*> \brief \b DLASD3 finds all square roots of the roots of the secular equation, as defined by the values in D and Z, and then updates the singular vectors by matrix multiplication. Used by sbdsdc.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLASD3 + dependencies
*>
*> [TGZ]
*>
*> [ZIP]
*>
*> [TXT]
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLASD3( NL, NR, SQRE, K, D, Q, LDQ, DSIGMA, U, LDU, U2,
* LDU2, VT, LDVT, VT2, LDVT2, IDXC, CTOT, Z,
* INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, K, LDQ, LDU, LDU2, LDVT, LDVT2, NL, NR,
* $ SQRE
* ..
* .. Array Arguments ..
* INTEGER CTOT( * ), IDXC( * )
* DOUBLE PRECISION D( * ), DSIGMA( * ), Q( LDQ, * ), U( LDU, * ),
* $ U2( LDU2, * ), VT( LDVT, * ), VT2( LDVT2, * ),
* $ Z( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLASD3 finds all the square roots of the roots of the secular
*> equation, as defined by the values in D and Z. It makes the
*> appropriate calls to DLASD4 and then updates the singular
*> vectors by matrix multiplication.
*>
*> This code makes very mild assumptions about floating point
*> arithmetic. It will work on machines with a guard digit in
*> add/subtract, or on those binary machines without guard digits
*> which subtract like the Cray XMP, Cray YMP, Cray C 90, or Cray 2.
*> It could conceivably fail on hexadecimal or decimal machines
*> without guard digits, but we know of none.
*>
*> DLASD3 is called from DLASD1.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] NL
*> \verbatim
*> NL is INTEGER
*> The row dimension of the upper block. NL >= 1.
*> \endverbatim
*>
*> \param[in] NR
*> \verbatim
*> NR is INTEGER
*> The row dimension of the lower block. NR >= 1.
*> \endverbatim
*>
*> \param[in] SQRE
*> \verbatim
*> SQRE is 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 N = NL + NR + 1 rows and
*> M = N + SQRE >= N columns.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*> K is INTEGER
*> The size of the secular equation, 1 =< K = < N.
*> \endverbatim
*>
*> \param[out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension(K)
*> On exit the square roots of the roots of the secular equation,
*> in ascending order.
*> \endverbatim
*>
*> \param[out] Q
*> \verbatim
*> Q is DOUBLE PRECISION array,
*> dimension at least (LDQ,K).
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*> LDQ is INTEGER
*> The leading dimension of the array Q. LDQ >= K.
*> \endverbatim
*>
*> \param[in] DSIGMA
*> \verbatim
*> DSIGMA is DOUBLE PRECISION array, dimension(K)
*> The first K elements of this array contain the old roots
*> of the deflated updating problem. These are the poles
*> of the secular equation.
*> \endverbatim
*>
*> \param[out] U
*> \verbatim
*> U is DOUBLE PRECISION array, dimension (LDU, N)
*> The last N - K columns of this matrix contain the deflated
*> left singular vectors.
*> \endverbatim
*>
*> \param[in] LDU
*> \verbatim
*> LDU is INTEGER
*> The leading dimension of the array U. LDU >= N.
*> \endverbatim
*>
*> \param[in,out] U2
*> \verbatim
*> U2 is DOUBLE PRECISION array, dimension (LDU2, N)
*> The first K columns of this matrix contain the non-deflated
*> left singular vectors for the split problem.
*> \endverbatim
*>
*> \param[in] LDU2
*> \verbatim
*> LDU2 is INTEGER
*> The leading dimension of the array U2. LDU2 >= N.
*> \endverbatim
*>
*> \param[out] VT
*> \verbatim
*> VT is DOUBLE PRECISION array, dimension (LDVT, M)
*> The last M - K columns of VT**T contain the deflated
*> right singular vectors.
*> \endverbatim
*>
*> \param[in] LDVT
*> \verbatim
*> LDVT is INTEGER
*> The leading dimension of the array VT. LDVT >= N.
*> \endverbatim
*>
*> \param[in,out] VT2
*> \verbatim
*> VT2 is DOUBLE PRECISION array, dimension (LDVT2, N)
*> The first K columns of VT2**T contain the non-deflated
*> right singular vectors for the split problem.
*> \endverbatim
*>
*> \param[in] LDVT2
*> \verbatim
*> LDVT2 is INTEGER
*> The leading dimension of the array VT2. LDVT2 >= N.
*> \endverbatim
*>
*> \param[in] IDXC
*> \verbatim
*> IDXC is INTEGER array, dimension ( N )
*> The permutation used to arrange the columns of U (and rows of
*> VT) into three groups: the first group contains non-zero
*> entries only at and above (or before) NL +1; the second
*> contains non-zero entries only at and below (or after) NL+2;
*> and the third is dense. The first column of U and the row of
*> VT are treated separately, however.
*>
*> The rows of the singular vectors found by DLASD4
*> must be likewise permuted before the matrix multiplies can
*> take place.
*> \endverbatim
*>
*> \param[in] CTOT
*> \verbatim
*> CTOT is INTEGER array, dimension ( 4 )
*> A count of the total number of the various types of columns
*> in U (or rows in VT), as described in IDXC. The fourth column
*> type is any column which has been deflated.
*> \endverbatim
*>
*> \param[in] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension (K)
*> The first K elements of this array contain the components
*> of the deflation-adjusted updating row vector.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit.
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> > 0: if INFO = 1, a singular value did not converge
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2015
*
*> \ingroup auxOTHERauxiliary
*
*> \par Contributors:
* ==================
*>
*> Ming Gu and Huan Ren, Computer Science Division, University of
*> California at Berkeley, USA
*>
* =====================================================================
SUBROUTINE DLASD3( NL, NR, SQRE, K, D, Q, LDQ, DSIGMA, U, LDU, U2,
$ LDU2, VT, LDVT, VT2, LDVT2, IDXC, CTOT, Z,
$ INFO )
*
* -- LAPACK auxiliary routine (version 3.6.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2015
*
* .. Scalar Arguments ..
INTEGER INFO, K, LDQ, LDU, LDU2, LDVT, LDVT2, NL, NR,
$ SQRE
* ..
* .. Array Arguments ..
INTEGER CTOT( * ), IDXC( * )
DOUBLE PRECISION D( * ), DSIGMA( * ), Q( LDQ, * ), U( LDU, * ),
$ U2( LDU2, * ), VT( LDVT, * ), VT2( LDVT2, * ),
$ Z( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO, NEGONE
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0,
$ NEGONE = -1.0D+0 )
* ..
* .. Local Scalars ..
INTEGER CTEMP, I, J, JC, KTEMP, M, N, NLP1, NLP2, NRP1
DOUBLE PRECISION RHO, TEMP
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMC3, DNRM2
EXTERNAL DLAMC3, DNRM2
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DGEMM, DLACPY, DLASCL, DLASD4, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, SIGN, SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
*
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
*
N = NL + NR + 1
M = N + SQRE
NLP1 = NL + 1
NLP2 = NL + 2
*
IF( ( K.LT.1 ) .OR. ( K.GT.N ) ) THEN
INFO = -4
ELSE IF( LDQ.LT.K ) THEN
INFO = -7
ELSE IF( LDU.LT.N ) THEN
INFO = -10
ELSE IF( LDU2.LT.N ) THEN
INFO = -12
ELSE IF( LDVT.LT.M ) THEN
INFO = -14
ELSE IF( LDVT2.LT.M ) THEN
INFO = -16
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DLASD3', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( K.EQ.1 ) THEN
D( 1 ) = ABS( Z( 1 ) )
CALL DCOPY( M, VT2( 1, 1 ), LDVT2, VT( 1, 1 ), LDVT )
IF( Z( 1 ).GT.ZERO ) THEN
CALL DCOPY( N, U2( 1, 1 ), 1, U( 1, 1 ), 1 )
ELSE
DO 10 I = 1, N
U( I, 1 ) = -U2( I, 1 )
10 CONTINUE
END IF
RETURN
END IF
*
* Modify values DSIGMA(i) to make sure all DSIGMA(i)-DSIGMA(j) can
* be computed with high relative accuracy (barring over/underflow).
* This is a problem on machines without a guard digit in
* add/subtract (Cray XMP, Cray YMP, Cray C 90 and Cray 2).
* The following code replaces DSIGMA(I) by 2*DSIGMA(I)-DSIGMA(I),
* which on any of these machines zeros out the bottommost
* bit of DSIGMA(I) if it is 1; this makes the subsequent
* subtractions DSIGMA(I)-DSIGMA(J) unproblematic when cancellation
* occurs. On binary machines with a guard digit (almost all
* machines) it does not change DSIGMA(I) at all. On hexadecimal
* and decimal machines with a guard digit, it slightly
* changes the bottommost bits of DSIGMA(I). It does not account
* for hexadecimal or decimal machines without guard digits
* (we know of none). We use a subroutine call to compute
* 2*DSIGMA(I) to prevent optimizing compilers from eliminating
* this code.
*
DO 20 I = 1, K
DSIGMA( I ) = DLAMC3( DSIGMA( I ), DSIGMA( I ) ) - DSIGMA( I )
20 CONTINUE
*
* Keep a copy of Z.
*
CALL DCOPY( K, Z, 1, Q, 1 )
*
* Normalize Z.
*
RHO = DNRM2( K, Z, 1 )
CALL DLASCL( 'G', 0, 0, RHO, ONE, K, 1, Z, K, INFO )
RHO = RHO*RHO
*
* Find the new singular values.
*
DO 30 J = 1, K
CALL DLASD4( K, J, DSIGMA, Z, U( 1, J ), RHO, D( J ),
$ VT( 1, J ), INFO )
*
* If the zero finder fails, report the convergence failure.
*
IF( INFO.NE.0 ) THEN
RETURN
END IF
30 CONTINUE
*
* Compute updated Z.
*
DO 60 I = 1, K
Z( I ) = U( I, K )*VT( I, K )
DO 40 J = 1, I - 1
Z( I ) = Z( I )*( U( I, J )*VT( I, J ) /
$ ( DSIGMA( I )-DSIGMA( J ) ) /
$ ( DSIGMA( I )+DSIGMA( J ) ) )
40 CONTINUE
DO 50 J = I, K - 1
Z( I ) = Z( I )*( U( I, J )*VT( I, J ) /
$ ( DSIGMA( I )-DSIGMA( J+1 ) ) /
$ ( DSIGMA( I )+DSIGMA( J+1 ) ) )
50 CONTINUE
Z( I ) = SIGN( SQRT( ABS( Z( I ) ) ), Q( I, 1 ) )
60 CONTINUE
*
* Compute left singular vectors of the modified diagonal matrix,
* and store related information for the right singular vectors.
*
DO 90 I = 1, K
VT( 1, I ) = Z( 1 ) / U( 1, I ) / VT( 1, I )
U( 1, I ) = NEGONE
DO 70 J = 2, K
VT( J, I ) = Z( J ) / U( J, I ) / VT( J, I )
U( J, I ) = DSIGMA( J )*VT( J, I )
70 CONTINUE
TEMP = DNRM2( K, U( 1, I ), 1 )
Q( 1, I ) = U( 1, I ) / TEMP
DO 80 J = 2, K
JC = IDXC( J )
Q( J, I ) = U( JC, I ) / TEMP
80 CONTINUE
90 CONTINUE
*
* Update the left singular vector matrix.
*
IF( K.EQ.2 ) THEN
CALL DGEMM( 'N', 'N', N, K, K, ONE, U2, LDU2, Q, LDQ, ZERO, U,
$ LDU )
GO TO 100
END IF
IF( CTOT( 1 ).GT.0 ) THEN
CALL DGEMM( 'N', 'N', NL, K, CTOT( 1 ), ONE, U2( 1, 2 ), LDU2,
$ Q( 2, 1 ), LDQ, ZERO, U( 1, 1 ), LDU )
IF( CTOT( 3 ).GT.0 ) THEN
KTEMP = 2 + CTOT( 1 ) + CTOT( 2 )
CALL DGEMM( 'N', 'N', NL, K, CTOT( 3 ), ONE, U2( 1, KTEMP ),
$ LDU2, Q( KTEMP, 1 ), LDQ, ONE, U( 1, 1 ), LDU )
END IF
ELSE IF( CTOT( 3 ).GT.0 ) THEN
KTEMP = 2 + CTOT( 1 ) + CTOT( 2 )
CALL DGEMM( 'N', 'N', NL, K, CTOT( 3 ), ONE, U2( 1, KTEMP ),
$ LDU2, Q( KTEMP, 1 ), LDQ, ZERO, U( 1, 1 ), LDU )
ELSE
CALL DLACPY( 'F', NL, K, U2, LDU2, U, LDU )
END IF
CALL DCOPY( K, Q( 1, 1 ), LDQ, U( NLP1, 1 ), LDU )
KTEMP = 2 + CTOT( 1 )
CTEMP = CTOT( 2 ) + CTOT( 3 )
CALL DGEMM( 'N', 'N', NR, K, CTEMP, ONE, U2( NLP2, KTEMP ), LDU2,
$ Q( KTEMP, 1 ), LDQ, ZERO, U( NLP2, 1 ), LDU )
*
* Generate the right singular vectors.
*
100 CONTINUE
DO 120 I = 1, K
TEMP = DNRM2( K, VT( 1, I ), 1 )
Q( I, 1 ) = VT( 1, I ) / TEMP
DO 110 J = 2, K
JC = IDXC( J )
Q( I, J ) = VT( JC, I ) / TEMP
110 CONTINUE
120 CONTINUE
*
* Update the right singular vector matrix.
*
IF( K.EQ.2 ) THEN
CALL DGEMM( 'N', 'N', K, M, K, ONE, Q, LDQ, VT2, LDVT2, ZERO,
$ VT, LDVT )
RETURN
END IF
KTEMP = 1 + CTOT( 1 )
CALL DGEMM( 'N', 'N', K, NLP1, KTEMP, ONE, Q( 1, 1 ), LDQ,
$ VT2( 1, 1 ), LDVT2, ZERO, VT( 1, 1 ), LDVT )
KTEMP = 2 + CTOT( 1 ) + CTOT( 2 )
IF( KTEMP.LE.LDVT2 )
$ CALL DGEMM( 'N', 'N', K, NLP1, CTOT( 3 ), ONE, Q( 1, KTEMP ),
$ LDQ, VT2( KTEMP, 1 ), LDVT2, ONE, VT( 1, 1 ),
$ LDVT )
*
KTEMP = CTOT( 1 ) + 1
NRP1 = NR + SQRE
IF( KTEMP.GT.1 ) THEN
DO 130 I = 1, K
Q( I, KTEMP ) = Q( I, 1 )
130 CONTINUE
DO 140 I = NLP2, M
VT2( KTEMP, I ) = VT2( 1, I )
140 CONTINUE
END IF
CTEMP = 1 + CTOT( 2 ) + CTOT( 3 )
CALL DGEMM( 'N', 'N', K, NRP1, CTEMP, ONE, Q( 1, KTEMP ), LDQ,
$ VT2( KTEMP, NLP2 ), LDVT2, ZERO, VT( 1, NLP2 ), LDVT )
*
RETURN
*
* End of DLASD3
*
END