# ?geql2

Computes the QL factorization of a general rectangular matrix using an unblocked algorithm.

## Syntax

call sgeql2( m, n, a, lda, tau, work, info )

call dgeql2( m, n, a, lda, tau, work, info )

call cgeql2( m, n, a, lda, tau, work, info )

call zgeql2( m, n, a, lda, tau, work, info )

## Description

The routine computes a QL factorization of a real/complex m-by-n matrix A as `A = Q*L`.

The routine does not form the matrix Q explicitly. Instead, Q is represented as a product of min(m, n) elementary reflectors :

`Q = H(k)* ... *H(2)*H(1)`, where `k = min(m, n)`

Each H(i) has the form

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

where tau is a real/complex scalar stored in tau(i), and v is a real/complex vector with `v(m-k+i+1:m) = 0` and `v(m-k+i) = 1`.

On exit, `v(1:m-k+i-1)` is stored in `a(1:m-k+i-1, n-k+i)`.

## 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 sgeql2

  DOUBLE PRECISION for dgeql2

  COMPLEX for cgeql2

  COMPLEX*16 for zgeql2.

  Arrays:

  `a(lda,*)` contains the m-by-n matrix A.

  The second dimension of a must be at least `max(1, n)`.

  work(m) is a workspace array.

lda

  INTEGER. The first dimension of a; at least `max(1, m)`.

## Output Parameters

a

  Overwritten by the factorization data as follows:

  on exit, if `m ≥ n`, the lower triangle of the subarray `a(m-n+1:m, 1:n)` 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.

tau

  REAL for sgeql2

  DOUBLE PRECISION for dgeql2

  COMPLEX for cgeql2

  COMPLEX*16 for zgeql2.

  Array, DIMENSION at least `max(1, min(m, n))`.

  Contains scalar factors of the elementary reflectors.

info

  INTEGER.

  If `info = 0`, the execution is successful.

  If `info = -i`, the i-th parameter had an illegal value.

Parent topic: Auxiliary Routines