SUBROUTINE DLALS0( ICOMPQ, NL, NR, SQRE, NRHS, B, LDB, BX, LDBX,
$ PERM, GIVPTR, GIVCOL, LDGCOL, GIVNUM, LDGNUM,
$ POLES, DIFL, DIFR, Z, K, C, S, WORK, INFO )
*
* -- LAPACK routine (version 3.1) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2006
*
* .. Scalar Arguments ..
INTEGER GIVPTR, ICOMPQ, INFO, K, LDB, LDBX, LDGCOL,
$ LDGNUM, NL, NR, NRHS, SQRE
DOUBLE PRECISION C, S
* ..
* .. Array Arguments ..
INTEGER GIVCOL( LDGCOL, * ), PERM( * )
DOUBLE PRECISION B( LDB, * ), BX( LDBX, * ), DIFL( * ),
$ DIFR( LDGNUM, * ), GIVNUM( LDGNUM, * ),
$ POLES( LDGNUM, * ), WORK( * ), Z( * )
* ..
*
* Purpose
* =======
*
* DLALS0 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).
*
* Arguments
* =========
*
* ICOMPQ (input) INTEGER
* Specifies whether singular vectors are to be computed in
* factored form:
* = 0: Left singular vector matrix.
* = 1: Right singular vector matrix.
*
* NL (input) INTEGER
* The row dimension of the upper block. NL >= 1.
*
* NR (input) INTEGER
* The row dimension of the lower block. NR >= 1.
*
* SQRE (input) 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.
*
* NRHS (input) INTEGER
* The number of columns of B and BX. NRHS must be at least 1.
*
* B (input/output) DOUBLE PRECISION 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.
*
* LDB (input) INTEGER
* The leading dimension of B. LDB must be at least
* max(1,MAX( M, N ) ).
*
* BX (workspace) DOUBLE PRECISION array, dimension ( LDBX, NRHS )
*
* LDBX (input) INTEGER
* The leading dimension of BX.
*
* PERM (input) INTEGER array, dimension ( N )
* The permutations (from deflation and sorting) applied
* to the two blocks.
*
* GIVPTR (input) INTEGER
* The number of Givens rotations which took place in this
* subproblem.
*
* GIVCOL (input) INTEGER array, dimension ( LDGCOL, 2 )
* Each pair of numbers indicates a pair of rows/columns
* involved in a Givens rotation.
*
* LDGCOL (input) INTEGER
* The leading dimension of GIVCOL, must be at least N.
*
* GIVNUM (input) DOUBLE PRECISION array, dimension ( LDGNUM, 2 )
* Each number indicates the C or S value used in the
* corresponding Givens rotation.
*
* LDGNUM (input) INTEGER
* The leading dimension of arrays DIFR, POLES and
* GIVNUM, must be at least K.
*
* POLES (input) DOUBLE PRECISION 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.
*
* DIFL (input) DOUBLE PRECISION 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.
*
* DIFR (input) DOUBLE PRECISION 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.
*
* Z (input) DOUBLE PRECISION array, dimension ( K )
* Contain the components of the deflation-adjusted updating row
* vector.
*
* K (input) INTEGER
* Contains the dimension of the non-deflated matrix,
* This is the order of the related secular equation. 1 <= K <=N.
*
* C (input) DOUBLE PRECISION
* C contains garbage if SQRE =0 and the C-value of a Givens
* rotation related to the right null space if SQRE = 1.
*
* S (input) DOUBLE PRECISION
* S contains garbage if SQRE =0 and the S-value of a Givens
* rotation related to the right null space if SQRE = 1.
*
* WORK (workspace) DOUBLE PRECISION array, dimension ( K )
*
* INFO (output) INTEGER
* = 0: successful exit.
* < 0: if INFO = -i, the i-th argument had an illegal value.
*
* Further Details
* ===============
*
* Based on contributions by
* Ming Gu and Ren-Cang Li, Computer Science Division, University of
* California at Berkeley, USA
* Osni Marques, LBNL/NERSC, USA
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO, NEGONE
PARAMETER ( ONE = 1.0D0, ZERO = 0.0D0, NEGONE = -1.0D0 )
* ..
* .. Local Scalars ..
INTEGER I, J, M, N, NLP1
DOUBLE PRECISION DIFLJ, DIFRJ, DJ, DSIGJ, DSIGJP, TEMP
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DGEMV, DLACPY, DLASCL, DROT, DSCAL,
$ XERBLA
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMC3, DNRM2
EXTERNAL DLAMC3, DNRM2
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
*
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
END IF
*
N = NL + NR + 1
*
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( 'DLALS0', -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 DROT( 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 DCOPY( NRHS, B( NLP1, 1 ), LDB, BX( 1, 1 ), LDBX )
DO 20 I = 2, N
CALL DCOPY( 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 DCOPY( NRHS, BX, LDBX, B, LDB )
IF( Z( 1 ).LT.ZERO ) THEN
CALL DSCAL( NRHS, NEGONE, B, LDB )
END IF
ELSE
DO 50 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
WORK( J ) = ZERO
ELSE
WORK( 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
WORK( I ) = ZERO
ELSE
WORK( I ) = POLES( I, 2 )*Z( I ) /
$ ( DLAMC3( 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
WORK( I ) = ZERO
ELSE
WORK( I ) = POLES( I, 2 )*Z( I ) /
$ ( DLAMC3( POLES( I, 2 ), DSIGJP )+
$ DIFRJ ) / ( POLES( I, 2 )+DJ )
END IF
40 CONTINUE
WORK( 1 ) = NEGONE
TEMP = DNRM2( K, WORK, 1 )
CALL DGEMV( 'T', K, NRHS, ONE, BX, LDBX, WORK, 1, ZERO,
$ B( J, 1 ), LDB )
CALL DLASCL( 'G', 0, 0, TEMP, ONE, 1, NRHS, B( J, 1 ),
$ LDB, INFO )
50 CONTINUE
END IF
*
* Move the deflated rows of BX to B also.
*
IF( K.LT.MAX( M, N ) )
$ CALL DLACPY( '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 DCOPY( NRHS, B, LDB, BX, LDBX )
ELSE
DO 80 J = 1, K
DSIGJ = POLES( J, 2 )
IF( Z( J ).EQ.ZERO ) THEN
WORK( J ) = ZERO
ELSE
WORK( J ) = -Z( J ) / DIFL( J ) /
$ ( DSIGJ+POLES( J, 1 ) ) / DIFR( J, 2 )
END IF
DO 60 I = 1, J - 1
IF( Z( J ).EQ.ZERO ) THEN
WORK( I ) = ZERO
ELSE
WORK( I ) = Z( J ) / ( DLAMC3( DSIGJ, -POLES( I+1,
$ 2 ) )-DIFR( I, 1 ) ) /
$ ( DSIGJ+POLES( I, 1 ) ) / DIFR( I, 2 )
END IF
60 CONTINUE
DO 70 I = J + 1, K
IF( Z( J ).EQ.ZERO ) THEN
WORK( I ) = ZERO
ELSE
WORK( I ) = Z( J ) / ( DLAMC3( DSIGJ, -POLES( I,
$ 2 ) )-DIFL( I ) ) /
$ ( DSIGJ+POLES( I, 1 ) ) / DIFR( I, 2 )
END IF
70 CONTINUE
CALL DGEMV( 'T', K, NRHS, ONE, B, LDB, WORK, 1, ZERO,
$ BX( J, 1 ), LDBX )
80 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 DCOPY( NRHS, B( M, 1 ), LDB, BX( M, 1 ), LDBX )
CALL DROT( NRHS, BX( 1, 1 ), LDBX, BX( M, 1 ), LDBX, C, S )
END IF
IF( K.LT.MAX( M, N ) )
$ CALL DLACPY( 'A', N-K, NRHS, B( K+1, 1 ), LDB, BX( K+1, 1 ),
$ LDBX )
*
* Step (3R): permute rows of B.
*
CALL DCOPY( NRHS, BX( 1, 1 ), LDBX, B( NLP1, 1 ), LDB )
IF( SQRE.EQ.1 ) THEN
CALL DCOPY( NRHS, BX( M, 1 ), LDBX, B( M, 1 ), LDB )
END IF
DO 90 I = 2, N
CALL DCOPY( NRHS, BX( I, 1 ), LDBX, B( PERM( I ), 1 ), LDB )
90 CONTINUE
*
* Step (4R): apply back the Givens rotations performed.
*
DO 100 I = GIVPTR, 1, -1
CALL DROT( NRHS, B( GIVCOL( I, 2 ), 1 ), LDB,
$ B( GIVCOL( I, 1 ), 1 ), LDB, GIVNUM( I, 2 ),
$ -GIVNUM( I, 1 ) )
100 CONTINUE
END IF
*
RETURN
*
* End of DLALS0
*
END