?orglq

Developer Reference for Intel® oneAPI Math Kernel Library for C

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ID 766684

Date 3/31/2023

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?orglq

Generates the real orthogonal matrix Q of the LQ factorization formed by ?gelqf.

Syntax

lapack_int LAPACKE_sorglq (int matrix_layout, lapack_int m, lapack_int n, lapack_int k, float* a, lapack_int lda, const float* tau);

lapack_int LAPACKE_dorglq (int matrix_layout, lapack_int m, lapack_int n, lapack_int k, double* a, lapack_int lda, const double* tau);

Include Files

mkl.h

Description

The routine generates the whole or part of n-by-n orthogonal matrix Q of the LQ factorization formed by the routines gelqf. Use this routine after a call to sgelqf/dgelqf.

Usually Q is determined from the LQ factorization of an p-by-n matrix A with n≥p. To compute the whole matrix Q, use:

info = LAPACKE_?orglq(matrix_layout, n, n, p, a, lda, tau)

To compute the leading p rows of Q, which form an orthonormal basis in the space spanned by the rows of A, use:

info = LAPACKE_?orglq(matrix_layout, p, n, p, a, lda, tau)

To compute the matrix Q^k of the LQ factorization of the leading k rows of A, use:

info = LAPACKE_?orglq(matrix_layout, n, n, k, a, lda, tau)

To compute the leading k rows of Q^k, which form an orthonormal basis in the space spanned by the leading k rows of A, use:

info = LAPACKE_?orgqr(matrix_layout, k, n, k, a, lda, tau)

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 of Q to be computed

(0 ≤m≤n).

n

The order of the orthogonal matrix Q (n≥m).

k

The number of elementary reflectors whose product defines the matrix Q (0 ≤k≤m).

a, tau

Arrays: a (size max(1, lda*n) for column major layout and max(1, lda*m) for row major layout) and tau are the arrays returned by sgelqf/dgelqf.

The size of tau must be at least max(1, k).

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 m leading rows of the n-by-n orthogonal matrix Q.

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 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 m = k, the number is approximately (2/3)*m²*(3n - m).

The complex counterpart of this routine is unglq.

Parent topic: Orthogonal Factorizations: LAPACK Computational Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for C

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