*> \brief \b DLASD6 computes the SVD of an updated upper bidiagonal matrix obtained by merging two smaller ones by appending a row. Used by sbdsdc.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
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*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLASD6( ICOMPQ, NL, NR, SQRE, D, VF, VL, ALPHA, BETA,
* IDXQ, PERM, GIVPTR, GIVCOL, LDGCOL, GIVNUM,
* LDGNUM, POLES, DIFL, DIFR, Z, K, C, S, WORK,
* IWORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER GIVPTR, ICOMPQ, INFO, K, LDGCOL, LDGNUM, NL,
* $ NR, SQRE
* DOUBLE PRECISION ALPHA, BETA, C, S
* ..
* .. Array Arguments ..
* INTEGER GIVCOL( LDGCOL, * ), IDXQ( * ), IWORK( * ),
* $ PERM( * )
* DOUBLE PRECISION D( * ), DIFL( * ), DIFR( * ),
* $ GIVNUM( LDGNUM, * ), POLES( LDGNUM, * ),
* $ VF( * ), VL( * ), WORK( * ), Z( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLASD6 computes the SVD of an updated upper bidiagonal matrix B
*> obtained by merging two smaller ones by appending a row. This
*> routine is used only for the problem which requires all singular
*> values and optionally singular vector matrices in factored form.
*> B is an N-by-M matrix with N = NL + NR + 1 and M = N + SQRE.
*> A related subroutine, DLASD1, handles the case in which all singular
*> values and singular vectors of the bidiagonal matrix are desired.
*>
*> DLASD6 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 singular values of B can be computed using D1, D2, the first
*> components of all the right singular vectors of the lower block, and
*> the last components of all the right singular vectors of the upper
*> block. These components are stored and updated in VF and VL,
*> respectively, in DLASD6. Hence U and VT are not explicitly
*> referenced.
*>
*> The singular values are stored in D. The algorithm consists of two
*> stages:
*>
*> The first stage consists of deflating the size of the problem
*> when there are multiple singular values or if there is a zero
*> 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 DLASD7.
*>
*> The second stage consists of calculating the updated
*> singular values. This is done by finding the roots of the
*> secular equation via the routine DLASD4 (as called by DLASD8).
*> This routine also updates VF and VL and computes the distances
*> between the updated singular values and the old singular
*> values.
*>
*> DLASD6 is called from DLASDA.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] ICOMPQ
*> \verbatim
*> ICOMPQ is INTEGER
*> Specifies whether singular vectors are to be computed in
*> factored form:
*> = 0: Compute singular values only.
*> = 1: Compute singular vectors in factored form as well.
*> \endverbatim
*>
*> \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 ( 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] VF
*> \verbatim
*> VF is DOUBLE PRECISION array, dimension ( M )
*> On entry, VF(1:NL+1) contains the first components of all
*> right singular vectors of the upper block; and VF(NL+2:M)
*> contains the first components of all right singular vectors
*> of the lower block. On exit, VF contains the first components
*> of all right singular vectors of the bidiagonal matrix.
*> \endverbatim
*>
*> \param[in,out] VL
*> \verbatim
*> VL is DOUBLE PRECISION array, dimension ( M )
*> On entry, VL(1:NL+1) contains the last components of all
*> right singular vectors of the upper block; and VL(NL+2:M)
*> contains the last components of all right singular vectors of
*> the lower block. On exit, VL contains the last components of
*> all right singular vectors of the bidiagonal 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] 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] PERM
*> \verbatim
*> PERM is INTEGER array, dimension ( N )
*> The permutations (from deflation and sorting) to be applied
*> to each block. Not referenced if ICOMPQ = 0.
*> \endverbatim
*>
*> \param[out] GIVPTR
*> \verbatim
*> GIVPTR is INTEGER
*> The number of Givens rotations which took place in this
*> subproblem. Not referenced if ICOMPQ = 0.
*> \endverbatim
*>
*> \param[out] GIVCOL
*> \verbatim
*> GIVCOL is INTEGER array, dimension ( LDGCOL, 2 )
*> Each pair of numbers indicates a pair of columns to take place
*> in a Givens rotation. Not referenced if ICOMPQ = 0.
*> \endverbatim
*>
*> \param[in] LDGCOL
*> \verbatim
*> LDGCOL is INTEGER
*> leading dimension of GIVCOL, must be at least N.
*> \endverbatim
*>
*> \param[out] GIVNUM
*> \verbatim
*> GIVNUM is DOUBLE PRECISION array, dimension ( LDGNUM, 2 )
*> Each number indicates the C or S value to be used in the
*> corresponding Givens rotation. Not referenced if ICOMPQ = 0.
*> \endverbatim
*>
*> \param[in] LDGNUM
*> \verbatim
*> LDGNUM is INTEGER
*> The leading dimension of GIVNUM and POLES, must be at least N.
*> \endverbatim
*>
*> \param[out] POLES
*> \verbatim
*> POLES is DOUBLE PRECISION array, dimension ( LDGNUM, 2 )
*> On exit, POLES(1,*) is an array containing the new singular
*> values obtained from solving the secular equation, and
*> POLES(2,*) is an array containing the poles in the secular
*> equation. Not referenced if ICOMPQ = 0.
*> \endverbatim
*>
*> \param[out] DIFL
*> \verbatim
*> DIFL is DOUBLE PRECISION array, dimension ( N )
*> On exit, DIFL(I) is the distance between I-th updated
*> (undeflated) singular value and the I-th (undeflated) old
*> singular value.
*> \endverbatim
*>
*> \param[out] DIFR
*> \verbatim
*> DIFR is DOUBLE PRECISION array,
*> dimension ( LDDIFR, 2 ) if ICOMPQ = 1 and
*> dimension ( K ) if ICOMPQ = 0.
*> On exit, DIFR(I,1) = D(I) - DSIGMA(I+1), DIFR(K,1) is not
*> defined and will not be referenced.
*>
*> If ICOMPQ = 1, DIFR(1:K,2) is an array containing the
*> normalizing factors for the right singular vector matrix.
*>
*> See DLASD8 for details on DIFL and DIFR.
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension ( M )
*> The first elements of this array contain the components
*> of the deflation-adjusted updating row vector.
*> \endverbatim
*>
*> \param[out] K
*> \verbatim
*> K is INTEGER
*> Contains the dimension of the non-deflated matrix,
*> This is the order of the related secular equation. 1 <= K <=N.
*> \endverbatim
*>
*> \param[out] C
*> \verbatim
*> C is DOUBLE PRECISION
*> C contains garbage if SQRE =0 and the C-value of a Givens
*> rotation related to the right null space if SQRE = 1.
*> \endverbatim
*>
*> \param[out] S
*> \verbatim
*> S is DOUBLE PRECISION
*> S contains garbage if SQRE =0 and the S-value of a Givens
*> rotation related to the right null space if SQRE = 1.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension ( 4 * M )
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension ( 3 * N )
*> \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 DLASD6( ICOMPQ, NL, NR, SQRE, D, VF, VL, ALPHA, BETA,
$ IDXQ, PERM, GIVPTR, GIVCOL, LDGCOL, GIVNUM,
$ LDGNUM, POLES, DIFL, DIFR, Z, K, C, S, WORK,
$ IWORK, 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 GIVPTR, ICOMPQ, INFO, K, LDGCOL, LDGNUM, NL,
$ NR, SQRE
DOUBLE PRECISION ALPHA, BETA, C, S
* ..
* .. Array Arguments ..
INTEGER GIVCOL( LDGCOL, * ), IDXQ( * ), IWORK( * ),
$ PERM( * )
DOUBLE PRECISION D( * ), DIFL( * ), DIFR( * ),
$ GIVNUM( LDGNUM, * ), POLES( LDGNUM, * ),
$ VF( * ), VL( * ), WORK( * ), Z( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I, IDX, IDXC, IDXP, ISIGMA, IVFW, IVLW, IW, M,
$ N, N1, N2
DOUBLE PRECISION ORGNRM
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DLAMRG, DLASCL, DLASD7, DLASD8, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
N = NL + NR + 1
M = N + SQRE
*
IF( ( ICOMPQ.LT.0 ) .OR. ( ICOMPQ.GT.1 ) ) THEN
INFO = -1
ELSE IF( NL.LT.1 ) THEN
INFO = -2
ELSE IF( NR.LT.1 ) THEN
INFO = -3
ELSE IF( ( SQRE.LT.0 ) .OR. ( SQRE.GT.1 ) ) THEN
INFO = -4
ELSE IF( LDGCOL.LT.N ) THEN
INFO = -14
ELSE IF( LDGNUM.LT.N ) THEN
INFO = -16
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DLASD6', -INFO )
RETURN
END IF
*
* 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 DLASD7 and DLASD8.
*
ISIGMA = 1
IW = ISIGMA + N
IVFW = IW + M
IVLW = IVFW + M
*
IDX = 1
IDXC = IDX + N
IDXP = IDXC + 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
*
* Sort and Deflate singular values.
*
CALL DLASD7( ICOMPQ, NL, NR, SQRE, K, D, Z, WORK( IW ), VF,
$ WORK( IVFW ), VL, WORK( IVLW ), ALPHA, BETA,
$ WORK( ISIGMA ), IWORK( IDX ), IWORK( IDXP ), IDXQ,
$ PERM, GIVPTR, GIVCOL, LDGCOL, GIVNUM, LDGNUM, C, S,
$ INFO )
*
* Solve Secular Equation, compute DIFL, DIFR, and update VF, VL.
*
CALL DLASD8( ICOMPQ, K, D, Z, VF, VL, DIFL, DIFR, LDGNUM,
$ WORK( ISIGMA ), WORK( IW ), INFO )
*
* Report the possible convergence failure.
*
IF( INFO.NE.0 ) THEN
RETURN
END IF
*
* Save the poles if ICOMPQ = 1.
*
IF( ICOMPQ.EQ.1 ) THEN
CALL DCOPY( K, D, 1, POLES( 1, 1 ), 1 )
CALL DCOPY( K, WORK( ISIGMA ), 1, POLES( 1, 2 ), 1 )
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 DLASD6
*
END