# p?geqr2

Computes a QR factorization of a general rectangular matrix (unblocked algorithm).

## Syntax

call psgeqr2(m, n, a, ia, ja, desca, tau, work, lwork, info)

call pdgeqr2(m, n, a, ia, ja, desca, tau, work, lwork, info)

call pcgeqr2(m, n, a, ia, ja, desca, tau, work, lwork, info)

call pzgeqr2(m, n, a, ia, ja, desca, tau, work, lwork, info)

## Description

This routine computes a QR factorization of a real/complex distributed m-by-n matrix `sub(A) = A(ia:ia+m-1, ja:ja+n-1)= Q*R`.

## Input Parameters

m

  (global). INTEGER.

  The number of rows to be operated on, that is, the number of rows of the distributed submatrix sub(A). `(m≥0)`.

n

  (global).INTEGER. The number of columns to be operated on, that is, the number of columns of the distributed submatrix sub(A). `(n≥0)`.

a

  (local).

  REAL for psgeqr2

  DOUBLE PRECISION for pdgeqr2

  COMPLEX for pcgeqr2

  COMPLEX*16 for pzgeqr2.

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

  On entry, this array contains the local pieces of the m-by-n distributed matrix sub(A) which is 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, respectively.

desca

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

work

  (local).

  REAL for psgeqr2

  DOUBLE PRECISION for pdgeqr2

  COMPLEX for pcgeqr2

  COMPLEX*16 for pzgeqr2.

  This is a workspace array of DIMENSION (lwork).

lwork

  (local or global). INTEGER.

  The dimension of the array work.

  lwork is local input and must be at least `lwork≥mp0+max(1, nq0)`,

  where `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)`.

  indxg2p and numroc 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

  (local).

  On exit, 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).

tau

  (local).

  REAL for psgeqr2

  DOUBLE PRECISION for pdgeqr2

  COMPLEX for pcgeqr2

  COMPLEX*16 for pzgeqr2.

  Array, DIMENSION LOCc(ja+min(m,n)-1). This array contains the scalar factors of the elementary reflectors. tau is tied to the distributed matrix A.

work

  On exit, `work(1)` returns the minimal and optimal lwork.

info

  (local). INTEGER.

  If `info = 0`, the execution is successful. if info < 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(ja)*H(ja+1)*. . .* H(ja+k-1)`, where `k = min(m,n)`.

Each `H(i)` has the form

`H(j)= 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:ia+m-1, ja+i-1), and tau in TAU(ja+i-1).

Parent topic: Auxiliary Routines