# ?geqlf

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

## Syntax

FORTRAN 77:

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

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

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

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

Fortran 95:

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

## Description

The routine forms the QL factorization of a general m-by-n matrix A. 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 sgeqlf

  DOUBLE PRECISION for dgeqlf

  COMPLEX for cgeqlf

  DOUBLE COMPLEX for zgeqlf.

  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, n).

  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 on exit by the factorization data as follows:

  if `m ≥ n`, the lower triangle of the subarray a(m-n+1:m, 1:n) contains the n-by-n lower triangular matrix L; if `m ≤ n`, the elements on and below the (n-m)-th superdiagonal contain the m-by-n lower trapezoidal matrix L; in both cases, the remaining elements, with the array tau, represent the orthogonal/unitary matrix Q as a product of elementary reflectors.

tau

  REAL for sgeqlf

  DOUBLE PRECISION for dgeqlf

  COMPLEX for cgeqlf

  DOUBLE COMPLEX for zgeqlf.

  Array, DIMENSION at least max(1, min(m, n)). Contains scalar factors of the elementary reflectors for the matrix Q.

`work(1)`

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

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 geqlf 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 =n*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.

Related routines include:

?orgql

  to generate matrix Q (for real matrices);

?ungql

  to generate matrix Q (for complex matrices);

?ormql

  to apply matrix Q (for real matrices);

?unmql

  to apply matrix Q (for complex matrices).

Parent topic: Orthogonal Factorizations