 CONTINUE CALL DGEMV( 'No transpose', N, N, -ONE, U, LDU, $ WORK( N+1 ), 1, ZERO, WORK, 1 ) WORK( J ) = WORK( J ) + D( J ) IF( J.LT.N ) THEN WORK( J+1 ) = WORK( J+1 ) + E( J ) BNORM = MAX( BNORM, ABS( D( J ) )+ABS( E( J ) ) ) ELSE BNORM = MAX( BNORM, ABS( D( J ) ) ) END IF RESID = MAX( RESID, DASUM( N, WORK, 1 ) ) 40 CONTINUE END IF ELSE * * B is diagonal. * DO 60 J = 1, N DO 50 I = 1, N WORK( N+I ) = S( I )*VT( I, J ) 50 CONTINUE CALL DGEMV( 'No transpose', N, N, -ONE, U, LDU, WORK( N+1 ), $ 1, ZERO, WORK, 1 ) WORK( J ) = WORK( J ) + D( J ) RESID = MAX( RESID, DASUM( N, WORK, 1 ) ) 60 CONTINUE J = IDAMAX( N, D, 1 ) BNORM = ABS( D( J ) ) END IF * * Compute norm(B - U * S * V') / ( n * norm(B) * EPS ) * EPS = DLAMCH( 'Precision' ) * IF( BNORM.LE.ZERO ) THEN IF( RESID.NE.ZERO ) $ RESID = ONE / EPS ELSE IF( BNORM.GE.RESID ) THEN RESID = ( RESID / BNORM ) / ( DBLE( N )*EPS ) ELSE IF( BNORM.LT.ONE ) THEN RESID = ( MIN( RESID, DBLE( N )*BNORM ) / BNORM ) / $ ( DBLE( N )*EPS ) ELSE RESID = MIN( RESID / BNORM, DBLE( N ) ) / $ ( DBLE( N )*EPS ) END IF END IF END IF * RETURN * * End of DBDT03 * END |