?geqpf

Developer Reference for Intel® oneAPI Math Kernel Library for C

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ID 766684

Date 3/31/2023

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?geqpf

Computes the QR factorization of a general m-by-n matrix with pivoting.

Syntax

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

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

lapack_int LAPACKE_cgeqpf (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_zgeqpf (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The routine is deprecated and has been replaced by routine geqp3.

The routine ?geqpf forms the QR factorization of a general m-by-n matrix A with column pivoting: A*P = Q*R (see Orthogonal Factorizations). Here P denotes an n-by-n permutation matrix.

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 A*P before the computation, and fixed in place during the computation.

If jpvt[i - 1] = 0, the ith 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 additional information on the matrix Q.

jpvt

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

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 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:

?geqpf (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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?geqpf