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SUBROUTINE <a name="DLASD1.1"></a><a href="dlasd1.f.html#DLASD1.1">DLASD1</a>( NL, NR, SQRE, D, ALPHA, BETA, U, LDU, VT, LDVT,
$ IDXQ, IWORK, WORK, 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 INFO, LDU, LDVT, NL, NR, SQRE
DOUBLE PRECISION ALPHA, BETA
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. Array Arguments ..
</span> INTEGER IDXQ( * ), IWORK( * )
DOUBLE PRECISION D( * ), U( LDU, * ), VT( LDVT, * ), WORK( * )
<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="DLASD1.20"></a><a href="dlasd1.f.html#DLASD1.1">DLASD1</a> computes the SVD of an upper bidiagonal N-by-M matrix B,
</span><span class="comment">*</span><span class="comment"> where N = NL + NR + 1 and M = N + SQRE. <a name="DLASD1.21"></a><a href="dlasd1.f.html#DLASD1.1">DLASD1</a> is called from <a name="DLASD0.21"></a><a href="dlasd0.f.html#DLASD0.1">DLASD0</a>.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> A related subroutine <a name="DLASD7.23"></a><a href="dlasd7.f.html#DLASD7.1">DLASD7</a> handles the case in which the singular
</span><span class="comment">*</span><span class="comment"> values (and the singular vectors in factored form) are desired.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> <a name="DLASD1.26"></a><a href="dlasd1.f.html#DLASD1.1">DLASD1</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 left singular vectors of the original matrix are stored in U, and
</span><span class="comment">*</span><span class="comment"> the transpose of the right singular vectors are stored in VT, and the
</span><span class="comment">*</span><span class="comment"> singular values are in D. The algorithm consists of three 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 when there are zeros in
</span><span class="comment">*</span><span class="comment"> 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="DLASD2.46"></a><a href="dlasd2.f.html#DLASD2.1">DLASD2</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 square roots of the
</span><span class="comment">*</span><span class="comment"> roots of the secular equation via the routine <a name="DLASD4.50"></a><a href="dlasd4.f.html#DLASD4.1">DLASD4</a> (as called
</span><span class="comment">*</span><span class="comment"> by <a name="DLASD3.51"></a><a href="dlasd3.f.html#DLASD3.1">DLASD3</a>). This routine also calculates the singular vectors of
</span><span class="comment">*</span><span class="comment"> the current problem.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> The final stage consists of computing the updated singular vectors
</span><span class="comment">*</span><span class="comment"> directly using the updated singular values. The singular vectors
</span><span class="comment">*</span><span class="comment"> for the current problem are multiplied with the singular vectors
</span><span class="comment">*</span><span class="comment"> from the overall problem.
</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"> 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,
</span><span class="comment">*</span><span class="comment"> dimension (N = 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 of
</span><span class="comment">*</span><span class="comment"> the lower block. On exit D(1:N) contains the singular values
</span><span class="comment">*</span><span class="comment"> of the modified 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"> U (input/output) DOUBLE PRECISION array, dimension(LDU,N)
</span><span class="comment">*</span><span class="comment"> On entry U(1:NL, 1:NL) contains the left singular vectors of
</span><span class="comment">*</span><span class="comment"> the upper block; U(NL+2:N, NL+2:N) contains the left singular
</span><span class="comment">*</span><span class="comment"> vectors of the lower block. On exit U contains the left
</span><span class="comment">*</span><span class="comment"> singular vectors of the bidiagonal matrix.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> LDU (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The leading dimension of the array U. LDU >= max( 1, N ).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> VT (input/output) DOUBLE PRECISION array, dimension(LDVT,M)
</span><span class="comment">*</span><span class="comment"> where M = N + SQRE.
</span><span class="comment">*</span><span class="comment"> On entry VT(1:NL+1, 1:NL+1)' contains the right singular
</span><span class="comment">*</span><span class="comment"> vectors of the upper block; VT(NL+2:M, NL+2:M)' contains
</span><span class="comment">*</span><span class="comment"> the right singular vectors of the lower block. On exit
</span><span class="comment">*</span><span class="comment"> VT' contains the right singular vectors of the
</span><span class="comment">*</span><span class="comment"> bidiagonal matrix.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> LDVT (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The leading dimension of the array VT. LDVT >= max( 1, M ).
</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
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