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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
Equation
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
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
,
Equation
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
Equation
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.
Computational Routines for Orthogonal Factorization
Matrix type, factorization
Factorize without pivoting
Factorize with pivoting
Generate matrix Q
Apply matrix Q
general matrices, QR factorization
general matrices, blocked QR factorization
 
 
general matrices, RQ factorization
 
general matrices, LQ factorization
 
general matrices, QL factorization
 
trapezoidal matrices, RZ factorization
 
 
pair of matrices, generalized QR factorization
 
 
 
pair of matrices, generalized RQ factorization
 
 
 
triangular-pentagonal matrices, blocked QR factorization
 
 

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