# ?unglq

Generates the complex unitary matrix Q of the LQ factorization formed by ?gelqf.

## Syntax

FORTRAN 77:

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

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

Fortran 95:

call unglq(a, tau [,info])

## Description

The routine generates the whole or part of n-by-n unitary matrix Q of the LQ factorization formed by the routines cgelqf/zgelqf. Use this routine after a call to cgelqf/zgelqf.

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 ?unglq(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 ?unglq(p, n, p, a, lda, tau, work, lwork, info)`

To compute the matrix Qk of the LQ factorization of the leading k rows of the matrix A, use:
`call ?unglq(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 the leading k rows of the matrix A, use:
`call ?ungqr(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 unitary matrix Q (`n ≥ m`).

k

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

a, tau, work

  COMPLEX for cunglq

  DOUBLE COMPLEX for zunglq

  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 unitary 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 unglq 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 it is not clear how much workspace to supply, use a generous value of lwork for the first run, or set `lwork = -1`.

In first case 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 `lwork = -1`, then 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 lwork is less than the minimal required value and is not equal to -1, then 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 unitary 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 `16*m*n*k - 8*(m + n)*k2 + (16/3)*k3`.

If `m = k`, the number is approximately `(8/3)*m2*(3n - m)` .

The real counterpart of this routine is ?orglq.

Parent topic: Orthogonal Factorizations