# p?geqlf

Computes the QL factorization of a general matrix.

## Syntax

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

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

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

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

## Description

The routine forms the Q L factorization of a real/complex distributed m-by-n matrix sub`(A) = A(ia:ia+m-1, ja:ja+n-1) = Q*L`.

## Input Parameters

m

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

n

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

a

  (local)

  REAL for psgeqlf

  DOUBLE PRECISION for pdgeqlf

  COMPLEX for pcgeqlf

  DOUBLE COMPLEX for pzgeqlf

  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`, `ia:ia+n-1)`, respectively.

desca

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

work

  (local)

  REAL for psgeqlf

  DOUBLE PRECISION for pdgeqlf

  COMPLEX for pcgeqlf

  DOUBLE COMPLEX for pzgeqlf

  Workspace array of dimension of lwork.

lwork

  (local or global) INTEGER, dimension of work, must be at least `lwork ≥ nb_a*(mp0 + nq0 + nb_a)`, 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)`

  numroc and 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

  On exit, if `m≥n`, the lower triangle of the distributed submatrix `A(ia+m-n:ia+m-1, ja:ja+n-1)` contains the n-by-n lower triangular matrix L; if `m≤n`, the elements on and below the (n-m)-th superdiagonal contain the m-by-n lower trapezoidal matrix L; the remaining elements, with the array tau, represent the orthogonal/unitary matrix Q as a product of elementary reflectors (see Application Notes below).

tau

  (local)

  REAL for psgeqlf

  DOUBLE PRECISION for pdgeqlf

  COMPLEX for pcgeqlf

  DOUBLE COMPLEX for pzgeqlf

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

  Contains the scalar factors 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(ja+k-1)*...*H(ja+1)*H(ja)`,

where `k = min(m,n)`

Each H(i) has the form

`H(i) = I - tau*v*v'`

where tau is a real/complex scalar, and v is a real/complex vector with `v(m-k+i+1:m) = 0` and `v(m-k+i) = 1`; `v(m-k+i-1)` is stored on exit in `A(ia+ia+m-k+i-2, ja+n-k+i-1)`, and tau in tau `(ja+n-k+i-1)`.

Parent topic: Orthogonal Factorizations