?orgqr

Developer Reference for Intel® oneAPI Math Kernel Library for C

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ID 766684

Date 12/16/2022

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?orgqr

Generates the real orthogonal matrix Q of the QR factorization formed by ?geqrf.

Syntax

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

lapack_int LAPACKE_dorgqr (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 m-by-m orthogonal matrix Q of the QR factorization formed by the routine ?geqrf or geqpf. Use this routine after a call to sgeqrf/dgeqrf or sgeqpf/dgeqpf.

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

 LAPACKE_?orgqr(matrix_layout, m, m, p, a, lda, tau)

To compute the leading p columns of Q (which form an orthonormal basis in the space spanned by the columns of A):

 LAPACKE_?orgqr(matrix_layout, m, p, p, a, lda)

To compute the matrix Q^k of the QR factorization of leading k columns of the matrix A:

 LAPACKE_?orgqr(matrix_layout, m, m, k, a, lda, tau)

To compute the leading k columns of Q^k (which form an orthonormal basis in the space spanned by leading k columns of the matrix A):

 LAPACKE_?orgqr(matrix_layout, m, k, 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 order of the orthogonal matrix Q (m≥ 0).

n

The number of columns of Q to be computed

(0 ≤n≤m).

k

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

a, tau

Arrays:

a and tau are the arrays returned by sgeqrf / dgeqrf or sgeqpf / dgeqpf.

The size of a is max(1, lda*n) for column major layout and max(1, lda*m) for row major layout .

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 n leading columns of the m-by-m 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 n = k, the number is approximately (2/3)*n²*(3m - n).

The complex counterpart of this routine is ungqr.

Parent topic: Orthogonal Factorizations: LAPACK Computational Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for C

?orgqr