# ?geqr2

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

## Syntax

call sgeqr2( m, n, a, lda, tau, work, info )

call dgeqr2( m, n, a, lda, tau, work, info )

call cgeqr2( m, n, a, lda, tau, work, info )

call zgeqr2( m, n, a, lda, tau, work, info )

• mkl.fi

## Description

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

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

`Q = H(1)*H(2)* ... *H(k)`, 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 `v1:i-1 = 0` and `vi = 1`.

On exit, `vi+1:m` is stored in `a(i+1:m, i)`.

## Input Parameters

The data types are given for the Fortran interface.

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 sgeqr2

DOUBLE PRECISION for dgeqr2

COMPLEX for cgeqr2

DOUBLE COMPLEX for zgeqr2.

Arrays:

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

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

work(n) 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, the elements on and above the diagonal of the array a contain the min(n,m)-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.

tau

REAL for sgeqr2

DOUBLE PRECISION for dgeqr2

COMPLEX for cgeqr2

DOUBLE COMPLEX for zgeqr2.

Array, size 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.

`If info = -1011`, memory allocation error occurred.

For more complete information about compiler optimizations, see our Optimization Notice.