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*> \brief \b SBDT01
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
* Definition:
* ===========
*
* SUBROUTINE SBDT01( M, N, KD, A, LDA, Q, LDQ, D, E, PT, LDPT, WORK,
* RESID )
*
* .. Scalar Arguments ..
* INTEGER KD, LDA, LDPT, LDQ, M, N
* REAL RESID
* ..
* .. Array Arguments ..
* REAL A( LDA, * ), D( * ), E( * ), PT( LDPT, * ),
* $ Q( LDQ, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> SBDT01 reconstructs a general matrix A from its bidiagonal form
*> A = Q * B * P'
*> where Q (m by min(m,n)) and P' (min(m,n) by n) are orthogonal
*> matrices and B is bidiagonal.
*>
*> The test ratio to test the reduction is
*> RESID = norm( A - Q * B * PT ) / ( n * norm(A) * EPS )
*> where PT = P' and EPS is the machine precision.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrices A and Q.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrices A and P'.
*> \endverbatim
*>
*> \param[in] KD
*> \verbatim
*> KD is INTEGER
*> If KD = 0, B is diagonal and the array E is not referenced.
*> If KD = 1, the reduction was performed by xGEBRD; B is upper
*> bidiagonal if M >= N, and lower bidiagonal if M < N.
*> If KD = -1, the reduction was performed by xGBBRD; B is
*> always upper bidiagonal.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is REAL array, dimension (LDA,N)
*> The m by n matrix A.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in] Q
*> \verbatim
*> Q is REAL array, dimension (LDQ,N)
*> The m by min(m,n) orthogonal matrix Q in the reduction
*> A = Q * B * P'.
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*> LDQ is INTEGER
*> The leading dimension of the array Q. LDQ >= max(1,M).
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*> D is REAL array, dimension (min(M,N))
*> The diagonal elements of the bidiagonal matrix B.
*> \endverbatim
*>
*> \param[in] E
*> \verbatim
*> E is REAL array, dimension (min(M,N)-1)
*> The superdiagonal elements of the bidiagonal matrix B if
*> m >= n, or the subdiagonal elements of B if m < n.
*> \endverbatim
*>
*> \param[in] PT
*> \verbatim
*> PT is REAL array, dimension (LDPT,N)
*> The min(m,n) by n orthogonal matrix P' in the reduction
*> A = Q * B * P'.
*> \endverbatim
*>
*> \param[in] LDPT
*> \verbatim
*> LDPT is INTEGER
*> The leading dimension of the array PT.
*> LDPT >= max(1,min(M,N)).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is REAL array, dimension (M+N)
*> \endverbatim
*>
*> \param[out] RESID
*> \verbatim
*> RESID is REAL
*> The test ratio: norm(A - Q * B * P') / ( n * norm(A) * EPS )
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2011
*
*> \ingroup single_eig
*
* =====================================================================
SUBROUTINE SBDT01( M, N, KD, A, LDA, Q, LDQ, D, E, PT, LDPT, WORK,
$ RESID )
*
* -- LAPACK test routine (version 3.4.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2011
*
* .. Scalar Arguments ..
INTEGER KD, LDA, LDPT, LDQ, M, N
REAL RESID
* ..
* .. Array Arguments ..
REAL A( LDA, * ), D( * ), E( * ), PT( LDPT, * ),
$ Q( LDQ, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO, ONE
PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 )
* ..
* .. Local Scalars ..
INTEGER I, J
REAL ANORM, EPS
* ..
* .. External Functions ..
REAL SASUM, SLAMCH, SLANGE
EXTERNAL SASUM, SLAMCH, SLANGE
* ..
* .. External Subroutines ..
EXTERNAL SCOPY, SGEMV
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN, REAL
* ..
* .. Executable Statements ..
*
* Quick return if possible
*
IF( M.LE.0 .OR. N.LE.0 ) THEN
RESID = ZERO
RETURN
END IF
*
* Compute A - Q * B * P' one column at a time.
*
RESID = ZERO
IF( KD.NE.0 ) THEN
*
* B is bidiagonal.
*
IF( KD.NE.0 .AND. M.GE.N ) THEN
*
* B is upper bidiagonal and M >= N.
*
DO 20 J = 1, N
CALL SCOPY( M, A( 1, J ), 1, WORK, 1 )
DO 10 I = 1, N - 1
WORK( M+I ) = D( I )*PT( I, J ) + E( I )*PT( I+1, J )
10 CONTINUE
WORK( M+N ) = D( N )*PT( N, J )
CALL SGEMV( 'No transpose', M, N, -ONE, Q, LDQ,
$ WORK( M+1 ), 1, ONE, WORK, 1 )
RESID = MAX( RESID, SASUM( M, WORK, 1 ) )
20 CONTINUE
ELSE IF( KD.LT.0 ) THEN
*
* B is upper bidiagonal and M < N.
*
DO 40 J = 1, N
CALL SCOPY( M, A( 1, J ), 1, WORK, 1 )
DO 30 I = 1, M - 1
WORK( M+I ) = D( I )*PT( I, J ) + E( I )*PT( I+1, J )
30 CONTINUE
WORK( M+M ) = D( M )*PT( M, J )
CALL SGEMV( 'No transpose', M, M, -ONE, Q, LDQ,
$ WORK( M+1 ), 1, ONE, WORK, 1 )
RESID = MAX( RESID, SASUM( M, WORK, 1 ) )
40 CONTINUE
ELSE
*
* B is lower bidiagonal.
*
DO 60 J = 1, N
CALL SCOPY( M, A( 1, J ), 1, WORK, 1 )
WORK( M+1 ) = D( 1 )*PT( 1, J )
DO 50 I = 2, M
WORK( M+I ) = E( I-1 )*PT( I-1, J ) +
$ D( I )*PT( I, J )
50 CONTINUE
CALL SGEMV( 'No transpose', M, M, -ONE, Q, LDQ,
$ WORK( M+1 ), 1, ONE, WORK, 1 )
RESID = MAX( RESID, SASUM( M, WORK, 1 ) )
60 CONTINUE
END IF
ELSE
*
* B is diagonal.
*
IF( M.GE.N ) THEN
DO 80 J = 1, N
CALL SCOPY( M, A( 1, J ), 1, WORK, 1 )
DO 70 I = 1, N
WORK( M+I ) = D( I )*PT( I, J )
70 CONTINUE
CALL SGEMV( 'No transpose', M, N, -ONE, Q, LDQ,
$ WORK( M+1 ), 1, ONE, WORK, 1 )
RESID = MAX( RESID, SASUM( M, WORK, 1 ) )
80 CONTINUE
ELSE
DO 100 J = 1, N
CALL SCOPY( M, A( 1, J ), 1, WORK, 1 )
DO 90 I = 1, M
WORK( M+I ) = D( I )*PT( I, J )
90 CONTINUE
CALL SGEMV( 'No transpose', M, M, -ONE, Q, LDQ,
$ WORK( M+1 ), 1, ONE, WORK, 1 )
RESID = MAX( RESID, SASUM( M, WORK, 1 ) )
100 CONTINUE
END IF
END IF
*
* Compute norm(A - Q * B * P') / ( n * norm(A) * EPS )
*
ANORM = SLANGE( '1', M, N, A, LDA, WORK )
EPS = SLAMCH( 'Precision' )
*
IF( ANORM.LE.ZERO ) THEN
IF( RESID.NE.ZERO )
$ RESID = ONE / EPS
ELSE
IF( ANORM.GE.RESID ) THEN
RESID = ( RESID / ANORM ) / ( REAL( N )*EPS )
ELSE
IF( ANORM.LT.ONE ) THEN
RESID = ( MIN( RESID, REAL( N )*ANORM ) / ANORM ) /
$ ( REAL( N )*EPS )
ELSE
RESID = MIN( RESID / ANORM, REAL( N ) ) /
$ ( REAL( N )*EPS )
END IF
END IF
END IF
*
RETURN
*
* End of SBDT01
*
END
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