# ?geqpf

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

## Syntax

FORTRAN 77:

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)

Fortran 95:

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

## Description

This routine is deprecated and has been replaced by routine ?geqp3.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 first dimension of a; at least max(1, m).

jpvt

  INTEGER. Array, DIMENSION 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:

  If `m ≥ n`, the elements below the diagonal are overwritten by the details of the unitary (orthogonal) matrix Q, and the upper triangle is overwritten by the corresponding elements of the upper triangular matrix R.

  If `m < n`, the strictly lower triangular part is overwritten by the details of the matrix Q, and the remaining elements are overwritten by the corresponding elements of the m-by-n upper trapezoidal matrix R.

tau

  REAL for sgeqpf

  DOUBLE PRECISION for dgeqpf

  COMPLEX for cgeqpf

  DOUBLE COMPLEX for zgeqpf.

  Array, DIMENSION 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.

## Fortran 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 Fortran 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)n3

  if `m = n`,

(2/3)n2(3m-n)

  if `m > n`,

(2/3)m2(3n-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 `A*P = Q*R`;

?ormqr

  to compute `C = QT*B` (for real matrices);

?unmqr

  to compute `C = QH*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