?ungqr

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 12/16/2022

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?ungqr

Generates the complex unitary matrix Q of the QR factorization formed by ?geqrf.

Syntax

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

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

call ungqr(a, tau [,info])

Include Files

mkl.fi, lapack.f90

Description

The routine generates the whole or part of m-by-m unitary matrix Q of the QR factorization formed by the routines ?geqrf or geqpf. Use this routine after a call to cgeqrf/zgeqrf or cgeqpf/zgeqpf.

Usually Q is determined from the QR factorization of an m by p matrix A with m≥p. To compute the whole matrix Q, use:

call ?ungqr(m, m, p, a, lda, tau, work, lwork, info)

To compute the leading p columns of Q (which form an orthonormal basis in the space spanned by the columns of A):

call ?ungqr(m, p, p, a, lda, tau, work, lwork, info)

To compute the matrix Q^k of the QR factorization of the leading k columns of the matrix A:

call ?ungqr(m, m, k, a, lda, tau, work, lwork, info)

To compute the leading k columns of Q^k (which form an orthonormal basis in the space spanned by the leading k columns of the matrix A):

call ?ungqr(m, k, k, a, lda, tau, work, lwork, info)

Input Parameters

m

INTEGER. The order of the unitary matrix Q (m≥ 0).

n

INTEGER. The number of columns of Q to be computed

(0 ≤n≤m).

k

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

a, tau, work

COMPLEX for cungqr

DOUBLE COMPLEX for zungqr

Arrays: a(lda,*) and tau(*) are the arrays returned by cgeqrf/zgeqrf or cgeqpf/zgeqpf.

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

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

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

lda

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

lwork

INTEGER. The size of the work array (lwork≥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

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.

LAPACK 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 LAPACK 95 Interface Conventions.

Specific details for the routine ungqr 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 =n*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 n = k, the number is approximately (8/3)*n²*(3m - n).

The real counterpart of this routine is orgqr.

Parent topic: Orthogonal Factorizations: LAPACK Computational Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

?ungqr