CLALSA - i 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.)

SUBROUTINE CLALSA(

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 )

INTEGER ICOMPQ, INFO, LDB, LDBX, LDGCOL, LDU, N, NRHS, SMLSIZ

INTEGER GIVCOL( LDGCOL, * ), GIVPTR( * ), IWORK( * ), K( * ), PERM( LDGCOL, * )

REAL C( * ), DIFL( LDU, * ), DIFR( LDU, * ), GIVNUM( LDU, * ), POLES( LDU, * ), RWORK( * ), S( * ), U( LDU, * ), VT( LDU, * ), Z( LDU, * )

COMPLEX B( LDB, * ), BX( LDBX, * )

CLALSA 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, CLALSA applies the inverse of the left singular vector matrix of an upper bidiagonal matrix to the right hand side; and if ICOMPQ = 1, CLALSA applies the right singular vector matrix to the right hand side. The singular vector matrices were generated in compact form by CLALSA.

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) 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.

LDB (input) INTEGER

The leading dimension of B in the calling subprogram. LDB must be at least max(1,MAX( M, N ) ).

BX (output) COMPLEX 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) REAL 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) REAL 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) REAL array, dimension ( LDU, NLVL ).

where NLVL = INT(log_2 (N/(SMLSIZ+1))) + 1.

DIFR (input) REAL 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) REAL 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) REAL 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) REAL 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) REAL 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) REAL 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.

RWORK (workspace) REAL array, dimension at least

max ( N, (SMLSZ+1)*NRHS*3 ).

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.

Based on contributions by

Ming Gu and Ren-Cang Li, Computer Science Division, University of California at Berkeley, USA

Osni Marques, LBNL/NERSC, USA