Orthogonal Factorizations: LAPACK Computational
Routines
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
). After factoring
the matrix, compute the solution
.
QR
factorization
for solving the following least squares problem: minimize
||
where
Ax
-
b
||2
A
is a full-rank
m
-by-n
matrix (m
≥n
x
by solving
Rx
= (Q
1
)T
b
If
,
the
m
<
n
QR
factorization
is given by
A
=
QR
=
Q
(R
1
R
2
)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
,
the
m
>
n
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
where
.
After factoring the matrix, compute the solution vector
for
.
LQ
factorization
to find the minimum-norm solution of an underdetermined system of linear
equations
Ax
=
b
A
is an
m
-by-n
matrix of rank
m
(m
<
n
)x
as follows:
solve
Ly
=
b
y
, and then
compute
x
= (Q
1
)H
y
Table
"Computational Routines for Orthogonal
Factorization"
lists LAPACK routines that perform orthogonal
factorization of matrices.
Matrix type, factorization
| Factorize without pivoting
| Factorize with pivoting
| Generate matrix Q
| Apply matrix Q
|
|---|---|---|---|---|