?gerqf

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 3/31/2023

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?gerqf

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

Syntax

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

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

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

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

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

Include Files

mkl.fi, lapack.f90

Description

The routine forms the RQ factorization of a general m-by-n matrix A(see Orthogonal Factorizations). 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.

NOTE:

This routine supports the Progress Routine feature. See Progress Function for details.

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 sgerqf

DOUBLE PRECISION for dgerqf

COMPLEX for cgerqf

DOUBLE COMPLEX for zgerqf.

Arrays:

Array a(lda,*) contains the m-by-n 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 leading dimension of a; at least max(1, m).

lwork

INTEGER. The size of the work array;

lwork≥ 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 on exit by the factorization data as follows:

if m≤n, the upper triangle of the subarray

a(1:m, n-m+1:n ) contains the m-by-m upper triangular matrix R;

if m≥n, the elements on and above the (m-n)th subdiagonal contain the m-by-n upper trapezoidal matrix R;

in both cases, the remaining elements, with the array tau, represent the orthogonal/unitary matrix Q as a product of min(m,n) elementary reflectors.

tau

REAL for sgerqf

DOUBLE PRECISION for dgerqf

COMPLEX for cgerqf

DOUBLE COMPLEX for zgerqf.

Array, size at least max (1, min(m, n)). (See Orthogonal Factorizations.)

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.

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 gerqf 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 =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.

Related routines include:

orgrq: to generate matrix Q (for real matrices);
ungrq: to generate matrix Q (for complex matrices);
ormrq: to apply matrix Q (for real matrices);
unmrq: to apply matrix Q (for complex matrices).

Parent topic: Orthogonal Factorizations: LAPACK Computational Routines

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