LALS0 - 5.2 English - 68552

LALS0 applies back multiplying factors in solving the least squares problem

using divide and conquer SVD approach. Used by sgelsd.

Purpose:

   LALS0 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).

Parameters:

• ICOMPQ – [in]

  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 – [in]

  NL is INTEGER

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

• NR – [in]

  NR is INTEGER

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

• SQRE – [in]

  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 – [in]

  NRHS is INTEGER

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

• B – [inout]

  B is REAL 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 – [in]

  LDB is INTEGER

  The leading dimension of B. LDB must be at least fla_max(1,MAX(M, N)).

• BX – [out] BX is REAL array, dimension (LDBX, NRHS)

• LDBX – [in]

  LDBX is INTEGER

  The leading dimension of BX.

• PERM – [in]

  PERM is INTEGER array, dimension (N)

  The permutations (from deflation and sorting) applied to the two blocks.

• GIVPTR – [in]

  GIVPTR is INTEGER

  The number of Givens rotations which took place in this subproblem.

• GIVCOL – [in]

  GIVCOL is INTEGER array, dimension (LDGCOL, 2)

  Each pair of numbers indicates a pair of rows/columns involved in a Givens rotation.

• LDGCOL – [in]

  LDGCOL is INTEGER

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

• GIVNUM – [in]

  GIVNUM is REAL array, dimension (LDGNUM, 2)

  Each number indicates the C or S value used in the corresponding Givens rotation.

• LDGNUM – [in]

  LDGNUM is INTEGER

  The leading dimension of arrays DIFR, POLES and GIVNUM, must be at least K.

• POLES – [in]

  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 – [in]

  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 – [in]

  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 – [in]

  Z is REAL array, dimension (K)

  Contain the components of the deflation-adjusted updating row vector.

• K – [in]

  K is INTEGER

  Contains the dimension of the non-deflated matrix, This is the order of the related secular equation. 1 <= K <=N.

• C – [in]

  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 – [in]

  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.

• WORK – [out] WORK is REAL array, dimension (K)

• INFO – [out]

  INFO is INTEGER

  = 0: successful exit.

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