# ?geqp3

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

## Syntax

FORTRAN 77:

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)

Fortran 95:

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

## 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 first 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, DIMENSION 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, DIMENSION at least max(1, 2*n). Used in complex flavors only.

## 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 sgeqp3

  DOUBLE PRECISION for dgeqp3

  COMPLEX for cgeqp3

  DOUBLE COMPLEX for zgeqp3.

  Array, DIMENSION 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.

## 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 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 `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).

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