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SUBROUTINE <a name="DLASD3.1"></a><a href="dlasd3.f.html#DLASD3.1">DLASD3</a>( NL, NR, SQRE, K, D, Q, LDQ, DSIGMA, U, LDU, U2,
$ LDU2, VT, LDVT, VT2, LDVT2, IDXC, CTOT, Z,
$ 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, K, LDQ, LDU, LDU2, LDVT, LDVT2, NL, NR,
$ SQRE
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. Array Arguments ..
</span> INTEGER CTOT( * ), IDXC( * )
DOUBLE PRECISION D( * ), DSIGMA( * ), Q( LDQ, * ), U( LDU, * ),
$ U2( LDU2, * ), VT( LDVT, * ), VT2( LDVT2, * ),
$ 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="DLASD3.23"></a><a href="dlasd3.f.html#DLASD3.1">DLASD3</a> finds all the square roots of the roots of the secular
</span><span class="comment">*</span><span class="comment"> equation, as defined by the values in D and Z. It makes the
</span><span class="comment">*</span><span class="comment"> appropriate calls to <a name="DLASD4.25"></a><a href="dlasd4.f.html#DLASD4.1">DLASD4</a> and then updates the singular
</span><span class="comment">*</span><span class="comment"> vectors by matrix multiplication.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> This code makes very mild assumptions about floating point
</span><span class="comment">*</span><span class="comment"> arithmetic. It will work on machines with a guard digit in
</span><span class="comment">*</span><span class="comment"> add/subtract, or on those binary machines without guard digits
</span><span class="comment">*</span><span class="comment"> which subtract like the Cray XMP, Cray YMP, Cray C 90, or Cray 2.
</span><span class="comment">*</span><span class="comment"> It could conceivably fail on hexadecimal or decimal machines
</span><span class="comment">*</span><span class="comment"> without guard digits, but we know of none.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> <a name="DLASD3.35"></a><a href="dlasd3.f.html#DLASD3.1">DLASD3</a> is called from <a name="DLASD1.35"></a><a href="dlasd1.f.html#DLASD1.1">DLASD1</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"> 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 N = NL + NR + 1 rows and
</span><span class="comment">*</span><span class="comment"> M = N + SQRE >= N columns.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> K (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The size of the secular equation, 1 =< K = < N.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> D (output) DOUBLE PRECISION array, dimension(K)
</span><span class="comment">*</span><span class="comment"> On exit the square roots of the roots of the secular equation,
</span><span class="comment">*</span><span class="comment"> in ascending order.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Q (workspace) DOUBLE PRECISION array,
</span><span class="comment">*</span><span class="comment"> dimension at least (LDQ,K).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> LDQ (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The leading dimension of the array Q. LDQ >= K.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> DSIGMA (input) DOUBLE PRECISION array, dimension(K)
</span><span class="comment">*</span><span class="comment"> The first K elements of this array contain the old roots
</span><span class="comment">*</span><span class="comment"> of the deflated updating problem. These are the poles
</span><span class="comment">*</span><span class="comment"> of the secular equation.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> U (output) DOUBLE PRECISION array, dimension (LDU, N)
</span><span class="comment">*</span><span class="comment"> The last N - K columns of this matrix contain the deflated
</span><span class="comment">*</span><span class="comment"> left singular vectors.
</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 >= N.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> U2 (input/output) DOUBLE PRECISION array, dimension (LDU2, N)
</span><span class="comment">*</span><span class="comment"> The first K columns of this matrix contain the non-deflated
</span><span class="comment">*</span><span class="comment"> left singular vectors for the split problem.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> LDU2 (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The leading dimension of the array U2. LDU2 >= N.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> VT (output) DOUBLE PRECISION array, dimension (LDVT, M)
</span><span class="comment">*</span><span class="comment"> The last M - K columns of VT' contain the deflated
</span><span class="comment">*</span><span class="comment"> right singular vectors.
</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 >= N.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> VT2 (input/output) DOUBLE PRECISION array, dimension (LDVT2, N)
</span><span class="comment">*</span><span class="comment"> The first K columns of VT2' contain the non-deflated
</span><span class="comment">*</span><span class="comment"> right singular vectors for the split problem.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> LDVT2 (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The leading dimension of the array VT2. LDVT2 >= N.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> IDXC (input) INTEGER array, dimension ( N )
</span><span class="comment">*</span><span class="comment"> The permutation used to arrange the columns of U (and rows of
</span><span class="comment">*</span><span class="comment"> VT) into three groups: the first group contains non-zero
</span><span class="comment">*</span><span class="comment"> entries only at and above (or before) NL +1; the second
</span><span class="comment">*</span><span class="comment"> contains non-zero entries only at and below (or after) NL+2;
</span><span class="comment">*</span><span class="comment"> and the third is dense. The first column of U and the row of
</span><span class="comment">*</span><span class="comment"> VT are treated separately, however.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> The rows of the singular vectors found by <a name="DLASD4.107"></a><a href="dlasd4.f.html#DLASD4.1">DLASD4</a>
</span><span class="comment">*</span><span class="comment"> must be likewise permuted before the matrix multiplies can
</span><span class="comment">*</span><span class="comment"> take place.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> CTOT (input) INTEGER array, dimension ( 4 )
</span><span class="comment">*</span><span class="comment"> A count of the total number of the various types of columns
</span><span class="comment">*</span><span class="comment"> in U (or rows in VT), as described in IDXC. The fourth column
</span><span class="comment">*</span><span class="comment"> type is any column which has been deflated.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Z (input) DOUBLE PRECISION array, dimension (K)
</span><span class="comment">*</span><span class="comment"> The first K elements of this array contain the components
</span><span class="comment">*</span><span class="comment"> of the deflation-adjusted updating row vector.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> INFO (output) INTEGER
</span><span class="comment">*</span><span class="comment"> = 0: successful exit.
</span><span class="comment">*</span><span class="comment"> < 0: if INFO = -i, the i-th argument had an illegal value.
</span><span class="comment">*</span><span class="comment"> > 0: if INFO = 1, an singular value did not converge
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Further Details
</span><span class="comment">*</span><span class="comment"> ===============
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Based on contributions by
</span><span class="comment">*</span><span class="comment"> Ming Gu and Huan Ren, Computer Science Division, University of
</span><span class="comment">*</span><span class="comment"> California at Berkeley, USA
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> =====================================================================
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> .. Parameters ..
</span> DOUBLE PRECISION ONE, ZERO, NEGONE
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0,
$ NEGONE = -1.0D+0 )
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. Local Scalars ..
</span> INTEGER CTEMP, I, J, JC, KTEMP, M, N, NLP1, NLP2, NRP1
DOUBLE PRECISION RHO, TEMP
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. External Functions ..
</span> DOUBLE PRECISION <a name="DLAMC3.144"></a><a href="dlamch.f.html#DLAMC3.574">DLAMC3</a>, DNRM2
EXTERNAL <a name="DLAMC3.145"></a><a href="dlamch.f.html#DLAMC3.574">DLAMC3</a>, DNRM2
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. External Subroutines ..
</span> EXTERNAL DCOPY, DGEMM, <a name="DLACPY.148"></a><a href="dlacpy.f.html#DLACPY.1">DLACPY</a>, <a name="DLASCL.148"></a><a href="dlascl.f.html#DLASCL.1">DLASCL</a>, <a name="DLASD4.148"></a><a href="dlasd4.f.html#DLASD4.1">DLASD4</a>, <a name="XERBLA.148"></a><a href="xerbla.f.html#XERBLA.1">XERBLA</a>
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. Intrinsic Functions ..
</span> INTRINSIC ABS, SIGN, SQRT
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. Executable Statements ..
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Test the input parameters.
</span><span class="comment">*</span><span class="comment">
</span> INFO = 0
<span class="comment">*</span><span class="comment">
</span> IF( NL.LT.1 ) THEN
INFO = -1
ELSE IF( NR.LT.1 ) THEN
INFO = -2
ELSE IF( ( SQRE.NE.1 ) .AND. ( SQRE.NE.0 ) ) THEN
INFO = -3
END IF
<span class="comment">*</span><span class="comment">
</span> N = NL + NR + 1
M = N + SQRE
NLP1 = NL + 1
NLP2 = NL + 2
<span class="comment">*</span><span class="comment">
</span> IF( ( K.LT.1 ) .OR. ( K.GT.N ) ) THEN
INFO = -4
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