This

topic

describes the LAPACK routines for the

QR (RQ)

and

LQ (QL)

factorization of matrices. Routines for the

RZ

factorization as well as for generalized

QR

and

RQ

factorizations are also included.

QR Factorization.

Assume that

A

is an

m

-by-

n

matrix to be factored.

If

m

≥

n

, the

QR

factorization is given by

where

R

is an

n

-by-

n

upper triangular matrix with real diagonal elements, and

Q

is an

m

-by-

m

orthogonal (or unitary) matrix.

You can use the

QR

factorization for solving the following least squares problem: minimize

||

Ax

-

b

||

2

where

A

is a full-rank

m

-by-

n

matrix (

m

≥

n

). After factoring the matrix, compute the solution

x

by solving

Rx

= (

Q

1

)

T

b

.

If

m

<

n

, the

QR

factorization is given by

where

R

is trapezoidal,

R

1

is upper triangular and

R

2

is rectangular.

Q

is represented as a product of min(

m

,

n

) elementary reflectors. Routines are provided to work with

Q

in this representation.

LQ Factorization

LQ factorization of an

m

-by-

n

matrix

A

is as follows. If

m

≤

n

,

where

L

is an

m

-by-

m

lower triangular matrix with real diagonal elements, and

Q

is an

n

-by-

n

orthogonal (or unitary) matrix.

If

m

>

n

, the

LQ

factorization is

where

L

1

is an

n

-by-

n

lower triangular matrix,

L

2

is rectangular, and

Q

is an

n

-by-

n

orthogonal (or unitary) matrix.

You can use the

LQ

factorization to find the minimum-norm solution of an underdetermined system of linear equations

Ax

=

b

where

A

is an

m

-by-

n

matrix of rank

m

(

m

<

n

)

. After factoring the matrix, compute the solution vector

x

as follows: solve

Ly

=

b

for

y

, and then compute

x

= (

Q

1

)

H

y

.

Table

"Computational Routines for Orthogonal Factorization"

lists LAPACK routines that perform orthogonal factorization of matrices.