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.

ICOMPQ

          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

SMLSIZ

          SMLSIZ is INTEGER


         The maximum size of the subproblems at the bottom of the


         computation tree.

N

          N is INTEGER


         The row and column dimensions of the upper bidiagonal matrix.

NRHS

          NRHS is INTEGER


         The number of columns of B and BX. NRHS must be at least 1.

B

          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.

LDB

          LDB is INTEGER


         The leading dimension of B in the calling subprogram.


         LDB must be at least max(1,MAX( M, N ) ).

BX

          BX is COMPLEX*16 array, dimension ( LDBX, NRHS )


         On exit, the result of applying the left or right singular


         vector matrix to B.

LDBX

          LDBX is INTEGER


         The leading dimension of BX.

U

          U is DOUBLE PRECISION array, dimension ( LDU, SMLSIZ ).


         On entry, U contains the left singular vector matrices of all


         subproblems at the bottom level.

LDU

          LDU is INTEGER, LDU = > N.


         The leading dimension of arrays U, VT, DIFL, DIFR,


         POLES, GIVNUM, and Z.

VT

          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.

K

          K is INTEGER array, dimension ( N ).

DIFL

          DIFL is DOUBLE PRECISION array, dimension ( LDU, NLVL ).


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

DIFR

          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.

Z

          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.

POLES

          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.

GIVPTR

          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.

GIVCOL

          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.

LDGCOL

          LDGCOL is INTEGER, LDGCOL = > N.


         The leading dimension of arrays GIVCOL and PERM.

PERM

          PERM is INTEGER array, dimension ( LDGCOL, NLVL ).


         On entry, PERM(*, I) records permutations done on the I-th


         level of the computation tree.

GIVNUM

          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.

C

          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.

S

          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.

RWORK

          RWORK is DOUBLE PRECISION array, dimension at least


         MAX( (SMLSZ+1)*NRHS*3, N*(1+NRHS) + 2*NRHS ).

IWORK

          IWORK is INTEGER array.


         The dimension must be at least 3 * N

INFO

          INFO is INTEGER


          = 0:  successful exit.


          < 0:  if INFO = -i, the i-th argument had an illegal value.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

September 2012

Ming Gu and Ren-Cang Li, Computer Science Division, University of California at Berkeley, USA
Osni Marques, LBNL/NERSC, USA