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).

ICOMPQ

          ICOMPQ is INTEGER


         Specifies whether singular vectors are to be computed in


         factored form:


         = 0: Left singular vector matrix.


         = 1: Right singular vector matrix.

NL

          NL is INTEGER


         The row dimension of the upper block. NL >= 1.

NR

          NR is INTEGER


         The row dimension of the lower block. NR >= 1.

SQRE

          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.

NRHS

          NRHS is INTEGER


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

B

          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.

LDB

          LDB is INTEGER


         The leading dimension of B. LDB must be at least


         max(1,MAX( M, N ) ).

BX

          BX is COMPLEX array, dimension ( LDBX, NRHS )

LDBX

          LDBX is INTEGER


         The leading dimension of BX.

PERM

          PERM is INTEGER array, dimension ( N )


         The permutations (from deflation and sorting) applied


         to the two blocks.

GIVPTR

          GIVPTR is INTEGER


         The number of Givens rotations which took place in this


         subproblem.

GIVCOL

          GIVCOL is INTEGER array, dimension ( LDGCOL, 2 )


         Each pair of numbers indicates a pair of rows/columns


         involved in a Givens rotation.

LDGCOL

          LDGCOL is INTEGER


         The leading dimension of GIVCOL, must be at least N.

GIVNUM

          GIVNUM is REAL array, dimension ( LDGNUM, 2 )


         Each number indicates the C or S value used in the


         corresponding Givens rotation.

LDGNUM

          LDGNUM is INTEGER


         The leading dimension of arrays DIFR, POLES and


         GIVNUM, must be at least K.

POLES

          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.

DIFL

          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.

DIFR

          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.

Z

          Z is REAL array, dimension ( K )


         Contain the components of the deflation-adjusted updating row


         vector.

K

          K is INTEGER


         Contains the dimension of the non-deflated matrix,


         This is the order of the related secular equation. 1 <= K <=N.

C

          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.

S

          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.

RWORK

          RWORK is REAL array, dimension


         ( K*(1+NRHS) + 2*NRHS )

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.

December 2016

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