SUBROUTINE DLALSA( ICOMPQ, SMLSIZ, N, NRHS, B, LDB, BX, LDBX, U,
$ LDU, VT, K, DIFL, DIFR, Z, POLES, GIVPTR,
$ GIVCOL, LDGCOL, PERM, GIVNUM, C, S, WORK,
$ IWORK, INFO )
*
* -- LAPACK routine (version 3.1) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2006
*
* .. Scalar Arguments ..
INTEGER ICOMPQ, INFO, LDB, LDBX, LDGCOL, LDU, N, NRHS,
$ SMLSIZ
* ..
* .. Array Arguments ..
INTEGER GIVCOL( LDGCOL, * ), GIVPTR( * ), IWORK( * ),
$ K( * ), PERM( LDGCOL, * )
DOUBLE PRECISION B( LDB, * ), BX( LDBX, * ), C( * ),
$ DIFL( LDU, * ), DIFR( LDU, * ),
$ GIVNUM( LDU, * ), POLES( LDU, * ), S( * ),
$ U( LDU, * ), VT( LDU, * ), WORK( * ),
$ Z( LDU, * )
* ..
*
* Purpose
* =======
*
* DLALSA 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, DLALSA applies the inverse of the left singular vector
* matrix of an upper bidiagonal matrix to the right hand side; and if
* ICOMPQ = 1, DLALSA applies the right singular vector matrix to the
* right hand side. The singular vector matrices were generated in
* compact form by DLALSA.
*
* Arguments
* =========
*
*
* ICOMPQ (input) INTEGER
* Specifies whether the left or the right singular vector
* matrix is involved.
* = 0: Left singular vector matrix
* = 1: Right singular vector matrix
*
* SMLSIZ (input) INTEGER
* The maximum size of the subproblems at the bottom of the
* computation tree.
*
* N (input) INTEGER
* The row and column dimensions of the upper bidiagonal matrix.
*
* 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 in the calling subprogram.
* LDB must be at least max(1,MAX( M, N ) ).
*
* BX (output) DOUBLE PRECISION array, dimension ( LDBX, NRHS )
* On exit, the result of applying the left or right singular
* vector matrix to B.
*
* LDBX (input) INTEGER
* The leading dimension of BX.
*
* U (input) DOUBLE PRECISION array, dimension ( LDU, SMLSIZ ).
* On entry, U contains the left singular vector matrices of all
* subproblems at the bottom level.
*
* LDU (input) INTEGER, LDU = > N.
* The leading dimension of arrays U, VT, DIFL, DIFR,
* POLES, GIVNUM, and Z.
*
* VT (input) DOUBLE PRECISION array, dimension ( LDU, SMLSIZ+1 ).
* On entry, VT' contains the right singular vector matrices of
* all subproblems at the bottom level.
*
* K (input) INTEGER array, dimension ( N ).
*
* DIFL (input) DOUBLE PRECISION array, dimension ( LDU, NLVL ).
* where NLVL = INT(log_2 (N/(SMLSIZ+1))) + 1.
*
* DIFR (input) 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.
*
* Z (input) 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.
*
* POLES (input) 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.
*
* GIVPTR (input) INTEGER array, dimension ( N ).
* On entry, GIVPTR( I ) records the number of Givens
* rotations performed on the I-th problem on the computation
* tree.
*
* GIVCOL (input) 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.
*
* LDGCOL (input) INTEGER, LDGCOL = > N.
* The leading dimension of arrays GIVCOL and PERM.
*
* PERM (input) INTEGER array, dimension ( LDGCOL, NLVL ).
* On entry, PERM(*, I) records permutations done on the I-th
* level of the computation tree.
*
* GIVNUM (input) 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.
*
* C (input) 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.
*
* S (input) 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.
*
* WORK (workspace) DOUBLE PRECISION array.
* The dimension must be at least N.
*
* IWORK (workspace) INTEGER array.
* The dimension must be at least 3 * N
*
* 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 ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
* ..
* .. Local Scalars ..
INTEGER I, I1, IC, IM1, INODE, J, LF, LL, LVL, LVL2,
$ ND, NDB1, NDIML, NDIMR, NL, NLF, NLP1, NLVL,
$ NR, NRF, NRP1, SQRE
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DGEMM, DLALS0, DLASDT, XERBLA
* ..
* .. 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( 'DLALSA', -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 50.
*
IF( ICOMPQ.EQ.1 ) THEN
GO TO 50
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 10 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
CALL DGEMM( 'T', 'N', NL, NRHS, NL, ONE, U( NLF, 1 ), LDU,
$ B( NLF, 1 ), LDB, ZERO, BX( NLF, 1 ), LDBX )
CALL DGEMM( 'T', 'N', NR, NRHS, NR, ONE, U( NRF, 1 ), LDU,
$ B( NRF, 1 ), LDB, ZERO, BX( NRF, 1 ), LDBX )
10 CONTINUE
*
* Next copy the rows of B that correspond to unchanged rows
* in the bidiagonal matrix to BX.
*
DO 20 I = 1, ND
IC = IWORK( INODE+I-1 )
CALL DCOPY( NRHS, B( IC, 1 ), LDB, BX( IC, 1 ), LDBX )
20 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 40 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 30 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 DLALS0( 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 ), WORK,
$ INFO )
30 CONTINUE
40 CONTINUE
GO TO 90
*
* ICOMPQ = 1: applying back the right singular vector factors.
*
50 CONTINUE
*
* First now go through the right singular vector matrices of all
* the tree nodes top-down.
*
J = 0
DO 70 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 60 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 DLALS0( 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 ), WORK,
$ INFO )
60 CONTINUE
70 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 80 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
CALL DGEMM( 'T', 'N', NLP1, NRHS, NLP1, ONE, VT( NLF, 1 ), LDU,
$ B( NLF, 1 ), LDB, ZERO, BX( NLF, 1 ), LDBX )
CALL DGEMM( 'T', 'N', NRP1, NRHS, NRP1, ONE, VT( NRF, 1 ), LDU,
$ B( NRF, 1 ), LDB, ZERO, BX( NRF, 1 ), LDBX )
80 CONTINUE
*
90 CONTINUE
*
RETURN
*
* End of DLALSA
*
END