?orgqr

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 12/16/2022

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?orgqr

Generates the real orthogonal matrix Q of the QR factorization formed by ?geqrf.

Syntax

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

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

call orgqr(a, tau [,info])

Include Files

mkl.fi, lapack.f90

Description

The routine generates the whole or part of m-by-m orthogonal matrix Q of the QR factorization formed by the routine ?geqrf or geqpf. Use this routine after a call to sgeqrf/dgeqrf or sgeqpf/dgeqpf.

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 ?orgqr(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 ?orgqr(m, p, p, a, lda, work, lwork, info)

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

call ?orgqr(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 leading k columns of the matrix A):

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

Input Parameters

m

INTEGER. The order of the orthogonal 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

REAL for sorgqr

DOUBLE PRECISION for dorgqr

Arrays:

a(lda,*) and tau(*) are the arrays returned by sgeqrf / dgeqrf or sgeqpf / dgeqpf.

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

Overwritten by n leading columns of the m-by-m 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.

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 orgqr 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 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 n = k, the number is approximately (2/3)*n²*(3m - n).

The complex counterpart of this routine is ungqr.

Parent topic: Orthogonal Factorizations: LAPACK Computational Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

?orgqr