*> \brief \b DLASD1 computes the SVD of an upper bidiagonal matrix B of the specified size. Used by sbdsdc.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
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*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLASD1( NL, NR, SQRE, D, ALPHA, BETA, U, LDU, VT, LDVT,
* IDXQ, IWORK, WORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDU, LDVT, NL, NR, SQRE
* DOUBLE PRECISION ALPHA, BETA
* ..
* .. Array Arguments ..
* INTEGER IDXQ( * ), IWORK( * )
* DOUBLE PRECISION D( * ), U( LDU, * ), VT( LDVT, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLASD1 computes the SVD of an upper bidiagonal N-by-M matrix B,
*> where N = NL + NR + 1 and M = N + SQRE. DLASD1 is called from DLASD0.
*>
*> A related subroutine DLASD7 handles the case in which the singular
*> values (and the singular vectors in factored form) are desired.
*>
*> DLASD1 computes the SVD as follows:
*>
*> ( D1(in) 0 0 0 )
*> B = U(in) * ( Z1**T a Z2**T b ) * VT(in)
*> ( 0 0 D2(in) 0 )
*>
*> = U(out) * ( D(out) 0) * VT(out)
*>
*> where Z**T = (Z1**T a Z2**T b) = u**T VT**T, and u is a vector of dimension M
*> with ALPHA and BETA in the NL+1 and NL+2 th entries and zeros
*> elsewhere; and the entry b is empty if SQRE = 0.
*>
*> The left singular vectors of the original matrix are stored in U, and
*> the transpose of the right singular vectors are stored in VT, and the
*> singular values are in D. The algorithm consists of three stages:
*>
*> The first stage consists of deflating the size of the problem
*> when there are multiple singular values or when there are zeros in
*> the Z vector. For each such occurrence the dimension of the
*> secular equation problem is reduced by one. This stage is
*> performed by the routine DLASD2.
*>
*> The second stage consists of calculating the updated
*> singular values. This is done by finding the square roots of the
*> roots of the secular equation via the routine DLASD4 (as called
*> by DLASD3). This routine also calculates the singular vectors of
*> the current problem.
*>
*> The final stage consists of computing the updated singular vectors
*> directly using the updated singular values. The singular vectors
*> for the current problem are multiplied with the singular vectors
*> from the overall problem.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] NL
*> \verbatim
*> NL is INTEGER
*> The row dimension of the upper block. NL >= 1.
*> \endverbatim
*>
*> \param[in] NR
*> \verbatim
*> NR is INTEGER
*> The row dimension of the lower block. NR >= 1.
*> \endverbatim
*>
*> \param[in] SQRE
*> \verbatim
*> SQRE is INTEGER
*> = 0: the lower block is an NR-by-NR square matrix.
*> = 1: the lower block is an NR-by-(NR+1) rectangular matrix.
*>
*> The bidiagonal matrix has row dimension N = NL + NR + 1,
*> and column dimension M = N + SQRE.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*> D is DOUBLE PRECISION array,
*> dimension (N = NL+NR+1).
*> On entry D(1:NL,1:NL) contains the singular values of the
*> upper block; and D(NL+2:N) contains the singular values of
*> the lower block. On exit D(1:N) contains the singular values
*> of the modified matrix.
*> \endverbatim
*>
*> \param[in,out] ALPHA
*> \verbatim
*> ALPHA is DOUBLE PRECISION
*> Contains the diagonal element associated with the added row.
*> \endverbatim
*>
*> \param[in,out] BETA
*> \verbatim
*> BETA is DOUBLE PRECISION
*> Contains the off-diagonal element associated with the added
*> row.
*> \endverbatim
*>
*> \param[in,out] U
*> \verbatim
*> U is DOUBLE PRECISION array, dimension(LDU,N)
*> On entry U(1:NL, 1:NL) contains the left singular vectors of
*> the upper block; U(NL+2:N, NL+2:N) contains the left singular
*> vectors of the lower block. On exit U contains the left
*> singular vectors of the bidiagonal matrix.
*> \endverbatim
*>
*> \param[in] LDU
*> \verbatim
*> LDU is INTEGER
*> The leading dimension of the array U. LDU >= max( 1, N ).
*> \endverbatim
*>
*> \param[in,out] VT
*> \verbatim
*> VT is DOUBLE PRECISION array, dimension(LDVT,M)
*> where M = N + SQRE.
*> On entry VT(1:NL+1, 1:NL+1)**T contains the right singular
*> vectors of the upper block; VT(NL+2:M, NL+2:M)**T contains
*> the right singular vectors of the lower block. On exit
*> VT**T contains the right singular vectors of the
*> bidiagonal matrix.
*> \endverbatim
*>
*> \param[in] LDVT
*> \verbatim
*> LDVT is INTEGER
*> The leading dimension of the array VT. LDVT >= max( 1, M ).
*> \endverbatim
*>
*> \param[in,out] IDXQ
*> \verbatim
*> IDXQ is INTEGER array, dimension(N)
*> This contains the permutation which will reintegrate the
*> subproblem just solved back into sorted order, i.e.
*> D( IDXQ( I = 1, N ) ) will be in ascending order.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension( 4 * 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 2016
*
*> \ingroup OTHERauxiliary
*
*> \par Contributors:
* ==================
*>
*> Ming Gu and Huan Ren, Computer Science Division, University of
*> California at Berkeley, USA
*>
* =====================================================================
SUBROUTINE DLASD1( NL, NR, SQRE, D, ALPHA, BETA, U, LDU, VT, LDVT,
$ IDXQ, IWORK, WORK, INFO )
*
* -- LAPACK auxiliary routine (version 3.7.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* June 2016
*
* .. Scalar Arguments ..
INTEGER INFO, LDU, LDVT, NL, NR, SQRE
DOUBLE PRECISION ALPHA, BETA
* ..
* .. Array Arguments ..
INTEGER IDXQ( * ), IWORK( * )
DOUBLE PRECISION D( * ), U( LDU, * ), VT( LDVT, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
*
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
INTEGER COLTYP, I, IDX, IDXC, IDXP, IQ, ISIGMA, IU2,
$ IVT2, IZ, K, LDQ, LDU2, LDVT2, M, N, N1, N2
DOUBLE PRECISION ORGNRM
* ..
* .. External Subroutines ..
EXTERNAL DLAMRG, DLASCL, DLASD2, DLASD3, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
*
IF( NL.LT.1 ) THEN
INFO = -1
ELSE IF( NR.LT.1 ) THEN
INFO = -2
ELSE IF( ( SQRE.LT.0 ) .OR. ( SQRE.GT.1 ) ) THEN
INFO = -3
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DLASD1', -INFO )
RETURN
END IF
*
N = NL + NR + 1
M = N + SQRE
*
* The following values are for bookkeeping purposes only. They are
* integer pointers which indicate the portion of the workspace
* used by a particular array in DLASD2 and DLASD3.
*
LDU2 = N
LDVT2 = M
*
IZ = 1
ISIGMA = IZ + M
IU2 = ISIGMA + N
IVT2 = IU2 + LDU2*N
IQ = IVT2 + LDVT2*M
*
IDX = 1
IDXC = IDX + N
COLTYP = IDXC + N
IDXP = COLTYP + N
*
* Scale.
*
ORGNRM = MAX( ABS( ALPHA ), ABS( BETA ) )
D( NL+1 ) = ZERO
DO 10 I = 1, N
IF( ABS( D( I ) ).GT.ORGNRM ) THEN
ORGNRM = ABS( D( I ) )
END IF
10 CONTINUE
CALL DLASCL( 'G', 0, 0, ORGNRM, ONE, N, 1, D, N, INFO )
ALPHA = ALPHA / ORGNRM
BETA = BETA / ORGNRM
*
* Deflate singular values.
*
CALL DLASD2( NL, NR, SQRE, K, D, WORK( IZ ), ALPHA, BETA, U, LDU,
$ VT, LDVT, WORK( ISIGMA ), WORK( IU2 ), LDU2,
$ WORK( IVT2 ), LDVT2, IWORK( IDXP ), IWORK( IDX ),
$ IWORK( IDXC ), IDXQ, IWORK( COLTYP ), INFO )
*
* Solve Secular Equation and update singular vectors.
*
LDQ = K
CALL DLASD3( NL, NR, SQRE, K, D, WORK( IQ ), LDQ, WORK( ISIGMA ),
$ U, LDU, WORK( IU2 ), LDU2, VT, LDVT, WORK( IVT2 ),
$ LDVT2, IWORK( IDXC ), IWORK( COLTYP ), WORK( IZ ),
$ INFO )
*
* Report the convergence failure.
*
IF( INFO.NE.0 ) THEN
RETURN
END IF
*
* Unscale.
*
CALL DLASCL( 'G', 0, 0, ONE, ORGNRM, N, 1, D, N, INFO )
*
* Prepare the IDXQ sorting permutation.
*
N1 = K
N2 = N - K
CALL DLAMRG( N1, N2, D, 1, -1, IDXQ )
*
RETURN
*
* End of DLASD1
*
END