## Developer Reference

• 2020.2
• 07/15/2020
• Public Content
Contents

# ?tplqt

Computes a blocked LQ factorization of a complex "triangular-pentagonal" matrix composed of a triangular block A and pentagonal block B, using the compact WY representation for Q.
Description
?tplqt
computes a blocked LQ factorization of a real or complex "triangular-pentagonal" matrix
C
, which is composed of a triangular block
A
and pentagonal block
B
, using the compact WY representation for
Q
.
The input matrix
C
is an
m
-by-(
m
+
n
) matrix:
C
= [
A
] [
B
]
where
A
is a lower triangular
m
-by-
m
matrix, and
B
is an
m
-by-
n
pentagonal matrix consisting of an
m
-by-(
n
-
l
) rectangular matrix
B1
to the left of an
m
-by-
l
lower trapezoidal matrix
B2
:
[
B
] = [
B1
] [
B2
]
[
B1
] <-
m
-by-(
n
-
l
) rectangular
[
B2
] <-
m
-by-
l
lower trapezoidal.
The lower trapezoidal matrix
B2
consists of the first
l
columns of an
m
-by-
m
lower triangular matrix, where 0
l
min(
m
,
n
). If
l
=0,
b
is rectangular
m
-by-
n
; if
m
=
l
=
n
,
b
is lower triangular.
The matrix
W
stores the elementary reflectors
H
(
i
) in the
i
-th row above the diagonal (of
A
) in the
m
-by-(
m
+
n
) input matrix
C
:
[
C
] = [
A
] [
B
]
[
A
] <- lower triangular
m
-by-
m
[
B
] <-
m
-by-
n
pentagonal
so that
W
can be represented as
[
W
] = [
I
] [
V
]
[
I
] <-
m
-by-
m
identity matrix
[
V
] <-
m
-by-
n
, same form as
B
.
Thus, all of information needed for
W
is contained on exit in the array
b
, called
V
in the preceding. Note that
V
has the same form as
B
; that is,
[
V
] = [
V1
] [
V2
]
[
V1
] <-
m
-by-(
n
-
l
) rectangular
[
V2
] <-
m
-by-
l
lower trapezoidal.
The rows of
V
represent the vectors which define the
H
(
i
) elementary reflectors .
The number of blocks is
B
= ceiling(
m
/