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 performed.

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