?geqp3

Developer Reference for Intel® oneAPI Math Kernel Library for C

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ID 766684

Date 12/16/2022

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?geqp3

Computes the QR factorization of a general m-by-n matrix with column pivoting using level 3 BLAS.

Syntax

lapack_int LAPACKE_sgeqp3 (int matrix_layout, lapack_int m, lapack_int n, float* a, lapack_int lda, lapack_int* jpvt, float* tau);

lapack_int LAPACKE_dgeqp3 (int matrix_layout, lapack_int m, lapack_int n, double* a, lapack_int lda, lapack_int* jpvt, double* tau);

lapack_int LAPACKE_cgeqp3 (int matrix_layout, lapack_int m, lapack_int n, lapack_complex_float* a, lapack_int lda, lapack_int* jpvt, lapack_complex_float* tau);

lapack_int LAPACKE_zgeqp3 (int matrix_layout, lapack_int m, lapack_int n, lapack_complex_double* a, lapack_int lda, lapack_int* jpvt, lapack_complex_double* tau);

Include Files

mkl.h

Description

The routine forms the QR factorization of a general m-by-n matrix A with column pivoting: A*P = Q*R (see Orthogonal Factorizations) using Level 3 BLAS. Here P denotes an n-by-n permutation matrix. Use this routine instead of geqpf for better performance.

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.

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.

jpvt

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

On entry, if jpvt[i - 1]≠ 0, the i-th column of A is moved to the beginning of AP before the computation, and fixed in place during the computation.

If jpvt[i - 1] = 0, the i-th column of A is a free column (that is, it may be interchanged during the computation with any other free column).

Output Parameters

a

Overwritten by the factorization data as follows:

The elements on and above the diagonal of the array contain the min(m,n)-by-n upper trapezoidal matrix R (R is upper triangular if m≥n); the elements below the diagonal, with the array tau, present the orthogonal matrix Q as a product of min(m,n) elementary reflectors (see Orthogonal Factorizations).

tau

Array, size at least max (1, min(m, n)). Contains scalar factors of the elementary reflectors for the matrix Q.

jpvt

Overwritten by details of the permutation matrix P in the factorization A*P = Q*R. More precisely, the columns of AP are the columns of A in the following order:

jpvt[0], jpvt[1], ..., jpvt[n - 1].

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

To solve a set of least squares problems minimizing ||A*x - b||₂ for all columns b of a given matrix B, you can call the following:

?geqp3 (this routine)	to factorize `AP = QR`;
ormqr	to compute `C = Q^T*B` (for real matrices);
unmqr	to compute `C = Q^H*B` (for complex matrices);
trsm (a BLAS routine)	to solve `R*X = C`.

(The columns of the computed X are the permuted least squares solution vectors x; the output array jpvt specifies the permutation order.)

To compute the elements of Q explicitly, call

orgqr: (for real matrices)
ungqr: (for complex matrices).

Parent topic: Orthogonal Factorizations: LAPACK Computational Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for C

?geqp3