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clals0.f
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*> \brief \b CLALS0 applies back multiplying factors in solving the least squares problem using divide and conquer SVD approach. Used by sgelsd.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download CLALS0 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/clals0.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/clals0.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/clals0.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE CLALS0( ICOMPQ, NL, NR, SQRE, NRHS, B, LDB, BX, LDBX,
* PERM, GIVPTR, GIVCOL, LDGCOL, GIVNUM, LDGNUM,
* POLES, DIFL, DIFR, Z, K, C, S, RWORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER GIVPTR, ICOMPQ, INFO, K, LDB, LDBX, LDGCOL,
* $ LDGNUM, NL, NR, NRHS, SQRE
* REAL C, S
* ..
* .. Array Arguments ..
* INTEGER GIVCOL( LDGCOL, * ), PERM( * )
* REAL DIFL( * ), DIFR( LDGNUM, * ),
* $ GIVNUM( LDGNUM, * ), POLES( LDGNUM, * ),
* $ RWORK( * ), Z( * )
* COMPLEX B( LDB, * ), BX( LDBX, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> CLALS0 applies back the multiplying factors of either the left or the
*> right singular vector matrix of a diagonal matrix appended by a row
*> to the right hand side matrix B in solving the least squares problem
*> using the divide-and-conquer SVD approach.
*>
*> For the left singular vector matrix, three types of orthogonal
*> matrices are involved:
*>
*> (1L) Givens rotations: the number of such rotations is GIVPTR; the
*> pairs of columns/rows they were applied to are stored in GIVCOL;
*> and the C- and S-values of these rotations are stored in GIVNUM.
*>
*> (2L) Permutation. The (NL+1)-st row of B is to be moved to the first
*> row, and for J=2:N, PERM(J)-th row of B is to be moved to the
*> J-th row.
*>
*> (3L) The left singular vector matrix of the remaining matrix.
*>
*> For the right singular vector matrix, four types of orthogonal
*> matrices are involved:
*>
*> (1R) The right singular vector matrix of the remaining matrix.
*>
*> (2R) If SQRE = 1, one extra Givens rotation to generate the right
*> null space.
*>
*> (3R) The inverse transformation of (2L).
*>
*> (4R) The inverse transformation of (1L).
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] ICOMPQ
*> \verbatim
*> ICOMPQ is INTEGER
*> Specifies whether singular vectors are to be computed in
*> factored form:
*> = 0: Left singular vector matrix.
*> = 1: Right singular vector matrix.
*> \endverbatim
*>
*> \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 row dimension N = NL + NR + 1,
*> and column dimension M = N + SQRE.
*> \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 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. LDB must be at least
*> max(1,MAX( M, N ) ).
*> \endverbatim
*>
*> \param[out] BX
*> \verbatim
*> BX is COMPLEX array, dimension ( LDBX, NRHS )
*> \endverbatim
*>
*> \param[in] LDBX
*> \verbatim
*> LDBX is INTEGER
*> The leading dimension of BX.
*> \endverbatim
*>
*> \param[in] PERM
*> \verbatim
*> PERM is INTEGER array, dimension ( N )
*> The permutations (from deflation and sorting) applied
*> to the two blocks.
*> \endverbatim
*>
*> \param[in] GIVPTR
*> \verbatim
*> GIVPTR is INTEGER
*> The number of Givens rotations which took place in this
*> subproblem.
*> \endverbatim
*>
*> \param[in] GIVCOL
*> \verbatim
*> GIVCOL is INTEGER array, dimension ( LDGCOL, 2 )
*> Each pair of numbers indicates a pair of rows/columns
*> involved in a Givens rotation.
*> \endverbatim
*>
*> \param[in] LDGCOL
*> \verbatim
*> LDGCOL is INTEGER
*> The leading dimension of GIVCOL, must be at least N.
*> \endverbatim
*>
*> \param[in] GIVNUM
*> \verbatim
*> GIVNUM is REAL array, dimension ( LDGNUM, 2 )
*> Each number indicates the C or S value used in the
*> corresponding Givens rotation.
*> \endverbatim
*>
*> \param[in] LDGNUM
*> \verbatim
*> LDGNUM is INTEGER
*> The leading dimension of arrays DIFR, POLES and
*> GIVNUM, must be at least K.
*> \endverbatim
*>
*> \param[in] POLES
*> \verbatim
*> POLES is REAL array, dimension ( LDGNUM, 2 )
*> On entry, POLES(1:K, 1) contains the new singular
*> values obtained from solving the secular equation, and
*> POLES(1:K, 2) is an array containing the poles in the secular
*> equation.
*> \endverbatim
*>
*> \param[in] DIFL
*> \verbatim
*> DIFL is REAL array, dimension ( K ).
*> On entry, DIFL(I) is the distance between I-th updated
*> (undeflated) singular value and the I-th (undeflated) old
*> singular value.
*> \endverbatim
*>
*> \param[in] DIFR
*> \verbatim
*> DIFR is REAL array, dimension ( LDGNUM, 2 ).
*> On entry, DIFR(I, 1) contains the distances between I-th
*> updated (undeflated) singular value and the I+1-th
*> (undeflated) old singular value. And DIFR(I, 2) is the
*> normalizing factor for the I-th right singular vector.
*> \endverbatim
*>
*> \param[in] Z
*> \verbatim
*> Z is REAL array, dimension ( K )
*> Contain the components of the deflation-adjusted updating row
*> vector.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*> K is INTEGER
*> Contains the dimension of the non-deflated matrix,
*> This is the order of the related secular equation. 1 <= K <=N.
*> \endverbatim
*>
*> \param[in] C
*> \verbatim
*> C is REAL
*> C contains garbage if SQRE =0 and the C-value of a Givens
*> rotation related to the right null space if SQRE = 1.
*> \endverbatim
*>
*> \param[in] S
*> \verbatim
*> S is REAL
*> S contains garbage if SQRE =0 and the S-value of a Givens
*> rotation related to the right null space if SQRE = 1.
*> \endverbatim
*>
*> \param[out] RWORK
*> \verbatim
*> RWORK is REAL array, dimension
*> ( K*(1+NRHS) + 2*NRHS )
*> \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.
*
*> \ingroup complexOTHERcomputational
*
*> \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 CLALS0( ICOMPQ, NL, NR, SQRE, NRHS, B, LDB, BX, LDBX,
$ PERM, GIVPTR, GIVCOL, LDGCOL, GIVNUM, LDGNUM,
$ POLES, DIFL, DIFR, Z, K, C, S, RWORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
INTEGER GIVPTR, ICOMPQ, INFO, K, LDB, LDBX, LDGCOL,
$ LDGNUM, NL, NR, NRHS, SQRE
REAL C, S
* ..
* .. Array Arguments ..
INTEGER GIVCOL( LDGCOL, * ), PERM( * )
REAL DIFL( * ), DIFR( LDGNUM, * ),
$ GIVNUM( LDGNUM, * ), POLES( LDGNUM, * ),
$ RWORK( * ), Z( * )
COMPLEX B( LDB, * ), BX( LDBX, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
REAL ONE, ZERO, NEGONE
PARAMETER ( ONE = 1.0E0, ZERO = 0.0E0, NEGONE = -1.0E0 )
* ..
* .. Local Scalars ..
INTEGER I, J, JCOL, JROW, M, N, NLP1
REAL DIFLJ, DIFRJ, DJ, DSIGJ, DSIGJP, TEMP
* ..
* .. External Subroutines ..
EXTERNAL CCOPY, CLACPY, CLASCL, CSROT, CSSCAL, SGEMV,
$ XERBLA
* ..
* .. External Functions ..
REAL SLAMC3, SNRM2
EXTERNAL SLAMC3, SNRM2
* ..
* .. Intrinsic Functions ..
INTRINSIC AIMAG, CMPLX, MAX, REAL
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
N = NL + NR + 1
*
IF( ( ICOMPQ.LT.0 ) .OR. ( ICOMPQ.GT.1 ) ) THEN
INFO = -1
ELSE IF( NL.LT.1 ) THEN
INFO = -2
ELSE IF( NR.LT.1 ) THEN
INFO = -3
ELSE IF( ( SQRE.LT.0 ) .OR. ( SQRE.GT.1 ) ) THEN
INFO = -4
ELSE IF( NRHS.LT.1 ) THEN
INFO = -5
ELSE IF( LDB.LT.N ) THEN
INFO = -7
ELSE IF( LDBX.LT.N ) THEN
INFO = -9
ELSE IF( GIVPTR.LT.0 ) THEN
INFO = -11
ELSE IF( LDGCOL.LT.N ) THEN
INFO = -13
ELSE IF( LDGNUM.LT.N ) THEN
INFO = -15
ELSE IF( K.LT.1 ) THEN
INFO = -20
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'CLALS0', -INFO )
RETURN
END IF
*
M = N + SQRE
NLP1 = NL + 1
*
IF( ICOMPQ.EQ.0 ) THEN
*
* Apply back orthogonal transformations from the left.
*
* Step (1L): apply back the Givens rotations performed.
*
DO 10 I = 1, GIVPTR
CALL CSROT( NRHS, B( GIVCOL( I, 2 ), 1 ), LDB,
$ B( GIVCOL( I, 1 ), 1 ), LDB, GIVNUM( I, 2 ),
$ GIVNUM( I, 1 ) )
10 CONTINUE
*
* Step (2L): permute rows of B.
*
CALL CCOPY( NRHS, B( NLP1, 1 ), LDB, BX( 1, 1 ), LDBX )
DO 20 I = 2, N
CALL CCOPY( NRHS, B( PERM( I ), 1 ), LDB, BX( I, 1 ), LDBX )
20 CONTINUE
*
* Step (3L): apply the inverse of the left singular vector
* matrix to BX.
*
IF( K.EQ.1 ) THEN
CALL CCOPY( NRHS, BX, LDBX, B, LDB )
IF( Z( 1 ).LT.ZERO ) THEN
CALL CSSCAL( NRHS, NEGONE, B, LDB )
END IF
ELSE
DO 100 J = 1, K
DIFLJ = DIFL( J )
DJ = POLES( J, 1 )
DSIGJ = -POLES( J, 2 )
IF( J.LT.K ) THEN
DIFRJ = -DIFR( J, 1 )
DSIGJP = -POLES( J+1, 2 )
END IF
IF( ( Z( J ).EQ.ZERO ) .OR. ( POLES( J, 2 ).EQ.ZERO ) )
$ THEN
RWORK( J ) = ZERO
ELSE
RWORK( J ) = -POLES( J, 2 )*Z( J ) / DIFLJ /
$ ( POLES( J, 2 )+DJ )
END IF
DO 30 I = 1, J - 1
IF( ( Z( I ).EQ.ZERO ) .OR.
$ ( POLES( I, 2 ).EQ.ZERO ) ) THEN
RWORK( I ) = ZERO
ELSE
*
* Use calls to the subroutine SLAMC3 to enforce the
* parentheses (x+y)+z. The goal is to prevent
* optimizing compilers from doing x+(y+z).
*
RWORK( I ) = POLES( I, 2 )*Z( I ) /
$ ( SLAMC3( POLES( I, 2 ), DSIGJ )-
$ DIFLJ ) / ( POLES( I, 2 )+DJ )
END IF
30 CONTINUE
DO 40 I = J + 1, K
IF( ( Z( I ).EQ.ZERO ) .OR.
$ ( POLES( I, 2 ).EQ.ZERO ) ) THEN
RWORK( I ) = ZERO
ELSE
RWORK( I ) = POLES( I, 2 )*Z( I ) /
$ ( SLAMC3( POLES( I, 2 ), DSIGJP )+
$ DIFRJ ) / ( POLES( I, 2 )+DJ )
END IF
40 CONTINUE
RWORK( 1 ) = NEGONE
TEMP = SNRM2( K, RWORK, 1 )
*
* Since B and BX are complex, the following call to SGEMV
* is performed in two steps (real and imaginary parts).
*
* CALL SGEMV( 'T', K, NRHS, ONE, BX, LDBX, WORK, 1, ZERO,
* $ B( J, 1 ), LDB )
*
I = K + NRHS*2
DO 60 JCOL = 1, NRHS
DO 50 JROW = 1, K
I = I + 1
RWORK( I ) = REAL( BX( JROW, JCOL ) )
50 CONTINUE
60 CONTINUE
CALL SGEMV( 'T', K, NRHS, ONE, RWORK( 1+K+NRHS*2 ), K,
$ RWORK( 1 ), 1, ZERO, RWORK( 1+K ), 1 )
I = K + NRHS*2
DO 80 JCOL = 1, NRHS
DO 70 JROW = 1, K
I = I + 1
RWORK( I ) = AIMAG( BX( JROW, JCOL ) )
70 CONTINUE
80 CONTINUE
CALL SGEMV( 'T', K, NRHS, ONE, RWORK( 1+K+NRHS*2 ), K,
$ RWORK( 1 ), 1, ZERO, RWORK( 1+K+NRHS ), 1 )
DO 90 JCOL = 1, NRHS
B( J, JCOL ) = CMPLX( RWORK( JCOL+K ),
$ RWORK( JCOL+K+NRHS ) )
90 CONTINUE
CALL CLASCL( 'G', 0, 0, TEMP, ONE, 1, NRHS, B( J, 1 ),
$ LDB, INFO )
100 CONTINUE
END IF
*
* Move the deflated rows of BX to B also.
*
IF( K.LT.MAX( M, N ) )
$ CALL CLACPY( 'A', N-K, NRHS, BX( K+1, 1 ), LDBX,
$ B( K+1, 1 ), LDB )
ELSE
*
* Apply back the right orthogonal transformations.
*
* Step (1R): apply back the new right singular vector matrix
* to B.
*
IF( K.EQ.1 ) THEN
CALL CCOPY( NRHS, B, LDB, BX, LDBX )
ELSE
DO 180 J = 1, K
DSIGJ = POLES( J, 2 )
IF( Z( J ).EQ.ZERO ) THEN
RWORK( J ) = ZERO
ELSE
RWORK( J ) = -Z( J ) / DIFL( J ) /
$ ( DSIGJ+POLES( J, 1 ) ) / DIFR( J, 2 )
END IF
DO 110 I = 1, J - 1
IF( Z( J ).EQ.ZERO ) THEN
RWORK( I ) = ZERO
ELSE
*
* Use calls to the subroutine SLAMC3 to enforce the
* parentheses (x+y)+z. The goal is to prevent optimizing
* compilers from doing x+(y+z).
*
RWORK( I ) = Z( J ) / ( SLAMC3( DSIGJ, -POLES( I+1,
$ 2 ) )-DIFR( I, 1 ) ) /
$ ( DSIGJ+POLES( I, 1 ) ) / DIFR( I, 2 )
END IF
110 CONTINUE
DO 120 I = J + 1, K
IF( Z( J ).EQ.ZERO ) THEN
RWORK( I ) = ZERO
ELSE
RWORK( I ) = Z( J ) / ( SLAMC3( DSIGJ, -POLES( I,
$ 2 ) )-DIFL( I ) ) /
$ ( DSIGJ+POLES( I, 1 ) ) / DIFR( I, 2 )
END IF
120 CONTINUE
*
* Since B and BX are complex, the following call to SGEMV
* is performed in two steps (real and imaginary parts).
*
* CALL SGEMV( 'T', K, NRHS, ONE, B, LDB, WORK, 1, ZERO,
* $ BX( J, 1 ), LDBX )
*
I = K + NRHS*2
DO 140 JCOL = 1, NRHS
DO 130 JROW = 1, K
I = I + 1
RWORK( I ) = REAL( B( JROW, JCOL ) )
130 CONTINUE
140 CONTINUE
CALL SGEMV( 'T', K, NRHS, ONE, RWORK( 1+K+NRHS*2 ), K,
$ RWORK( 1 ), 1, ZERO, RWORK( 1+K ), 1 )
I = K + NRHS*2
DO 160 JCOL = 1, NRHS
DO 150 JROW = 1, K
I = I + 1
RWORK( I ) = AIMAG( B( JROW, JCOL ) )
150 CONTINUE
160 CONTINUE
CALL SGEMV( 'T', K, NRHS, ONE, RWORK( 1+K+NRHS*2 ), K,
$ RWORK( 1 ), 1, ZERO, RWORK( 1+K+NRHS ), 1 )
DO 170 JCOL = 1, NRHS
BX( J, JCOL ) = CMPLX( RWORK( JCOL+K ),
$ RWORK( JCOL+K+NRHS ) )
170 CONTINUE
180 CONTINUE
END IF
*
* Step (2R): if SQRE = 1, apply back the rotation that is
* related to the right null space of the subproblem.
*
IF( SQRE.EQ.1 ) THEN
CALL CCOPY( NRHS, B( M, 1 ), LDB, BX( M, 1 ), LDBX )
CALL CSROT( NRHS, BX( 1, 1 ), LDBX, BX( M, 1 ), LDBX, C, S )
END IF
IF( K.LT.MAX( M, N ) )
$ CALL CLACPY( 'A', N-K, NRHS, B( K+1, 1 ), LDB,
$ BX( K+1, 1 ), LDBX )
*
* Step (3R): permute rows of B.
*
CALL CCOPY( NRHS, BX( 1, 1 ), LDBX, B( NLP1, 1 ), LDB )
IF( SQRE.EQ.1 ) THEN
CALL CCOPY( NRHS, BX( M, 1 ), LDBX, B( M, 1 ), LDB )
END IF
DO 190 I = 2, N
CALL CCOPY( NRHS, BX( I, 1 ), LDBX, B( PERM( I ), 1 ), LDB )
190 CONTINUE
*
* Step (4R): apply back the Givens rotations performed.
*
DO 200 I = GIVPTR, 1, -1
CALL CSROT( NRHS, B( GIVCOL( I, 2 ), 1 ), LDB,
$ B( GIVCOL( I, 1 ), 1 ), LDB, GIVNUM( I, 2 ),
$ -GIVNUM( I, 1 ) )
200 CONTINUE
END IF
*
RETURN
*
* End of CLALS0
*
END