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 (seeOrthogonal 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 mn); 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)n3

if m = n,

(2/3)n2(3m-n)

if m > n,

(2/3)m2(3n-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).

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