# slaorhr_col_getrfnp.f - Man Page

SRC/slaorhr_col_getrfnp.f

## Synopsis

### Functions/Subroutines

subroutine slaorhr_col_getrfnp (m, n, a, lda, d, info)
SLAORHR_COL_GETRFNP

## Function/Subroutine Documentation

### subroutine slaorhr_col_getrfnp (integer m, integer n, real, dimension( lda, * ) a, integer lda, real, dimension( * ) d, integer info)

SLAORHR_COL_GETRFNP

Purpose:

``` SLAORHR_COL_GETRFNP computes the modified LU factorization without
pivoting of a real general M-by-N matrix A. The factorization has
the form:

A - S = L * U,

where:
S is a m-by-n diagonal sign matrix with the diagonal D, so that
D(i) = S(i,i), 1 <= i <= min(M,N). The diagonal D is constructed
as D(i)=-SIGN(A(i,i)), where A(i,i) is the value after performing
i-1 steps of Gaussian elimination. This means that the diagonal
element at each step of 'modified' Gaussian elimination is
at least one in absolute value (so that division-by-zero not
not possible during the division by the diagonal element);

L is a M-by-N lower triangular matrix with unit diagonal elements
(lower trapezoidal if M > N);

and U is a M-by-N upper triangular matrix
(upper trapezoidal if M < N).

This routine is an auxiliary routine used in the Householder
reconstruction routine SORHR_COL. In SORHR_COL, this routine is
applied to an M-by-N matrix A with orthonormal columns, where each
element is bounded by one in absolute value. With the choice of
the matrix S above, one can show that the diagonal element at each
step of Gaussian elimination is the largest (in absolute value) in
the column on or below the diagonal, so that no pivoting is required
for numerical stability [1].

For more details on the Householder reconstruction algorithm,
including the modified LU factorization, see [1].

This is the blocked right-looking version of the algorithm,
calling Level 3 BLAS to update the submatrix. To factorize a block,
this routine calls the recursive routine SLAORHR_COL_GETRFNP2.

[1] 'Reconstructing Householder vectors from tall-skinny QR',
G. Ballard, J. Demmel, L. Grigori, M. Jacquelin, H.D. Nguyen,
E. Solomonik, J. Parallel Distrib. Comput.,
vol. 85, pp. 3-31, 2015.```
Parameters

M

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

N

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

A

```          A is REAL array, dimension (LDA,N)
On entry, the M-by-N matrix to be factored.
On exit, the factors L and U from the factorization
A-S=L*U; the unit diagonal elements of L are not stored.```

LDA

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

D

```          D is REAL array, dimension min(M,N)
The diagonal elements of the diagonal M-by-N sign matrix S,
D(i) = S(i,i), where 1 <= i <= min(M,N). The elements can
be only plus or minus one.```

INFO

```          INFO is INTEGER
= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal value```
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

``` November 2019, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley```

Definition at line 145 of file slaorhr_col_getrfnp.f.

## Author

Generated automatically by Doxygen for LAPACK from the source code.

## Referenced By

The man page slaorhr_col_getrfnp(3) is an alias of slaorhr_col_getrfnp.f(3).

Tue Nov 28 2023 12:08:42 Version 3.12.0 LAPACK