*> \brief \b DLASD0 computes the singular values of a real upper bidiagonal n-by-m matrix B with diagonal d and off-diagonal e. Used by sbdsdc.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
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*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLASD0( N, SQRE, D, E, U, LDU, VT, LDVT, SMLSIZ, IWORK,
* WORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDU, LDVT, N, SMLSIZ, SQRE
* ..
* .. Array Arguments ..
* INTEGER IWORK( * )
* DOUBLE PRECISION D( * ), E( * ), U( LDU, * ), VT( LDVT, * ),
* $ WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> Using a divide and conquer approach, DLASD0 computes the singular
*> value decomposition (SVD) of a real upper bidiagonal N-by-M
*> matrix B with diagonal D and offdiagonal E, where M = N + SQRE.
*> The algorithm computes orthogonal matrices U and VT such that
*> B = U * S * VT. The singular values S are overwritten on D.
*>
*> A related subroutine, DLASDA, computes only the singular values,
*> and optionally, the singular vectors in compact form.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> On entry, the row dimension of the upper bidiagonal matrix.
*> This is also the dimension of the main diagonal array D.
*> \endverbatim
*>
*> \param[in] SQRE
*> \verbatim
*> SQRE is INTEGER
*> Specifies the column dimension of the bidiagonal matrix.
*> = 0: The bidiagonal matrix has column dimension M = N;
*> = 1: The bidiagonal matrix has column dimension M = N+1;
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> On entry D contains the main diagonal of the bidiagonal
*> matrix.
*> On exit D, if INFO = 0, contains its singular values.
*> \endverbatim
*>
*> \param[in,out] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (M-1)
*> Contains the subdiagonal entries of the bidiagonal matrix.
*> On exit, E has been destroyed.
*> \endverbatim
*>
*> \param[out] U
*> \verbatim
*> U is DOUBLE PRECISION array, dimension (LDU, N)
*> On exit, U contains the left singular vectors.
*> \endverbatim
*>
*> \param[in] LDU
*> \verbatim
*> LDU is INTEGER
*> On entry, leading dimension of U.
*> \endverbatim
*>
*> \param[out] VT
*> \verbatim
*> VT is DOUBLE PRECISION array, dimension (LDVT, M)
*> On exit, VT**T contains the right singular vectors.
*> \endverbatim
*>
*> \param[in] LDVT
*> \verbatim
*> LDVT is INTEGER
*> On entry, leading dimension of VT.
*> \endverbatim
*>
*> \param[in] SMLSIZ
*> \verbatim
*> SMLSIZ is INTEGER
*> On entry, maximum size of the subproblems at the
*> bottom of the computation tree.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (8*N)
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (3*M**2+2*M)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit.
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> > 0: if INFO = 1, a singular value did not converge
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date June 2017
*
*> \ingroup OTHERauxiliary
*
*> \par Contributors:
* ==================
*>
*> Ming Gu and Huan Ren, Computer Science Division, University of
*> California at Berkeley, USA
*>
* =====================================================================
SUBROUTINE DLASD0( N, SQRE, D, E, U, LDU, VT, LDVT, SMLSIZ, IWORK,
$ WORK, INFO )
*
* -- LAPACK auxiliary routine (version 3.7.1) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* June 2017
*
* .. Scalar Arguments ..
INTEGER INFO, LDU, LDVT, N, SMLSIZ, SQRE
* ..
* .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION D( * ), E( * ), U( LDU, * ), VT( LDVT, * ),
$ WORK( * )
* ..
*
* =====================================================================
*
* .. Local Scalars ..
INTEGER I, I1, IC, IDXQ, IDXQC, IM1, INODE, ITEMP, IWK,
$ J, LF, LL, LVL, M, NCC, ND, NDB1, NDIML, NDIMR,
$ NL, NLF, NLP1, NLVL, NR, NRF, NRP1, SQREI
DOUBLE PRECISION ALPHA, BETA
* ..
* .. External Subroutines ..
EXTERNAL DLASD1, DLASDQ, DLASDT, XERBLA
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
*
IF( N.LT.0 ) THEN
INFO = -1
ELSE IF( ( SQRE.LT.0 ) .OR. ( SQRE.GT.1 ) ) THEN
INFO = -2
END IF
*
M = N + SQRE
*
IF( LDU.LT.N ) THEN
INFO = -6
ELSE IF( LDVT.LT.M ) THEN
INFO = -8
ELSE IF( SMLSIZ.LT.3 ) THEN
INFO = -9
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DLASD0', -INFO )
RETURN
END IF
*
* If the input matrix is too small, call DLASDQ to find the SVD.
*
IF( N.LE.SMLSIZ ) THEN
CALL DLASDQ( 'U', SQRE, N, M, N, 0, D, E, VT, LDVT, U, LDU, U,
$ LDU, WORK, INFO )
RETURN
END IF
*
* Set up the computation tree.
*
INODE = 1
NDIML = INODE + N
NDIMR = NDIML + N
IDXQ = NDIMR + N
IWK = IDXQ + N
CALL DLASDT( N, NLVL, ND, IWORK( INODE ), IWORK( NDIML ),
$ IWORK( NDIMR ), SMLSIZ )
*
* For the nodes on bottom level of the tree, solve
* their subproblems by DLASDQ.
*
NDB1 = ( ND+1 ) / 2
NCC = 0
DO 30 I = NDB1, ND
*
* IC : center row of each node
* NL : number of rows of left subproblem
* NR : number of rows of right subproblem
* NLF: starting row of the left subproblem
* NRF: starting row of the right subproblem
*
I1 = I - 1
IC = IWORK( INODE+I1 )
NL = IWORK( NDIML+I1 )
NLP1 = NL + 1
NR = IWORK( NDIMR+I1 )
NRP1 = NR + 1
NLF = IC - NL
NRF = IC + 1
SQREI = 1
CALL DLASDQ( 'U', SQREI, NL, NLP1, NL, NCC, D( NLF ), E( NLF ),
$ VT( NLF, NLF ), LDVT, U( NLF, NLF ), LDU,
$ U( NLF, NLF ), LDU, WORK, INFO )
IF( INFO.NE.0 ) THEN
RETURN
END IF
ITEMP = IDXQ + NLF - 2
DO 10 J = 1, NL
IWORK( ITEMP+J ) = J
10 CONTINUE
IF( I.EQ.ND ) THEN
SQREI = SQRE
ELSE
SQREI = 1
END IF
NRP1 = NR + SQREI
CALL DLASDQ( 'U', SQREI, NR, NRP1, NR, NCC, D( NRF ), E( NRF ),
$ VT( NRF, NRF ), LDVT, U( NRF, NRF ), LDU,
$ U( NRF, NRF ), LDU, WORK, INFO )
IF( INFO.NE.0 ) THEN
RETURN
END IF
ITEMP = IDXQ + IC
DO 20 J = 1, NR
IWORK( ITEMP+J-1 ) = J
20 CONTINUE
30 CONTINUE
*
* Now conquer each subproblem bottom-up.
*
DO 50 LVL = NLVL, 1, -1
*
* Find the first node LF and last node LL on the
* current level LVL.
*
IF( LVL.EQ.1 ) THEN
LF = 1
LL = 1
ELSE
LF = 2**( LVL-1 )
LL = 2*LF - 1
END IF
DO 40 I = LF, LL
IM1 = I - 1
IC = IWORK( INODE+IM1 )
NL = IWORK( NDIML+IM1 )
NR = IWORK( NDIMR+IM1 )
NLF = IC - NL
IF( ( SQRE.EQ.0 ) .AND. ( I.EQ.LL ) ) THEN
SQREI = SQRE
ELSE
SQREI = 1
END IF
IDXQC = IDXQ + NLF - 1
ALPHA = D( IC )
BETA = E( IC )
CALL DLASD1( NL, NR, SQREI, D( NLF ), ALPHA, BETA,
$ U( NLF, NLF ), LDU, VT( NLF, NLF ), LDVT,
$ IWORK( IDXQC ), IWORK( IWK ), WORK, INFO )
*
* Report the possible convergence failure.
*
IF( INFO.NE.0 ) THEN
RETURN
END IF
40 CONTINUE
50 CONTINUE
*
RETURN
*
* End of DLASD0
*
END