?geqp3

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 3/31/2023

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?geqp3

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

Syntax

call sgeqp3(m, n, a, lda, jpvt, tau, work, lwork, info)

call dgeqp3(m, n, a, lda, jpvt, tau, work, lwork, info)

call cgeqp3(m, n, a, lda, jpvt, tau, work, lwork, rwork, info)

call zgeqp3(m, n, a, lda, jpvt, tau, work, lwork, rwork, info)

call geqp3(a, jpvt [,tau] [,info])

Include Files

mkl.fi, lapack.f90

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

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 sgeqp3

DOUBLE PRECISION for dgeqp3

COMPLEX for cgeqp3

DOUBLE COMPLEX for zgeqp3.

Arrays:

a (lda,*) contains the matrix A.

The second dimension of a must be at least max(1, n).

work is a workspace array, its dimension max(1, lwork).

lda

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

lwork

INTEGER. The size of the work array; must be at least max(1, 3*n+1) for real flavors, and at least max(1, n+1) for complex flavors.

If lwork = -1, then a workspace query is assumed; the routine only calculates the optimal size of the work array, returns this value as the first entry of the work array, and no error message related to lwork is issued by xerbla. See Application Notes below for details.

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 AP before the computation, and fixed in place during the computation.

If jpvt(i) = 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).

rwork

REAL for cgeqp3

DOUBLE PRECISION for zgeqp3.

A workspace array, size at least max(1, 2*n). Used in complex flavors only.

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 sgeqp3

DOUBLE PRECISION for dgeqp3

COMPLEX for cgeqp3

DOUBLE COMPLEX for zgeqp3.

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(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 geqp3 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

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).

If you are in doubt how much workspace to supply, use a generous value of lwork for the first run or set lwork = -1.

If you choose the first option and set any of admissible lwork sizes, which is no less than the minimal value described, the routine completes the task, though probably not so fast as with a recommended workspace, and provides the recommended workspace in the first element of the corresponding array work on exit. Use this value (work(1)) for subsequent runs.

If you set lwork = -1, the routine returns immediately and provides the recommended workspace in the first element of the corresponding array (work). This operation is called a workspace query.

Note that if you set lwork to less than the minimal required value and not -1, the routine returns immediately with an error exit and does not provide any information on the recommended workspace.

Parent topic: Orthogonal Factorizations: LAPACK Computational Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

?geqp3