SGSVJ1(1LAPACK routine (version 3.2)SGSVJ1(1)NAME
SGSVJ1 - is called from SGESVJ as a pre-processor and that is its main
purpose
SYNOPSIS
SUBROUTINE SGSVJ1( JOBV, M, N, N1, A, LDA, D, SVA, MV, V, LDV,
+ EPS, SFMIN, TOL, NSWEEP, WORK, LWORK, INFO )
IMPLICIT NONE
REAL EPS, SFMIN, TOL
INTEGER INFO, LDA, LDV, LWORK, M, MV, N, N1, NSWEEP
CHARACTER*1 JOBV
REAL A( LDA, * ), D( N ), SVA( N ), V( LDV, * ),
+ WORK( LWORK )
PURPOSE
SGSVJ1 is called from SGESVJ as a pre-processor and that is its main
purpose. It applies Jacobi rotations in the same way as SGESVJ does,
but it targets only particular pivots and it does not check convergence
(stopping criterion). Few tunning parameters (marked by [TP]) are
available for the implementer.
Further Details
SGSVJ1 applies few sweeps of Jacobi rotations in the column space of
the input M-by-N matrix A. The pivot pairs are taken from the (1,2)
off-diagonal block in the corresponding N-by-N Gram matrix A^T * A. The
block-entries (tiles) of the (1,2) off-diagonal block are marked by the
[x]'s in the following scheme:
| * * * [x] [x] [x]|
| * * * [x] [x] [x]| Row-cycling in the nblr-by-nblc [x]
blocks.
| * * * [x] [x] [x]| Row-cyclic pivoting inside each [x]
block.
|[x] [x] [x] * * * |
|[x] [x] [x] * * * |
|[x] [x] [x] * * * |
In terms of the columns of A, the first N1 columns are rotated
'against' the remaining N-N1 columns, trying to increase the angle
between the corresponding subspaces. The off-diagonal block is N1-by(N-
N1) and it is tiled using quadratic tiles of side KBL. Here, KBL is a
tunning parmeter. The number of sweeps is given in NSWEEP and the
orthogonality threshold is given in TOL.
Contributors
~~~~~~~~~~~~
Zlatko Drmac (Zagreb, Croatia) and Kresimir Veselic (Hagen, Germany)
ARGUMENTS
JOBV (input) CHARACTER*1
Specifies whether the output from this procedure is used to
compute the matrix V:
= 'V': the product of the Jacobi rotations is accumulated by
postmulyiplying the N-by-N array V. (See the description of
V.) = 'A': the product of the Jacobi rotations is accumulated
by postmulyiplying the MV-by-N array V. (See the descriptions
of MV and V.) = 'N': the Jacobi rotations are not accumulated.
M (input) INTEGER
The number of rows of the input matrix A. M >= 0.
N (input) INTEGER
The number of columns of the input matrix A. M >= N >= 0.
N1 (input) INTEGER
N1 specifies the 2 x 2 block partition, the first N1 columns
are rotated 'against' the remaining N-N1 columns of A.
A (input/output) REAL array, dimension (LDA,N)
On entry, M-by-N matrix A, such that A*diag(D) represents the
input matrix. On exit, A_onexit * D_onexit represents the
input matrix A*diag(D) post-multiplied by a sequence of Jacobi
rotations, where the rotation threshold and the total number of
sweeps are given in TOL and NSWEEP, respectively. (See the
descriptions of N1, D, TOL and NSWEEP.)
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,M).
D (input/workspace/output) REAL array, dimension (N)
The array D accumulates the scaling factors from the fast
scaled Jacobi rotations. On entry, A*diag(D) represents the
input matrix. On exit, A_onexit*diag(D_onexit) represents the
input matrix post-multiplied by a sequence of Jacobi rotations,
where the rotation threshold and the total number of sweeps are
given in TOL and NSWEEP, respectively. (See the descriptions
of N1, A, TOL and NSWEEP.)
SVA (input/workspace/output) REAL array, dimension (N)
On entry, SVA contains the Euclidean norms of the columns of
the matrix A*diag(D). On exit, SVA contains the Euclidean
norms of the columns of the matrix onexit*diag(D_onexit).
MV (input) INTEGER
If JOBV .EQ. 'A', then MV rows of V are post-multipled by a
sequence of Jacobi rotations. If JOBV = 'N', then MV is not
referenced.
V (input/output) REAL array, dimension (LDV,N)
If JOBV .EQ. 'V' then N rows of V are post-multipled by a
sequence of Jacobi rotations. If JOBV .EQ. 'A' then MV rows of
V are post-multipled by a sequence of Jacobi rotations. If
JOBV = 'N', then V is not referenced.
LDV (input) INTEGER
The leading dimension of the array V, LDV >= 1. If JOBV =
'V', LDV .GE. N. If JOBV = 'A', LDV .GE. MV.
EPS (input) INTEGER
EPS = SLAMCH('Epsilon')
SFMIN (input) INTEGER
SFMIN = SLAMCH('Safe Minimum')
TOL (input) REAL
TOL is the threshold for Jacobi rotations. For a pair A(:,p),
A(:,q) of pivot columns, the Jacobi rotation is
applied only if ABS(COS(angle(A(:,p),A(:,q)))) .GT. TOL.
NSWEEP (input) INTEGER
NSWEEP is the number of sweeps of Jacobi rotations to be per‐
formed.
WORK (workspace) REAL array, dimension LWORK.
LWORK (input) INTEGER
LWORK is the dimension of WORK. LWORK .GE. M.
INFO (output) INTEGER
= 0 : successful exit.
< 0 : if INFO = -i, then the i-th argument had an illegal value
LAPACK routine (version 3.2) November 2008 SGSVJ1(1)
