SYNOPSIS

SUBROUTINE SLAQPS(

M, N, OFFSET, NB, KB, A, LDA, JPVT, TAU, VN1, VN2, AUXV, F, LDF )

    

INTEGER KB, LDA, LDF, M, N, NB, OFFSET

    

INTEGER JPVT( * )

    

REAL A( LDA, * ), AUXV( * ), F( LDF, * ), TAU( * ), VN1( * ), VN2( * )

PURPOSE

SLAQPS computes a step of QR factorization with column pivoting of a real M-by-N matrix A by using Blas-3. It tries to factorize NB columns from A starting from the row OFFSET+1, and updates all of the matrix with Blas-3 xGEMM.
In some cases, due to catastrophic cancellations, it cannot factorize NB columns. Hence, the actual number of factorized columns is returned in KB.
Block A(1:OFFSET,1:N) is accordingly pivoted, but not factorized.

ARGUMENTS

M (input) INTEGER

The number of rows of the matrix A. M >= 0.

N (input) INTEGER

The number of columns of the matrix A. N >= 0

OFFSET (input) INTEGER

The number of rows of A that have been factorized in previous steps.

NB (input) INTEGER

The number of columns to factorize.

KB (output) INTEGER

The number of columns actually factorized.

A (input/output) REAL array, dimension (LDA,N)

On entry, the M-by-N matrix A. On exit, block A(OFFSET+1:M,1:KB) is the triangular
factor obtained and block A(1:OFFSET,1:N) has been accordingly pivoted, but no factorized. The rest of the matrix, block A(OFFSET+1:M,KB+1:N) has been updated.

LDA (input) INTEGER

The leading dimension of the array A. LDA >= max(1,M).

JPVT (input/output) INTEGER array, dimension (N)

JPVT(I) = K <==> Column K of the full matrix A has been permuted into position I in AP.

TAU (output) REAL array, dimension (KB)

The scalar factors of the elementary reflectors.

VN1 (input/output) REAL array, dimension (N)

The vector with the partial column norms.

VN2 (input/output) REAL array, dimension (N)

The vector with the exact column norms.

AUXV (input/output) REAL array, dimension (NB)

Auxiliar vector.

F (input/output) REAL array, dimension (LDF,NB)

Matrix F' = L*Y'*A.

LDF (input) INTEGER

The leading dimension of the array F. LDF >= max(1,N).

FURTHER DETAILS

Based on contributions by

  G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
  X. Sun, Computer Science Dept., Duke University, USA
Partial column norm updating strategy modified by

  Z. Drmac and Z. Bujanovic, Dept. of Mathematics,

  University of Zagreb, Croatia.

  June 2006.
For more details see LAPACK Working Note 176.