SUBROUTINE SLASD6( 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 3.1) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2006
*
* .. Scalar Arguments ..
INTEGER GIVPTR, ICOMPQ, INFO, K, LDGCOL, LDGNUM, NL,
$ NR, SQRE
REAL ALPHA, BETA, C, S
* ..
* .. Array Arguments ..
INTEGER GIVCOL( LDGCOL, * ), IDXQ( * ), IWORK( * ),
$ PERM( * )
REAL D( * ), DIFL( * ), DIFR( * ),
$ GIVNUM( LDGNUM, * ), POLES( LDGNUM, * ),
$ VF( * ), VL( * ), WORK( * ), Z( * )
* ..
*
* Purpose
* =======
*
* SLASD6 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, SLASD1, handles the case in which all singular
* values and singular vectors of the bidiagonal matrix are desired.
*
* SLASD6 computes the SVD as follows:
*
* ( D1(in) 0 0 0 )
* B = U(in) * ( Z1' a Z2' b ) * VT(in)
* ( 0 0 D2(in) 0 )
*
* = U(out) * ( D(out) 0) * VT(out)
*
* where Z' = (Z1' a Z2' b) = u' VT', 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 SLASD6. 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 occurence the dimension of the
* secular equation problem is reduced by one. This stage is
* performed by the routine SLASD7.
*
* The second stage consists of calculating the updated
* singular values. This is done by finding the roots of the
* secular equation via the routine SLASD4 (as called by SLASD8).
* This routine also updates VF and VL and computes the distances
* between the updated singular values and the old singular
* values.
*
* SLASD6 is called from SLASDA.
*
* Arguments
* =========
*
* ICOMPQ (input) 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.
*
* NL (input) INTEGER
* The row dimension of the upper block. NL >= 1.
*
* NR (input) INTEGER
* The row dimension of the lower block. NR >= 1.
*
* SQRE (input) 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.
*
* D (input/output) REAL 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.
*
* VF (input/output) REAL 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.
*
* VL (input/output) REAL 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.
*
* ALPHA (input/output) REAL
* Contains the diagonal element associated with the added row.
*
* BETA (input/output) REAL
* Contains the off-diagonal element associated with the added
* row.
*
* IDXQ (output) 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.
*
* PERM (output) INTEGER array, dimension ( N )
* The permutations (from deflation and sorting) to be applied
* to each block. Not referenced if ICOMPQ = 0.
*
* GIVPTR (output) INTEGER
* The number of Givens rotations which took place in this
* subproblem. Not referenced if ICOMPQ = 0.
*
* GIVCOL (output) 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.
*
* LDGCOL (input) INTEGER
* leading dimension of GIVCOL, must be at least N.
*
* GIVNUM (output) REAL 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.
*
* LDGNUM (input) INTEGER
* The leading dimension of GIVNUM and POLES, must be at least N.
*
* POLES (output) REAL 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.
*
* DIFL (output) REAL 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.
*
* DIFR (output) REAL array,
* dimension ( LDGNUM, 2 ) if ICOMPQ = 1 and
* dimension ( N ) if ICOMPQ = 0.
* On exit, DIFR(I, 1) is the distance between I-th updated
* (undeflated) singular value and the I+1-th (undeflated) old
* singular value.
*
* If ICOMPQ = 1, DIFR(1:K,2) is an array containing the
* normalizing factors for the right singular vector matrix.
*
* See SLASD8 for details on DIFL and DIFR.
*
* Z (output) REAL array, dimension ( M )
* The first elements of this array contain the components
* of the deflation-adjusted updating row vector.
*
* K (output) INTEGER
* Contains the dimension of the non-deflated matrix,
* This is the order of the related secular equation. 1 <= K <=N.
*
* C (output) REAL
* C contains garbage if SQRE =0 and the C-value of a Givens
* rotation related to the right null space if SQRE = 1.
*
* S (output) REAL
* S contains garbage if SQRE =0 and the S-value of a Givens
* rotation related to the right null space if SQRE = 1.
*
* WORK (workspace) REAL array, dimension ( 4 * M )
*
* IWORK (workspace) INTEGER array, dimension ( 3 * N )
*
* INFO (output) INTEGER
* = 0: successful exit.
* < 0: if INFO = -i, the i-th argument had an illegal value.
* > 0: if INFO = 1, an singular value did not converge
*
* Further Details
* ===============
*
* Based on contributions by
* Ming Gu and Huan Ren, Computer Science Division, University of
* California at Berkeley, USA
*
* =====================================================================
*
* .. Parameters ..
REAL ONE, ZERO
PARAMETER ( ONE = 1.0E+0, ZERO = 0.0E+0 )
* ..
* .. Local Scalars ..
INTEGER I, IDX, IDXC, IDXP, ISIGMA, IVFW, IVLW, IW, M,
$ N, N1, N2
REAL ORGNRM
* ..
* .. External Subroutines ..
EXTERNAL SCOPY, SLAMRG, SLASCL, SLASD7, SLASD8, 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( 'SLASD6', -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 SLASD7 and SLASD8.
*
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 SLASCL( 'G', 0, 0, ORGNRM, ONE, N, 1, D, N, INFO )
ALPHA = ALPHA / ORGNRM
BETA = BETA / ORGNRM
*
* Sort and Deflate singular values.
*
CALL SLASD7( 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 SLASD8( ICOMPQ, K, D, Z, VF, VL, DIFL, DIFR, LDGNUM,
$ WORK( ISIGMA ), WORK( IW ), INFO )
*
* Save the poles if ICOMPQ = 1.
*
IF( ICOMPQ.EQ.1 ) THEN
CALL SCOPY( K, D, 1, POLES( 1, 1 ), 1 )
CALL SCOPY( K, WORK( ISIGMA ), 1, POLES( 1, 2 ), 1 )
END IF
*
* Unscale.
*
CALL SLASCL( 'G', 0, 0, ONE, ORGNRM, N, 1, D, N, INFO )
*
* Prepare the IDXQ sorting permutation.
*
N1 = K
N2 = N - K
CALL SLAMRG( N1, N2, D, 1, -1, IDXQ )
*
RETURN
*
* End of SLASD6
*
END