?gelqf

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 12/16/2022

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?gelqf

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

Syntax

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

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

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

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

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

Include Files

mkl.fi, lapack.f90

Description

The routine forms the LQ 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 sgelqf

DOUBLE PRECISION for dgelqf

COMPLEX for cgelqf

DOUBLE COMPLEX for zgelqf.

Arrays:

Array a(lda,*) contains the 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; 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 the factorization data as follows:

The elements on and below the diagonal of the array contain the m-by-min(m,n) lower trapezoidal matrix L (L is lower triangular if m≤n); the elements above the diagonal, with the array tau, represent the orthogonal matrix Q as a product of elementary reflectors.

tau

REAL for sgelqf

DOUBLE PRECISION for dgelqf

COMPLEX for cgelqf

DOUBLE COMPLEX for zgelqf.

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

Contains scalars that define elementary reflectors for the matrix Q (see Orthogonal Factorizations).

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 gelqf 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.

The computed factorization is the exact factorization of a matrix A + E, where

||E||₂ = O(ε) ||A||₂.

The approximate number of floating-point operations for real flavors is

(4/3)`n`³	if `m = n`,
(2/3)`n`²(3`m`-`n`)	if `m > n`,
(2/3)`m`²(3`n`-`m`)	if `m < n`.

The number of operations for complex flavors is 4 times greater.

To find the minimum-norm solution of an underdetermined least squares problem minimizing ||A*x - b||₂ for all columns b of a given matrix B, you can call the following:

?gelqf (this routine)	to factorize `A = L*Q`;
trsm (a BLAS routine)	to solve `L*Y = B` for `Y`;
ormlq	to compute `X = (Q₁)^T*Y` (for real matrices);
unmlq	to compute `X = (Q₁)^H*Y` (for complex matrices).

(The columns of the computed X are the minimum-norm solution vectors x. Here A is an m-by-n matrix with m < n; Q₁ denotes the first m columns of Q).

To compute the elements of Q explicitly, call

orglq	(for real matrices)
unglq	(for complex matrices).

Parent topic: Orthogonal Factorizations: LAPACK Computational Routines

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