*> \brief \b ZLALSA computes the SVD of the coefficient matrix in compact form. Used by sgelsd.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download ZLALSA + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zlalsa.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zlalsa.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zlalsa.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE ZLALSA( ICOMPQ, SMLSIZ, N, NRHS, B, LDB, BX, LDBX, U,
* LDU, VT, K, DIFL, DIFR, Z, POLES, GIVPTR,
* GIVCOL, LDGCOL, PERM, GIVNUM, C, S, RWORK,
* IWORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER ICOMPQ, INFO, LDB, LDBX, LDGCOL, LDU, N, NRHS,
* $ SMLSIZ
* ..
* .. Array Arguments ..
* INTEGER GIVCOL( LDGCOL, * ), GIVPTR( * ), IWORK( * ),
* $ K( * ), PERM( LDGCOL, * )
* DOUBLE PRECISION C( * ), DIFL( LDU, * ), DIFR( LDU, * ),
* $ GIVNUM( LDU, * ), POLES( LDU, * ), RWORK( * ),
* $ S( * ), U( LDU, * ), VT( LDU, * ), Z( LDU, * )
* COMPLEX*16 B( LDB, * ), BX( LDBX, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> ZLALSA is an itermediate step in solving the least squares problem
*> by computing the SVD of the coefficient matrix in compact form (The
*> singular vectors are computed as products of simple orthorgonal
*> matrices.).
*>
*> If ICOMPQ = 0, ZLALSA applies the inverse of the left singular vector
*> matrix of an upper bidiagonal matrix to the right hand side; and if
*> ICOMPQ = 1, ZLALSA applies the right singular vector matrix to the
*> right hand side. The singular vector matrices were generated in
*> compact form by ZLALSA.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] ICOMPQ
*> \verbatim
*> ICOMPQ is INTEGER
*> Specifies whether the left or the right singular vector
*> matrix is involved.
*> = 0: Left singular vector matrix
*> = 1: Right singular vector matrix
*> \endverbatim
*>
*> \param[in] SMLSIZ
*> \verbatim
*> SMLSIZ is INTEGER
*> The maximum size of the subproblems at the bottom of the
*> computation tree.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The row and column dimensions of the upper bidiagonal matrix.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*> NRHS is INTEGER
*> The number of columns of B and BX. NRHS must be at least 1.
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is COMPLEX*16 array, dimension ( LDB, NRHS )
*> On input, B contains the right hand sides of the least
*> squares problem in rows 1 through M.
*> On output, B contains the solution X in rows 1 through N.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of B in the calling subprogram.
*> LDB must be at least max(1,MAX( M, N ) ).
*> \endverbatim
*>
*> \param[out] BX
*> \verbatim
*> BX is COMPLEX*16 array, dimension ( LDBX, NRHS )
*> On exit, the result of applying the left or right singular
*> vector matrix to B.
*> \endverbatim
*>
*> \param[in] LDBX
*> \verbatim
*> LDBX is INTEGER
*> The leading dimension of BX.
*> \endverbatim
*>
*> \param[in] U
*> \verbatim
*> U is DOUBLE PRECISION array, dimension ( LDU, SMLSIZ ).
*> On entry, U contains the left singular vector matrices of all
*> subproblems at the bottom level.
*> \endverbatim
*>
*> \param[in] LDU
*> \verbatim
*> LDU is INTEGER, LDU = > N.
*> The leading dimension of arrays U, VT, DIFL, DIFR,
*> POLES, GIVNUM, and Z.
*> \endverbatim
*>
*> \param[in] VT
*> \verbatim
*> VT is DOUBLE PRECISION array, dimension ( LDU, SMLSIZ+1 ).
*> On entry, VT**H contains the right singular vector matrices of
*> all subproblems at the bottom level.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*> K is INTEGER array, dimension ( N ).
*> \endverbatim
*>
*> \param[in] DIFL
*> \verbatim
*> DIFL is DOUBLE PRECISION array, dimension ( LDU, NLVL ).
*> where NLVL = INT(log_2 (N/(SMLSIZ+1))) + 1.
*> \endverbatim
*>
*> \param[in] DIFR
*> \verbatim
*> DIFR is DOUBLE PRECISION array, dimension ( LDU, 2 * NLVL ).
*> On entry, DIFL(*, I) and DIFR(*, 2 * I -1) record
*> distances between singular values on the I-th level and
*> singular values on the (I -1)-th level, and DIFR(*, 2 * I)
*> record the normalizing factors of the right singular vectors
*> matrices of subproblems on I-th level.
*> \endverbatim
*>
*> \param[in] Z
*> \verbatim
*> Z is DOUBLE PRECISION array, dimension ( LDU, NLVL ).
*> On entry, Z(1, I) contains the components of the deflation-
*> adjusted updating row vector for subproblems on the I-th
*> level.
*> \endverbatim
*>
*> \param[in] POLES
*> \verbatim
*> POLES is DOUBLE PRECISION array, dimension ( LDU, 2 * NLVL ).
*> On entry, POLES(*, 2 * I -1: 2 * I) contains the new and old
*> singular values involved in the secular equations on the I-th
*> level.
*> \endverbatim
*>
*> \param[in] GIVPTR
*> \verbatim
*> GIVPTR is INTEGER array, dimension ( N ).
*> On entry, GIVPTR( I ) records the number of Givens
*> rotations performed on the I-th problem on the computation
*> tree.
*> \endverbatim
*>
*> \param[in] GIVCOL
*> \verbatim
*> GIVCOL is INTEGER array, dimension ( LDGCOL, 2 * NLVL ).
*> On entry, for each I, GIVCOL(*, 2 * I - 1: 2 * I) records the
*> locations of Givens rotations performed on the I-th level on
*> the computation tree.
*> \endverbatim
*>
*> \param[in] LDGCOL
*> \verbatim
*> LDGCOL is INTEGER, LDGCOL = > N.
*> The leading dimension of arrays GIVCOL and PERM.
*> \endverbatim
*>
*> \param[in] PERM
*> \verbatim
*> PERM is INTEGER array, dimension ( LDGCOL, NLVL ).
*> On entry, PERM(*, I) records permutations done on the I-th
*> level of the computation tree.
*> \endverbatim
*>
*> \param[in] GIVNUM
*> \verbatim
*> GIVNUM is DOUBLE PRECISION array, dimension ( LDU, 2 * NLVL ).
*> On entry, GIVNUM(*, 2 *I -1 : 2 * I) records the C- and S-
*> values of Givens rotations performed on the I-th level on the
*> computation tree.
*> \endverbatim
*>
*> \param[in] C
*> \verbatim
*> C is DOUBLE PRECISION array, dimension ( N ).
*> On entry, if the I-th subproblem is not square,
*> C( I ) contains the C-value of a Givens rotation related to
*> the right null space of the I-th subproblem.
*> \endverbatim
*>
*> \param[in] S
*> \verbatim
*> S is DOUBLE PRECISION array, dimension ( N ).
*> On entry, if the I-th subproblem is not square,
*> S( I ) contains the S-value of a Givens rotation related to
*> the right null space of the I-th subproblem.
*> \endverbatim
*>
*> \param[out] RWORK
*> \verbatim
*> RWORK is DOUBLE PRECISION array, dimension at least
*> MAX( (SMLSZ+1)*NRHS*3, N*(1+NRHS) + 2*NRHS ).
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (3*N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit.
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date June 2017
*
*> \ingroup complex16OTHERcomputational
*
*> \par Contributors:
* ==================
*>
*> Ming Gu and Ren-Cang Li, Computer Science Division, University of
*> California at Berkeley, USA \n
*> Osni Marques, LBNL/NERSC, USA \n
*
* =====================================================================
SUBROUTINE ZLALSA( ICOMPQ, SMLSIZ, N, NRHS, B, LDB, BX, LDBX, U,
$ LDU, VT, K, DIFL, DIFR, Z, POLES, GIVPTR,
$ GIVCOL, LDGCOL, PERM, GIVNUM, C, S, RWORK,
$ IWORK, INFO )
*
* -- LAPACK computational routine (version 3.7.1) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* June 2017
*
* .. Scalar Arguments ..
INTEGER ICOMPQ, INFO, LDB, LDBX, LDGCOL, LDU, N, NRHS,
$ SMLSIZ
* ..
* .. Array Arguments ..
INTEGER GIVCOL( LDGCOL, * ), GIVPTR( * ), IWORK( * ),
$ K( * ), PERM( LDGCOL, * )
DOUBLE PRECISION C( * ), DIFL( LDU, * ), DIFR( LDU, * ),
$ GIVNUM( LDU, * ), POLES( LDU, * ), RWORK( * ),
$ S( * ), U( LDU, * ), VT( LDU, * ), Z( LDU, * )
COMPLEX*16 B( LDB, * ), BX( LDBX, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
* ..
* .. Local Scalars ..
INTEGER I, I1, IC, IM1, INODE, J, JCOL, JIMAG, JREAL,
$ JROW, LF, LL, LVL, LVL2, ND, NDB1, NDIML,
$ NDIMR, NL, NLF, NLP1, NLVL, NR, NRF, NRP1, SQRE
* ..
* .. External Subroutines ..
EXTERNAL DGEMM, DLASDT, XERBLA, ZCOPY, ZLALS0
* ..
* .. Intrinsic Functions ..
INTRINSIC DBLE, DCMPLX, DIMAG
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
*
IF( ( ICOMPQ.LT.0 ) .OR. ( ICOMPQ.GT.1 ) ) THEN
INFO = -1
ELSE IF( SMLSIZ.LT.3 ) THEN
INFO = -2
ELSE IF( N.LT.SMLSIZ ) THEN
INFO = -3
ELSE IF( NRHS.LT.1 ) THEN
INFO = -4
ELSE IF( LDB.LT.N ) THEN
INFO = -6
ELSE IF( LDBX.LT.N ) THEN
INFO = -8
ELSE IF( LDU.LT.N ) THEN
INFO = -10
ELSE IF( LDGCOL.LT.N ) THEN
INFO = -19
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'ZLALSA', -INFO )
RETURN
END IF
*
* Book-keeping and setting up the computation tree.
*
INODE = 1
NDIML = INODE + N
NDIMR = NDIML + N
*
CALL DLASDT( N, NLVL, ND, IWORK( INODE ), IWORK( NDIML ),
$ IWORK( NDIMR ), SMLSIZ )
*
* The following code applies back the left singular vector factors.
* For applying back the right singular vector factors, go to 170.
*
IF( ICOMPQ.EQ.1 ) THEN
GO TO 170
END IF
*
* The nodes on the bottom level of the tree were solved
* by DLASDQ. The corresponding left and right singular vector
* matrices are in explicit form. First apply back the left
* singular vector matrices.
*
NDB1 = ( ND+1 ) / 2
DO 130 I = NDB1, ND
*
* IC : center row of each node
* NL : number of rows of left subproblem
* NR : number of rows of right subproblem
* NLF: starting row of the left subproblem
* NRF: starting row of the right subproblem
*
I1 = I - 1
IC = IWORK( INODE+I1 )
NL = IWORK( NDIML+I1 )
NR = IWORK( NDIMR+I1 )
NLF = IC - NL
NRF = IC + 1
*
* Since B and BX are complex, the following call to DGEMM
* is performed in two steps (real and imaginary parts).
*
* CALL DGEMM( 'T', 'N', NL, NRHS, NL, ONE, U( NLF, 1 ), LDU,
* $ B( NLF, 1 ), LDB, ZERO, BX( NLF, 1 ), LDBX )
*
J = NL*NRHS*2
DO 20 JCOL = 1, NRHS
DO 10 JROW = NLF, NLF + NL - 1
J = J + 1
RWORK( J ) = DBLE( B( JROW, JCOL ) )
10 CONTINUE
20 CONTINUE
CALL DGEMM( 'T', 'N', NL, NRHS, NL, ONE, U( NLF, 1 ), LDU,
$ RWORK( 1+NL*NRHS*2 ), NL, ZERO, RWORK( 1 ), NL )
J = NL*NRHS*2
DO 40 JCOL = 1, NRHS
DO 30 JROW = NLF, NLF + NL - 1
J = J + 1
RWORK( J ) = DIMAG( B( JROW, JCOL ) )
30 CONTINUE
40 CONTINUE
CALL DGEMM( 'T', 'N', NL, NRHS, NL, ONE, U( NLF, 1 ), LDU,
$ RWORK( 1+NL*NRHS*2 ), NL, ZERO, RWORK( 1+NL*NRHS ),
$ NL )
JREAL = 0
JIMAG = NL*NRHS
DO 60 JCOL = 1, NRHS
DO 50 JROW = NLF, NLF + NL - 1
JREAL = JREAL + 1
JIMAG = JIMAG + 1
BX( JROW, JCOL ) = DCMPLX( RWORK( JREAL ),
$ RWORK( JIMAG ) )
50 CONTINUE
60 CONTINUE
*
* Since B and BX are complex, the following call to DGEMM
* is performed in two steps (real and imaginary parts).
*
* CALL DGEMM( 'T', 'N', NR, NRHS, NR, ONE, U( NRF, 1 ), LDU,
* $ B( NRF, 1 ), LDB, ZERO, BX( NRF, 1 ), LDBX )
*
J = NR*NRHS*2
DO 80 JCOL = 1, NRHS
DO 70 JROW = NRF, NRF + NR - 1
J = J + 1
RWORK( J ) = DBLE( B( JROW, JCOL ) )
70 CONTINUE
80 CONTINUE
CALL DGEMM( 'T', 'N', NR, NRHS, NR, ONE, U( NRF, 1 ), LDU,
$ RWORK( 1+NR*NRHS*2 ), NR, ZERO, RWORK( 1 ), NR )
J = NR*NRHS*2
DO 100 JCOL = 1, NRHS
DO 90 JROW = NRF, NRF + NR - 1
J = J + 1
RWORK( J ) = DIMAG( B( JROW, JCOL ) )
90 CONTINUE
100 CONTINUE
CALL DGEMM( 'T', 'N', NR, NRHS, NR, ONE, U( NRF, 1 ), LDU,
$ RWORK( 1+NR*NRHS*2 ), NR, ZERO, RWORK( 1+NR*NRHS ),
$ NR )
JREAL = 0
JIMAG = NR*NRHS
DO 120 JCOL = 1, NRHS
DO 110 JROW = NRF, NRF + NR - 1
JREAL = JREAL + 1
JIMAG = JIMAG + 1
BX( JROW, JCOL ) = DCMPLX( RWORK( JREAL ),
$ RWORK( JIMAG ) )
110 CONTINUE
120 CONTINUE
*
130 CONTINUE
*
* Next copy the rows of B that correspond to unchanged rows
* in the bidiagonal matrix to BX.
*
DO 140 I = 1, ND
IC = IWORK( INODE+I-1 )
CALL ZCOPY( NRHS, B( IC, 1 ), LDB, BX( IC, 1 ), LDBX )
140 CONTINUE
*
* Finally go through the left singular vector matrices of all
* the other subproblems bottom-up on the tree.
*
J = 2**NLVL
SQRE = 0
*
DO 160 LVL = NLVL, 1, -1
LVL2 = 2*LVL - 1
*
* find the first node LF and last node LL on
* the current level LVL
*
IF( LVL.EQ.1 ) THEN
LF = 1
LL = 1
ELSE
LF = 2**( LVL-1 )
LL = 2*LF - 1
END IF
DO 150 I = LF, LL
IM1 = I - 1
IC = IWORK( INODE+IM1 )
NL = IWORK( NDIML+IM1 )
NR = IWORK( NDIMR+IM1 )
NLF = IC - NL
NRF = IC + 1
J = J - 1
CALL ZLALS0( ICOMPQ, NL, NR, SQRE, NRHS, BX( NLF, 1 ), LDBX,
$ B( NLF, 1 ), LDB, PERM( NLF, LVL ),
$ GIVPTR( J ), GIVCOL( NLF, LVL2 ), LDGCOL,
$ GIVNUM( NLF, LVL2 ), LDU, POLES( NLF, LVL2 ),
$ DIFL( NLF, LVL ), DIFR( NLF, LVL2 ),
$ Z( NLF, LVL ), K( J ), C( J ), S( J ), RWORK,
$ INFO )
150 CONTINUE
160 CONTINUE
GO TO 330
*
* ICOMPQ = 1: applying back the right singular vector factors.
*
170 CONTINUE
*
* First now go through the right singular vector matrices of all
* the tree nodes top-down.
*
J = 0
DO 190 LVL = 1, NLVL
LVL2 = 2*LVL - 1
*
* Find the first node LF and last node LL on
* the current level LVL.
*
IF( LVL.EQ.1 ) THEN
LF = 1
LL = 1
ELSE
LF = 2**( LVL-1 )
LL = 2*LF - 1
END IF
DO 180 I = LL, LF, -1
IM1 = I - 1
IC = IWORK( INODE+IM1 )
NL = IWORK( NDIML+IM1 )
NR = IWORK( NDIMR+IM1 )
NLF = IC - NL
NRF = IC + 1
IF( I.EQ.LL ) THEN
SQRE = 0
ELSE
SQRE = 1
END IF
J = J + 1
CALL ZLALS0( ICOMPQ, NL, NR, SQRE, NRHS, B( NLF, 1 ), LDB,
$ BX( NLF, 1 ), LDBX, PERM( NLF, LVL ),
$ GIVPTR( J ), GIVCOL( NLF, LVL2 ), LDGCOL,
$ GIVNUM( NLF, LVL2 ), LDU, POLES( NLF, LVL2 ),
$ DIFL( NLF, LVL ), DIFR( NLF, LVL2 ),
$ Z( NLF, LVL ), K( J ), C( J ), S( J ), RWORK,
$ INFO )
180 CONTINUE
190 CONTINUE
*
* The nodes on the bottom level of the tree were solved
* by DLASDQ. The corresponding right singular vector
* matrices are in explicit form. Apply them back.
*
NDB1 = ( ND+1 ) / 2
DO 320 I = NDB1, ND
I1 = I - 1
IC = IWORK( INODE+I1 )
NL = IWORK( NDIML+I1 )
NR = IWORK( NDIMR+I1 )
NLP1 = NL + 1
IF( I.EQ.ND ) THEN
NRP1 = NR
ELSE
NRP1 = NR + 1
END IF
NLF = IC - NL
NRF = IC + 1
*
* Since B and BX are complex, the following call to DGEMM is
* performed in two steps (real and imaginary parts).
*
* CALL DGEMM( 'T', 'N', NLP1, NRHS, NLP1, ONE, VT( NLF, 1 ), LDU,
* $ B( NLF, 1 ), LDB, ZERO, BX( NLF, 1 ), LDBX )
*
J = NLP1*NRHS*2
DO 210 JCOL = 1, NRHS
DO 200 JROW = NLF, NLF + NLP1 - 1
J = J + 1
RWORK( J ) = DBLE( B( JROW, JCOL ) )
200 CONTINUE
210 CONTINUE
CALL DGEMM( 'T', 'N', NLP1, NRHS, NLP1, ONE, VT( NLF, 1 ), LDU,
$ RWORK( 1+NLP1*NRHS*2 ), NLP1, ZERO, RWORK( 1 ),
$ NLP1 )
J = NLP1*NRHS*2
DO 230 JCOL = 1, NRHS
DO 220 JROW = NLF, NLF + NLP1 - 1
J = J + 1
RWORK( J ) = DIMAG( B( JROW, JCOL ) )
220 CONTINUE
230 CONTINUE
CALL DGEMM( 'T', 'N', NLP1, NRHS, NLP1, ONE, VT( NLF, 1 ), LDU,
$ RWORK( 1+NLP1*NRHS*2 ), NLP1, ZERO,
$ RWORK( 1+NLP1*NRHS ), NLP1 )
JREAL = 0
JIMAG = NLP1*NRHS
DO 250 JCOL = 1, NRHS
DO 240 JROW = NLF, NLF + NLP1 - 1
JREAL = JREAL + 1
JIMAG = JIMAG + 1
BX( JROW, JCOL ) = DCMPLX( RWORK( JREAL ),
$ RWORK( JIMAG ) )
240 CONTINUE
250 CONTINUE
*
* Since B and BX are complex, the following call to DGEMM is
* performed in two steps (real and imaginary parts).
*
* CALL DGEMM( 'T', 'N', NRP1, NRHS, NRP1, ONE, VT( NRF, 1 ), LDU,
* $ B( NRF, 1 ), LDB, ZERO, BX( NRF, 1 ), LDBX )
*
J = NRP1*NRHS*2
DO 270 JCOL = 1, NRHS
DO 260 JROW = NRF, NRF + NRP1 - 1
J = J + 1
RWORK( J ) = DBLE( B( JROW, JCOL ) )
260 CONTINUE
270 CONTINUE
CALL DGEMM( 'T', 'N', NRP1, NRHS, NRP1, ONE, VT( NRF, 1 ), LDU,
$ RWORK( 1+NRP1*NRHS*2 ), NRP1, ZERO, RWORK( 1 ),
$ NRP1 )
J = NRP1*NRHS*2
DO 290 JCOL = 1, NRHS
DO 280 JROW = NRF, NRF + NRP1 - 1
J = J + 1
RWORK( J ) = DIMAG( B( JROW, JCOL ) )
280 CONTINUE
290 CONTINUE
CALL DGEMM( 'T', 'N', NRP1, NRHS, NRP1, ONE, VT( NRF, 1 ), LDU,
$ RWORK( 1+NRP1*NRHS*2 ), NRP1, ZERO,
$ RWORK( 1+NRP1*NRHS ), NRP1 )
JREAL = 0
JIMAG = NRP1*NRHS
DO 310 JCOL = 1, NRHS
DO 300 JROW = NRF, NRF + NRP1 - 1
JREAL = JREAL + 1
JIMAG = JIMAG + 1
BX( JROW, JCOL ) = DCMPLX( RWORK( JREAL ),
$ RWORK( JIMAG ) )
300 CONTINUE
310 CONTINUE
*
320 CONTINUE
*
330 CONTINUE
*
RETURN
*
* End of ZLALSA
*
END