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SUBROUTINE ZBDT03( UPLO, N, KD, D, E, U, LDU, S, VT, LDVT, WORK,
$ RESID )
*
* -- LAPACK test routine (version 3.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* February 29, 1992
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER KD, LDU, LDVT, N
DOUBLE PRECISION RESID
* ..
* .. Array Arguments ..
DOUBLE PRECISION D( * ), E( * ), S( * )
COMPLEX*16 U( LDU, * ), VT( LDVT, * ), WORK( * )
* ..
*
* Purpose
* =======
*
* ZBDT03 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) COMPLEX*16 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) COMPLEX*16 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) COMPLEX*16 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 DLAMCH, DZASUM
EXTERNAL LSAME, IDAMAX, DLAMCH, DZASUM
* ..
* .. External Subroutines ..
EXTERNAL ZGEMV
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, DBLE, DCMPLX, 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 ZGEMV( 'No transpose', N, N, -DCMPLX( ONE ), U, LDU,
$ WORK( N+1 ), 1, DCMPLX( 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, DZASUM( 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 ZGEMV( 'No transpose', N, N, -DCMPLX( ONE ), U, LDU,
$ WORK( N+1 ), 1, DCMPLX( 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, DZASUM( 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 ZGEMV( 'No transpose', N, N, -DCMPLX( ONE ), U, LDU,
$ WORK( N+1 ), 1, DCMPLX( ZERO ), WORK, 1 )
WORK( J ) = WORK( J ) + D( J )
RESID = MAX( RESID, DZASUM( 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 ZBDT03
*
END
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