SLALS0
November 2006
Purpose
SLALS0 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).
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).
Arguments
| ICOMPQ |
(input) INTEGER
Specifies whether singular vectors are to be computed in
factored form: = 0: Left singular vector matrix. = 1: Right singular vector matrix. |
| NL |
(input) INTEGER
The row dimension of the upper block. NL >= 1.
|
| NR |
(input) INTEGER
The row dimension of the lower block. NR >= 1.
|
| SQRE |
(input) 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 |
(input) INTEGER
The number of columns of B and BX. NRHS must be at least 1.
|
| B |
(input/output) 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 |
(input) INTEGER
The leading dimension of B. LDB must be at least
max(1,MAX( M, N ) ). |
| BX |
(workspace) REAL array, dimension ( LDBX, NRHS )
|
| LDBX |
(input) INTEGER
The leading dimension of BX.
|
| PERM |
(input) INTEGER array, dimension ( N )
The permutations (from deflation and sorting) applied
to the two blocks. |
| GIVPTR |
(input) INTEGER
The number of Givens rotations which took place in this
subproblem. |
| GIVCOL |
(input) INTEGER array, dimension ( LDGCOL, 2 )
Each pair of numbers indicates a pair of rows/columns
involved in a Givens rotation. |
| LDGCOL |
(input) INTEGER
The leading dimension of GIVCOL, must be at least N.
|
| GIVNUM |
(input) REAL array, dimension ( LDGNUM, 2 )
Each number indicates the C or S value used in the
corresponding Givens rotation. |
| LDGNUM |
(input) INTEGER
The leading dimension of arrays DIFR, POLES and
GIVNUM, must be at least K. |
| POLES |
(input) 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 |
(input) 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 |
(input) 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 |
(input) REAL array, dimension ( K )
Contain the components of the deflation-adjusted updating row
vector. |
| K |
(input) INTEGER
Contains the dimension of the non-deflated matrix,
This is the order of the related secular equation. 1 <= K <=N. |
| C |
(input) 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 |
(input) 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 |
(workspace) REAL array, dimension ( K )
|
| INFO |
(output) INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value. |
Further Details
Based on contributions by
Ming Gu and Ren-Cang Li, Computer Science Division, University of
California at Berkeley, USA
Osni Marques, LBNL/NERSC, USA
Ming Gu and Ren-Cang Li, Computer Science Division, University of
California at Berkeley, USA
Osni Marques, LBNL/NERSC, USA