306 lines
11 KiB
FortranFixed
306 lines
11 KiB
FortranFixed
SUBROUTINE SLASD6( ICOMPQ, NL, NR, SQRE, D, VF, VL, ALPHA, BETA,
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$ IDXQ, PERM, GIVPTR, GIVCOL, LDGCOL, GIVNUM,
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$ LDGNUM, POLES, DIFL, DIFR, Z, K, C, S, WORK,
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$ IWORK, INFO )
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*
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* -- LAPACK auxiliary routine (version 3.1) --
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* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
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* November 2006
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*
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* .. Scalar Arguments ..
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INTEGER GIVPTR, ICOMPQ, INFO, K, LDGCOL, LDGNUM, NL,
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$ NR, SQRE
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REAL ALPHA, BETA, C, S
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* ..
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* .. Array Arguments ..
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INTEGER GIVCOL( LDGCOL, * ), IDXQ( * ), IWORK( * ),
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$ PERM( * )
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REAL D( * ), DIFL( * ), DIFR( * ),
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$ GIVNUM( LDGNUM, * ), POLES( LDGNUM, * ),
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$ VF( * ), VL( * ), WORK( * ), Z( * )
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* ..
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*
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* Purpose
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* =======
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*
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* SLASD6 computes the SVD of an updated upper bidiagonal matrix B
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* obtained by merging two smaller ones by appending a row. This
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* routine is used only for the problem which requires all singular
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* values and optionally singular vector matrices in factored form.
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* B is an N-by-M matrix with N = NL + NR + 1 and M = N + SQRE.
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* A related subroutine, SLASD1, handles the case in which all singular
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* values and singular vectors of the bidiagonal matrix are desired.
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*
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* SLASD6 computes the SVD as follows:
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*
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* ( D1(in) 0 0 0 )
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* B = U(in) * ( Z1' a Z2' b ) * VT(in)
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* ( 0 0 D2(in) 0 )
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*
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* = U(out) * ( D(out) 0) * VT(out)
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*
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* where Z' = (Z1' a Z2' b) = u' VT', and u is a vector of dimension M
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* with ALPHA and BETA in the NL+1 and NL+2 th entries and zeros
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* elsewhere; and the entry b is empty if SQRE = 0.
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*
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* The singular values of B can be computed using D1, D2, the first
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* components of all the right singular vectors of the lower block, and
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* the last components of all the right singular vectors of the upper
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* block. These components are stored and updated in VF and VL,
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* respectively, in SLASD6. Hence U and VT are not explicitly
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* referenced.
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*
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* The singular values are stored in D. The algorithm consists of two
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* stages:
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*
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* The first stage consists of deflating the size of the problem
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* when there are multiple singular values or if there is a zero
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* in the Z vector. For each such occurence the dimension of the
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* secular equation problem is reduced by one. This stage is
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* performed by the routine SLASD7.
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*
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* The second stage consists of calculating the updated
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* singular values. This is done by finding the roots of the
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* secular equation via the routine SLASD4 (as called by SLASD8).
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* This routine also updates VF and VL and computes the distances
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* between the updated singular values and the old singular
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* values.
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*
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* SLASD6 is called from SLASDA.
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*
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* Arguments
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* =========
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*
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* ICOMPQ (input) INTEGER
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* Specifies whether singular vectors are to be computed in
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* factored form:
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* = 0: Compute singular values only.
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* = 1: Compute singular vectors in factored form as well.
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*
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* NL (input) INTEGER
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* The row dimension of the upper block. NL >= 1.
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*
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* NR (input) INTEGER
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* The row dimension of the lower block. NR >= 1.
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*
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* SQRE (input) INTEGER
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* = 0: the lower block is an NR-by-NR square matrix.
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* = 1: the lower block is an NR-by-(NR+1) rectangular matrix.
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*
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* The bidiagonal matrix has row dimension N = NL + NR + 1,
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* and column dimension M = N + SQRE.
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*
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* D (input/output) REAL array, dimension (NL+NR+1).
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* On entry D(1:NL,1:NL) contains the singular values of the
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* upper block, and D(NL+2:N) contains the singular values
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* of the lower block. On exit D(1:N) contains the singular
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* values of the modified matrix.
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*
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* VF (input/output) REAL array, dimension (M)
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* On entry, VF(1:NL+1) contains the first components of all
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* right singular vectors of the upper block; and VF(NL+2:M)
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* contains the first components of all right singular vectors
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* of the lower block. On exit, VF contains the first components
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* of all right singular vectors of the bidiagonal matrix.
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*
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* VL (input/output) REAL array, dimension (M)
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* On entry, VL(1:NL+1) contains the last components of all
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* right singular vectors of the upper block; and VL(NL+2:M)
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* contains the last components of all right singular vectors of
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* the lower block. On exit, VL contains the last components of
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* all right singular vectors of the bidiagonal matrix.
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*
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* ALPHA (input/output) REAL
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* Contains the diagonal element associated with the added row.
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*
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* BETA (input/output) REAL
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* Contains the off-diagonal element associated with the added
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* row.
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*
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* IDXQ (output) INTEGER array, dimension (N)
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* This contains the permutation which will reintegrate the
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* subproblem just solved back into sorted order, i.e.
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* D( IDXQ( I = 1, N ) ) will be in ascending order.
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*
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* PERM (output) INTEGER array, dimension ( N )
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* The permutations (from deflation and sorting) to be applied
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* to each block. Not referenced if ICOMPQ = 0.
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*
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* GIVPTR (output) INTEGER
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* The number of Givens rotations which took place in this
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* subproblem. Not referenced if ICOMPQ = 0.
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*
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* GIVCOL (output) INTEGER array, dimension ( LDGCOL, 2 )
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* Each pair of numbers indicates a pair of columns to take place
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* in a Givens rotation. Not referenced if ICOMPQ = 0.
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*
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* LDGCOL (input) INTEGER
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* leading dimension of GIVCOL, must be at least N.
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*
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* GIVNUM (output) REAL array, dimension ( LDGNUM, 2 )
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* Each number indicates the C or S value to be used in the
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* corresponding Givens rotation. Not referenced if ICOMPQ = 0.
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*
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* LDGNUM (input) INTEGER
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* The leading dimension of GIVNUM and POLES, must be at least N.
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*
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* POLES (output) REAL array, dimension ( LDGNUM, 2 )
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* On exit, POLES(1,*) is an array containing the new singular
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* values obtained from solving the secular equation, and
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* POLES(2,*) is an array containing the poles in the secular
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* equation. Not referenced if ICOMPQ = 0.
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*
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* DIFL (output) REAL array, dimension ( N )
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* On exit, DIFL(I) is the distance between I-th updated
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* (undeflated) singular value and the I-th (undeflated) old
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* singular value.
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*
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* DIFR (output) REAL array,
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* dimension ( LDGNUM, 2 ) if ICOMPQ = 1 and
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* dimension ( N ) if ICOMPQ = 0.
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* On exit, DIFR(I, 1) is the distance between I-th updated
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* (undeflated) singular value and the I+1-th (undeflated) old
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* singular value.
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*
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* If ICOMPQ = 1, DIFR(1:K,2) is an array containing the
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* normalizing factors for the right singular vector matrix.
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*
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* See SLASD8 for details on DIFL and DIFR.
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*
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* Z (output) REAL array, dimension ( M )
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* The first elements of this array contain the components
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* of the deflation-adjusted updating row vector.
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*
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* K (output) INTEGER
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* Contains the dimension of the non-deflated matrix,
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* This is the order of the related secular equation. 1 <= K <=N.
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*
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* C (output) REAL
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* C contains garbage if SQRE =0 and the C-value of a Givens
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* rotation related to the right null space if SQRE = 1.
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*
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* S (output) REAL
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* S contains garbage if SQRE =0 and the S-value of a Givens
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* rotation related to the right null space if SQRE = 1.
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*
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* WORK (workspace) REAL array, dimension ( 4 * M )
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*
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* IWORK (workspace) INTEGER array, dimension ( 3 * N )
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*
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* INFO (output) INTEGER
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* = 0: successful exit.
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* < 0: if INFO = -i, the i-th argument had an illegal value.
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* > 0: if INFO = 1, an singular value did not converge
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*
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* Further Details
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* ===============
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*
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* Based on contributions by
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* Ming Gu and Huan Ren, Computer Science Division, University of
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* California at Berkeley, USA
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*
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* =====================================================================
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*
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* .. Parameters ..
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REAL ONE, ZERO
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PARAMETER ( ONE = 1.0E+0, ZERO = 0.0E+0 )
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* ..
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* .. Local Scalars ..
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INTEGER I, IDX, IDXC, IDXP, ISIGMA, IVFW, IVLW, IW, M,
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$ N, N1, N2
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REAL ORGNRM
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* ..
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* .. External Subroutines ..
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EXTERNAL SCOPY, SLAMRG, SLASCL, SLASD7, SLASD8, XERBLA
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC ABS, MAX
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* ..
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* .. Executable Statements ..
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*
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* Test the input parameters.
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*
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INFO = 0
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N = NL + NR + 1
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M = N + SQRE
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*
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IF( ( ICOMPQ.LT.0 ) .OR. ( ICOMPQ.GT.1 ) ) THEN
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INFO = -1
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ELSE IF( NL.LT.1 ) THEN
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INFO = -2
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ELSE IF( NR.LT.1 ) THEN
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INFO = -3
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ELSE IF( ( SQRE.LT.0 ) .OR. ( SQRE.GT.1 ) ) THEN
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INFO = -4
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ELSE IF( LDGCOL.LT.N ) THEN
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INFO = -14
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ELSE IF( LDGNUM.LT.N ) THEN
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INFO = -16
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END IF
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IF( INFO.NE.0 ) THEN
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CALL XERBLA( 'SLASD6', -INFO )
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RETURN
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END IF
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*
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* The following values are for bookkeeping purposes only. They are
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* integer pointers which indicate the portion of the workspace
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* used by a particular array in SLASD7 and SLASD8.
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*
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ISIGMA = 1
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IW = ISIGMA + N
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IVFW = IW + M
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IVLW = IVFW + M
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*
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IDX = 1
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IDXC = IDX + N
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IDXP = IDXC + N
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*
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* Scale.
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*
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ORGNRM = MAX( ABS( ALPHA ), ABS( BETA ) )
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D( NL+1 ) = ZERO
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DO 10 I = 1, N
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IF( ABS( D( I ) ).GT.ORGNRM ) THEN
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ORGNRM = ABS( D( I ) )
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END IF
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10 CONTINUE
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CALL SLASCL( 'G', 0, 0, ORGNRM, ONE, N, 1, D, N, INFO )
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ALPHA = ALPHA / ORGNRM
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BETA = BETA / ORGNRM
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*
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* Sort and Deflate singular values.
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*
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CALL SLASD7( ICOMPQ, NL, NR, SQRE, K, D, Z, WORK( IW ), VF,
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$ WORK( IVFW ), VL, WORK( IVLW ), ALPHA, BETA,
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$ WORK( ISIGMA ), IWORK( IDX ), IWORK( IDXP ), IDXQ,
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$ PERM, GIVPTR, GIVCOL, LDGCOL, GIVNUM, LDGNUM, C, S,
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$ INFO )
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*
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* Solve Secular Equation, compute DIFL, DIFR, and update VF, VL.
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*
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CALL SLASD8( ICOMPQ, K, D, Z, VF, VL, DIFL, DIFR, LDGNUM,
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$ WORK( ISIGMA ), WORK( IW ), INFO )
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*
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* Save the poles if ICOMPQ = 1.
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*
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IF( ICOMPQ.EQ.1 ) THEN
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CALL SCOPY( K, D, 1, POLES( 1, 1 ), 1 )
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CALL SCOPY( K, WORK( ISIGMA ), 1, POLES( 1, 2 ), 1 )
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END IF
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*
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* Unscale.
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*
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CALL SLASCL( 'G', 0, 0, ONE, ORGNRM, N, 1, D, N, INFO )
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*
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* Prepare the IDXQ sorting permutation.
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*
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N1 = K
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N2 = N - K
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CALL SLAMRG( N1, N2, D, 1, -1, IDXQ )
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*
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RETURN
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*
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* End of SLASD6
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*
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END
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