DGSVJ1(3) LAPACK routine of NEC Numeric Library Collection DGSVJ1(3)
NAME
DGSVJ1
SYNOPSIS
SUBROUTINE DGSVJ1 (JOBV, M, N, N1, A, LDA, D, SVA, MV, V, LDV, EPS,
SFMIN, TOL, NSWEEP, WORK, LWORK, INFO)
PURPOSE
DGSVJ1 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
~~~~~~~~~~~~~~~
DGSVJ1 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.
ARGUMENTS
JOBV (input)
JOBV is 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)
M is INTEGER
The number of rows of the input matrix A. M >= 0.
N (input)
N is INTEGER
The number of columns of the input matrix A.
M >= N >= 0.
N1 (input)
N1 is 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)
A is DOUBLE PRECISION 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)
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
D (input/output)
D is DOUBLE PRECISION 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/output)
SVA is DOUBLE PRECISION 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)
MV is 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)
V is DOUBLE PRECISION 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)
LDV is INTEGER
The leading dimension of the array V, LDV >= 1.
If JOBV = 'V', LDV .GE. N.
If JOBV = 'A', LDV .GE. MV.
EPS (input)
EPS is DOUBLE PRECISION
EPS = DLAMCH('Epsilon')
SFMIN (input)
SFMIN is DOUBLE PRECISION
SFMIN = DLAMCH('Safe Minimum')
TOL (input)
TOL is DOUBLE PRECISION
TOL is the threshold for Jacobi rotations. For a pair
A(:,p), A(:,q) of pivot columns, the Jacobi rotation is
applied only if DABS(COS(angle(A(:,p),A(:,q)))) .GT. TOL.
NSWEEP (input)
NSWEEP is INTEGER
NSWEEP is the number of sweeps of Jacobi rotations to be
performed.
WORK (output)
WORK is DOUBLE PRECISION array, dimension (LWORK)
LWORK (input)
LWORK is INTEGER
LWORK is the dimension of WORK. LWORK .GE. M.
INFO (output)
INFO is INTEGER
= 0 : successful exit.
< 0 : if INFO = -i, then the i-th argument had an illegal value
LAPACK routine 31 October 2017 DGSVJ1(3)