# ?orglq

Generates the real orthogonal matrix Q of the LQ factorization formed by ?gelqf.

## Syntax

FORTRAN 77:

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

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

Fortran 95:

call orglq(a, tau [,info])

## Description

The routine generates the whole or part of n-by-n orthogonal matrix Q of the LQ factorization formed by the routines sgelqf/gelqf. Use this routine after a call to sgelqf/dgelqf.

Usually Q is determined from the LQ factorization of an p-by-n matrix A with `n ≥ p`. To compute the whole matrix Q, use:
`call ?orglq(n, n, p, a, lda, tau, work, lwork, info)`

To compute the leading p rows of Q, which form an orthonormal basis in the space spanned by the rows of A, use:
`call ?orglq(p, n, p, a, lda, tau, work, lwork, info)`

To compute the matrix Qk of the LQ factorization of A's leading k rows, use:
`call ?orglq(n, n, k, a, lda, tau, work, lwork, info)`

To compute the leading k rows of Qk, which form an orthonormal basis in the space spanned by A's leading k rows, use:
`call ?orgqr(k, n, k, a, lda, tau, work, lwork, info)`

## Input Parameters

m

  INTEGER. The number of rows of Q to be computed

  (`0 ≤ m ≤ n`).

n

  INTEGER. The order of the orthogonal matrix Q (`n ≥ m`).

k

  INTEGER. The number of elementary reflectors whose product defines the matrix Q (`0 ≤ k ≤ m`).

a, tau, work

  REAL for sorglq

  DOUBLE PRECISION for dorglq

  Arrays: a(lda,*) and tau(*) are the arrays returned by sgelqf/dgelqf.

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

  The dimension of tau must be at least max(1, k).

  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 m leading rows of the n-by-n orthogonal 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 orglq interface are the following:

a

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

tau

  Holds the vector of length (k).

## 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 Q differs from an exactly orthogonal matrix by a matrix E such that `||E||2 = O(ε)*||A||2`, where ε is the machine precision.

The total number of floating-point operations is approximately `4*m*n*k - 2*(m + n)*k2 + (4/3)*k3`.

If `m = k`, the number is approximately `(2/3)*m2*(3n - m)`.

The complex counterpart of this routine is ?unglq.

Parent topic: Orthogonal Factorizations