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SUBROUTINE <a name="DLASD6.1"></a><a href="dlasd6.f.html#DLASD6.1">DLASD6</a>( 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 )
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> -- LAPACK auxiliary routine (version 3.1) --
</span><span class="comment">*</span><span class="comment"> Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
</span><span class="comment">*</span><span class="comment"> November 2006
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> .. Scalar Arguments ..
</span> INTEGER GIVPTR, ICOMPQ, INFO, K, LDGCOL, LDGNUM, NL,
$ NR, SQRE
DOUBLE PRECISION ALPHA, BETA, C, S
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. Array Arguments ..
</span> INTEGER GIVCOL( LDGCOL, * ), IDXQ( * ), IWORK( * ),
$ PERM( * )
DOUBLE PRECISION D( * ), DIFL( * ), DIFR( * ),
$ GIVNUM( LDGNUM, * ), POLES( LDGNUM, * ),
$ VF( * ), VL( * ), WORK( * ), Z( * )
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Purpose
</span><span class="comment">*</span><span class="comment"> =======
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> <a name="DLASD6.26"></a><a href="dlasd6.f.html#DLASD6.1">DLASD6</a> computes the SVD of an updated upper bidiagonal matrix B
</span><span class="comment">*</span><span class="comment"> obtained by merging two smaller ones by appending a row. This
</span><span class="comment">*</span><span class="comment"> routine is used only for the problem which requires all singular
</span><span class="comment">*</span><span class="comment"> values and optionally singular vector matrices in factored form.
</span><span class="comment">*</span><span class="comment"> B is an N-by-M matrix with N = NL + NR + 1 and M = N + SQRE.
</span><span class="comment">*</span><span class="comment"> A related subroutine, <a name="DLASD1.31"></a><a href="dlasd1.f.html#DLASD1.1">DLASD1</a>, handles the case in which all singular
</span><span class="comment">*</span><span class="comment"> values and singular vectors of the bidiagonal matrix are desired.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> <a name="DLASD6.34"></a><a href="dlasd6.f.html#DLASD6.1">DLASD6</a> computes the SVD as follows:
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> ( D1(in) 0 0 0 )
</span><span class="comment">*</span><span class="comment"> B = U(in) * ( Z1' a Z2' b ) * VT(in)
</span><span class="comment">*</span><span class="comment"> ( 0 0 D2(in) 0 )
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> = U(out) * ( D(out) 0) * VT(out)
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> where Z' = (Z1' a Z2' b) = u' VT', and u is a vector of dimension M
</span><span class="comment">*</span><span class="comment"> with ALPHA and BETA in the NL+1 and NL+2 th entries and zeros
</span><span class="comment">*</span><span class="comment"> elsewhere; and the entry b is empty if SQRE = 0.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> The singular values of B can be computed using D1, D2, the first
</span><span class="comment">*</span><span class="comment"> components of all the right singular vectors of the lower block, and
</span><span class="comment">*</span><span class="comment"> the last components of all the right singular vectors of the upper
</span><span class="comment">*</span><span class="comment"> block. These components are stored and updated in VF and VL,
</span><span class="comment">*</span><span class="comment"> respectively, in <a name="DLASD6.50"></a><a href="dlasd6.f.html#DLASD6.1">DLASD6</a>. Hence U and VT are not explicitly
</span><span class="comment">*</span><span class="comment"> referenced.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> The singular values are stored in D. The algorithm consists of two
</span><span class="comment">*</span><span class="comment"> stages:
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> The first stage consists of deflating the size of the problem
</span><span class="comment">*</span><span class="comment"> when there are multiple singular values or if there is a zero
</span><span class="comment">*</span><span class="comment"> in the Z vector. For each such occurence the dimension of the
</span><span class="comment">*</span><span class="comment"> secular equation problem is reduced by one. This stage is
</span><span class="comment">*</span><span class="comment"> performed by the routine <a name="DLASD7.60"></a><a href="dlasd7.f.html#DLASD7.1">DLASD7</a>.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> The second stage consists of calculating the updated
</span><span class="comment">*</span><span class="comment"> singular values. This is done by finding the roots of the
</span><span class="comment">*</span><span class="comment"> secular equation via the routine <a name="DLASD4.64"></a><a href="dlasd4.f.html#DLASD4.1">DLASD4</a> (as called by <a name="DLASD8.64"></a><a href="dlasd8.f.html#DLASD8.1">DLASD8</a>).
</span><span class="comment">*</span><span class="comment"> This routine also updates VF and VL and computes the distances
</span><span class="comment">*</span><span class="comment"> between the updated singular values and the old singular
</span><span class="comment">*</span><span class="comment"> values.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> <a name="DLASD6.69"></a><a href="dlasd6.f.html#DLASD6.1">DLASD6</a> is called from <a name="DLASDA.69"></a><a href="dlasda.f.html#DLASDA.1">DLASDA</a>.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Arguments
</span><span class="comment">*</span><span class="comment"> =========
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> ICOMPQ (input) INTEGER
</span><span class="comment">*</span><span class="comment"> Specifies whether singular vectors are to be computed in
</span><span class="comment">*</span><span class="comment"> factored form:
</span><span class="comment">*</span><span class="comment"> = 0: Compute singular values only.
</span><span class="comment">*</span><span class="comment"> = 1: Compute singular vectors in factored form as well.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> NL (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The row dimension of the upper block. NL >= 1.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> NR (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The row dimension of the lower block. NR >= 1.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> SQRE (input) INTEGER
</span><span class="comment">*</span><span class="comment"> = 0: the lower block is an NR-by-NR square matrix.
</span><span class="comment">*</span><span class="comment"> = 1: the lower block is an NR-by-(NR+1) rectangular matrix.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> The bidiagonal matrix has row dimension N = NL + NR + 1,
</span><span class="comment">*</span><span class="comment"> and column dimension M = N + SQRE.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> D (input/output) DOUBLE PRECISION array, dimension ( NL+NR+1 ).
</span><span class="comment">*</span><span class="comment"> On entry D(1:NL,1:NL) contains the singular values of the
</span><span class="comment">*</span><span class="comment"> upper block, and D(NL+2:N) contains the singular values
</span><span class="comment">*</span><span class="comment"> of the lower block. On exit D(1:N) contains the singular
</span><span class="comment">*</span><span class="comment"> values of the modified matrix.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> VF (input/output) DOUBLE PRECISION array, dimension ( M )
</span><span class="comment">*</span><span class="comment"> On entry, VF(1:NL+1) contains the first components of all
</span><span class="comment">*</span><span class="comment"> right singular vectors of the upper block; and VF(NL+2:M)
</span><span class="comment">*</span><span class="comment"> contains the first components of all right singular vectors
</span><span class="comment">*</span><span class="comment"> of the lower block. On exit, VF contains the first components
</span><span class="comment">*</span><span class="comment"> of all right singular vectors of the bidiagonal matrix.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> VL (input/output) DOUBLE PRECISION array, dimension ( M )
</span><span class="comment">*</span><span class="comment"> On entry, VL(1:NL+1) contains the last components of all
</span><span class="comment">*</span><span class="comment"> right singular vectors of the upper block; and VL(NL+2:M)
</span><span class="comment">*</span><span class="comment"> contains the last components of all right singular vectors of
</span><span class="comment">*</span><span class="comment"> the lower block. On exit, VL contains the last components of
</span><span class="comment">*</span><span class="comment"> all right singular vectors of the bidiagonal matrix.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> ALPHA (input/output) DOUBLE PRECISION
</span><span class="comment">*</span><span class="comment"> Contains the diagonal element associated with the added row.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> BETA (input/output) DOUBLE PRECISION
</span><span class="comment">*</span><span class="comment"> Contains the off-diagonal element associated with the added
</span><span class="comment">*</span><span class="comment"> row.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> IDXQ (output) INTEGER array, dimension ( N )
</span><span class="comment">*</span><span class="comment"> This contains the permutation which will reintegrate the
</span><span class="comment">*</span><span class="comment"> subproblem just solved back into sorted order, i.e.
</span><span class="comment">*</span><span class="comment"> D( IDXQ( I = 1, N ) ) will be in ascending order.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> PERM (output) INTEGER array, dimension ( N )
</span><span class="comment">*</span><span class="comment"> The permutations (from deflation and sorting) to be applied
</span><span class="comment">*</span><span class="comment"> to each block. Not referenced if ICOMPQ = 0.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> GIVPTR (output) INTEGER
</span><span class="comment">*</span><span class="comment"> The number of Givens rotations which took place in this
</span><span class="comment">*</span><span class="comment"> subproblem. Not referenced if ICOMPQ = 0.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> GIVCOL (output) INTEGER array, dimension ( LDGCOL, 2 )
</span><span class="comment">*</span><span class="comment"> Each pair of numbers indicates a pair of columns to take place
</span><span class="comment">*</span><span class="comment"> in a Givens rotation. Not referenced if ICOMPQ = 0.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> LDGCOL (input) INTEGER
</span><span class="comment">*</span><span class="comment"> leading dimension of GIVCOL, must be at least N.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> GIVNUM (output) DOUBLE PRECISION array, dimension ( LDGNUM, 2 )
</span><span class="comment">*</span><span class="comment"> Each number indicates the C or S value to be used in the
</span><span class="comment">*</span><span class="comment"> corresponding Givens rotation. Not referenced if ICOMPQ = 0.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> LDGNUM (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The leading dimension of GIVNUM and POLES, must be at least N.
</span><span class="comment">*</span><span class="comment">
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