# p?geqpf

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

## Syntax

call psgeqpf( m, n, a, ia, ja, desca, ipiv, tau, work, lwork, info)

call pdgeqpf( m, n, a, ia, ja, desca, ipiv, tau, work, lwork, info)

call pcgeqpf( m, n, a, ia, ja, desca, ipiv, tau, work, lwork, info)

call pzgeqpf( m, n, a, ia, ja, desca, ipiv, tau, work, lwork, info)

## Description

The routine forms the QR factorization with column pivoting of a general m-by-n distributed matrix `sub(A)= A(ia:ia+m-1,ja:ja+n-1)` as

`sub(A)*P=Q*R`

## Input Parameters

m

  (global) INTEGER. The number of rows in the submatrix sub(A) `(m ≥ 0)`.

n

  (global) INTEGER. The number of columns in the submatrix sub(A) `(n ≥ 0)`.

a

  (local)

  REAL for psgeqpf

  DOUBLE PRECISION for pdgeqpf

  COMPLEX for pcgeqpf

  DOUBLE COMPLEX for pzgeqpf.

  Pointer into the local memory to an array of local dimension `(lld_a, LOCc(ja+n-1))`.

  Contains the local pieces of the distributed matrix sub(A) to be factored.

ia, ja

  (global) INTEGER. The row and column indices in the global array a indicating the first row and the first column of the submatrix `A(ia:ia+m-1,ja:ja+n-1)`, respectively.

desca

  (global and local) INTEGER array, dimension (dlen_). The array descriptor for the distributed matrix A.

work

  (local).

  REAL for psgeqpf

  DOUBLE PRECISION for pdgeqpf.

  COMPLEX for pcgeqpf.

  DOUBLE COMPLEX for pzgeqpf

  Workspace array of dimension lwork.

lwork

  (local or global) INTEGER, dimension of work, must be at least

  For real flavors:

  `lwork ≥ max(3,mp0+nq0) + LOCc (ja+n-1) + nq0`.

  For complex flavors:

  `lwork ≥ max(3,mp0+nq0)` .

  Here

  `iroff = mod(ia-1, mb_a), icoff = mod(ja-1, nb_a)`,

  `iarow = indxg2p(ia, mb_a, MYROW, rsrc_a, NPROW)`,

  `iacol = indxg2p(ja, nb_a, MYCOL, csrc_a, NPCOL)`,

  `mp0 = numroc(m+iroff, mb_a, MYROW, iarow, NPROW )`,

  `nq0 = numroc(n+icoff, nb_a, MYCOL, iacol, NPCOL)`,

  `LOCc (ja+n-1) = numroc(ja+n-1, nb_a, MYCOL,csrc_a, NPCOL)`, and numroc, indxg2p are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine blacs_gridinfo.

  If `lwork = -1`, then lwork is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by pxerbla.

## Output Parameters

a

  The elements on and above the diagonal of sub(A)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, represent the orthogonal/unitary matrix Q as a product of elementary reflectors (see Application Notes below).

ipiv

  (local) INTEGER. Array, DIMENSION `LOCc(ja+n-1)`.

  `ipiv(i) = k`, the local i-th column of sub(A)*P was the global k-th column of sub(A). ipiv is tied to the distributed matrix A.

tau

  (local)

  REAL for psgeqpf

  DOUBLE PRECISION for pdgeqpf

  COMPLEX for pcgeqpf

  DOUBLE COMPLEX for pzgeqpf.

  Array, DIMENSION `LOCc(ja+min(m, n)-1))`.

  Contains the scalar factor tau of elementary reflectors. tau is tied to the distributed matrix A.

`work(1)`

  On exit, `work(1)` contains the minimum value of lwork required for optimum performance.

info

  (global) INTEGER.

  `= 0`, the execution is successful.

  `< 0`, if the i-th argument is an array and the j-entry had an illegal value, then `info = - (i* 100+j)`, if the i-th argument is a scalar and had an illegal value, then `info = -i`.

## Application Notes

The matrix Q is represented as a product of elementary reflectors

`Q = H(1)*H(2)*...*H(n)`,

Each H(i) has the form

`H = I - tau*v*v'`

where tau is a real/complex scalar, and v is a real/complex vector with `v(1:i-1) = 0 and v(i) = 1; v(i+1:m)` is stored on exit in `A(ia+i-1:ia+m-1,ja+i-1)`.

The matrix P is represented in jpvt as follows: if jpvt(j)= i then the j-th column of P is the i-th canonical unit vector.

Parent topic: Orthogonal Factorizations