# ?gelqf

Computes the LQ factorization of a general m-by-n matrix.

## Syntax

FORTRAN 77:

call sgelqf(m, n, a, lda, tau, work, lwork, info)

call dgelqf(m, n, a, lda, tau, work, lwork, info)

call cgelqf(m, n, a, lda, tau, work, lwork, info)

call zgelqf(m, n, a, lda, tau, work, lwork, info)

Fortran 95:

call gelqf(a [, tau] [,info])

## Description

The routine forms the LQ factorization of a general m-by-n matrix A (see Orthogonal Factorizations). No pivoting is performed.

The routine does not form the matrix Q explicitly. Instead, Q is represented as a product of min(m, n) elementary reflectors. Routines are provided to work with Q in this representation.

## Input Parameters

m

  INTEGER. The number of rows in the matrix A (`m ≥ 0`).

n

  INTEGER. The number of columns in A (`n ≥ 0`).

a, work

  REAL for sgelqf

  DOUBLE PRECISION for dgelqf

  COMPLEX for cgelqf

  DOUBLE COMPLEX for zgelqf.

  Arrays:

  a(lda,*) contains the matrix A.

  The second dimension of a must be at least max(1, n).

  work is a workspace array, its dimension `max(1, lwork)`.

lda

  INTEGER. The first dimension of a; at least max(1, m).

lwork

  INTEGER. The size of the work array; at least max(1, m).

  If `lwork = -1`, then a workspace query is assumed; the routine only calculates the optimal size of the work array, returns this value as the first entry of the work array, and no error message related to lwork is issued by xerbla.

  See Application Notes for the suggested value of lwork.

## Output Parameters

a

  Overwritten by the factorization data as follows:

  If m ≤ n, the elements above the diagonal are overwritten by the details of the unitary (orthogonal) matrix Q, and the lower triangle is overwritten by the corresponding elements of the lower triangular matrix L.

  If m > n, the strictly upper triangular part is overwritten by the details of the matrix Q, and the remaining elements are overwritten by the corresponding elements of the m-by-n lower trapezoidal matrix L.

tau

  REAL for sgelqf

  DOUBLE PRECISION for dgelqf

  COMPLEX for cgelqf

  DOUBLE COMPLEX for zgelqf.

  Array, DIMENSION at least `max(1, min(m, n))`.

  Contains additional information on the matrix Q.

`work(1)`

  If `info = 0`, on exit `work(1)` contains the minimum value of lwork required for optimum performance. Use this lwork for subsequent runs.

info

  INTEGER.

  If `info = 0`, the execution is successful.

  If `info = -i`, the i-th parameter had an illegal value.

## Fortran 95 Interface Notes

Routines in Fortran 95 interface have fewer arguments in the calling sequence than their FORTRAN 77 counterparts. For general conventions applied to skip redundant or restorable arguments, see Fortran 95 Interface Conventions.

Specific details for the routine gelqf interface are the following:

a

  Holds the matrix A of size (m,n).

tau

  Holds the vector of length min(m,n).

## Application Notes

For better performance, try using `lwork =m*blocksize`, where blocksize is a machine-dependent value (typically, 16 to 64) required for optimum performance of the blocked algorithm.

If you are in doubt how much workspace to supply, use a generous value of lwork for the first run or set `lwork = -1`.

If you choose the first option and set any of admissible lwork sizes, which is no less than the minimal value described, the routine completes the task, though probably not so fast as with a recommended workspace, and provides the recommended workspace in the first element of the corresponding array work on exit. Use this value (`work(1)`) for subsequent runs.

If you set `lwork = -1`, the routine returns immediately and provides the recommended workspace in the first element of the corresponding array (work). This operation is called a workspace query.

Note that if you set lwork to less than the minimal required value and not -1, the routine returns immediately with an error exit and does not provide any information on the recommended workspace.

The computed factorization is the exact factorization of a matrix A + E, where

`||E||2 = O(ε) ||A||2`.

The approximate number of floating-point operations for real flavors is

(4/3)n3

  if `m = n`,

(2/3)n2(3m-n)

  if `m > n`,

(2/3)m2(3n-m)

  if `m < n`.

The number of operations for complex flavors is 4 times greater.

To find the minimum-norm solution of an underdetermined least squares problem minimizing `||A*x - b||2` for all columns b of a given matrix B, you can call the following:

?gelqf (this routine)

  to factorize `A = L*Q`;

?trsm (a BLAS routine)

  to solve `L*Y = B` for Y;

?ormlq

  to compute `X = (Q1)T*Y` (for real matrices);

?unmlq

  to compute `X = (Q1)H*Y` (for complex matrices).

(The columns of the computed X are the minimum-norm solution vectors x. Here A is an m-by-n matrix with `m < n`; Q1 denotes the first m columns of Q).

To compute the elements of Q explicitly, call

?orglq

  (for real matrices)

?unglq

  (for complex matrices).

Parent topic: Orthogonal Factorizations