?gelqf

Developer Reference for Intel® oneAPI Math Kernel Library for C

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ID 766684

Date 12/16/2022

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?gelqf

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

Syntax

lapack_int LAPACKE_sgelqf (int matrix_layout, lapack_int m, lapack_int n, float* a, lapack_int lda, float* tau);

lapack_int LAPACKE_dgelqf (int matrix_layout, lapack_int m, lapack_int n, double* a, lapack_int lda, double* tau);

lapack_int LAPACKE_cgelqf (int matrix_layout, lapack_int m, lapack_int n, lapack_complex_float* a, lapack_int lda, lapack_complex_float* tau);

lapack_int LAPACKE_zgelqf (int matrix_layout, lapack_int m, lapack_int n, lapack_complex_double* a, lapack_int lda, lapack_complex_double* tau);

Include Files

mkl.h

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

matrix_layout: Specifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).
m: The number of rows in the matrix A (m≥ 0).
n: The number of columns in A (n≥ 0).
a: Array a of size max(1, lda*n) for column major layout and max(1, lda*m) for row major layout contains the matrix A.
lda: The leading dimension of a; at least max(1, m) for column major layout and max(1, n) for row major layout.

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

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

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

Return Values

This function returns a value info.

If info=0, the execution is successful.

If info = -i, the i-th parameter had an illegal value.

Application Notes

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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