?geqpf

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 3/31/2023

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?geqpf

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

Syntax

call sgeqpf(m, n, a, lda, jpvt, tau, work, info)

call dgeqpf(m, n, a, lda, jpvt, tau, work, info)

call cgeqpf(m, n, a, lda, jpvt, tau, work, rwork, info)

call zgeqpf(m, n, a, lda, jpvt, tau, work, rwork, info)

call geqpf(a, jpvt [,tau] [,info])

Include Files

mkl.fi, lapack.f90

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The routine is deprecated and has been replaced by routine geqp3.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

m

INTEGER. The number of rows in the matrix A (m≥ 0).

n

INTEGER. The number of columns in A (n≥ 0).

a, work

REAL for sgeqpf

DOUBLE PRECISION for dgeqpf

COMPLEX for cgeqpf

DOUBLE COMPLEX for zgeqpf.

Arrays: a (lda,*) contains the matrix A. The second dimension of a must be at least max(1, n).

work (lwork) is a workspace array. The size of the work array must be at least max(1, 3*n) for real flavors and at least max(1, n) for complex flavors.

lda

INTEGER. The leading dimension of a; at least max(1, m).

jpvt

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

On entry, if jpvt(i) > 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) = 0, the ith column of A is a free column (that is, it may be interchanged during the computation with any other free column).

rwork

REAL for cgeqpf

DOUBLE PRECISION for zgeqpf.

A workspace array, DIMENSION at least max(1, 2*n).

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

REAL for sgeqpf

DOUBLE PRECISION for dgeqpf

COMPLEX for cgeqpf

DOUBLE COMPLEX for zgeqpf.

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(1), jpvt(2), ..., jpvt(n).

info

INTEGER.

If info = 0, the execution is successful.

If info = -i, the i-th parameter had an illegal value.

LAPACK 95 Interface Notes

Routines in Fortran 95 interface have fewer arguments in the calling sequence than their FORTRAN 77 counterparts. For general conventions applied to skip redundant or restorable arguments, see LAPACK 95 Interface Conventions.

Specific details for the routine geqpf interface are the following:

a: Holds the matrix A of size (m,n).
jpvt: Holds the vector of length n.
tau: Holds the vector of length min(m,n)

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