# ?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 )

• mkl.fi

## 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*vT for real flavors, or

H(i) = I - tau*v*vH for complex flavors

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

DOUBLE COMPLEX 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 leading dimension of a; at least max(1, m).

## Output Parameters

a

Overwritten by the factorization data as follows:

on exit, if mn, 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

DOUBLE COMPLEX 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.