ZLALS0 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*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. LDB must be at least
         max(1,MAX( M, N ) ).

BX

          BX is COMPLEX*16 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 DOUBLE PRECISION 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 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

          DIFL is 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

          DIFR is 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

          Z is DOUBLE PRECISION 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 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

          S is 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.

RWORK

          RWORK is DOUBLE PRECISION 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.

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