## Developer Reference

• 2021.2
• 03/26/2021
• Public Content
Contents

# ?ggrqf

Computes the generalized RQ factorization of two matrices.

## Syntax

Include Files
• mkl.h
Description
The routine forms the generalized
RQ
factorization of an
m
-by-
n
matrix
A
and an
p
-by-
n
matrix
B
as
A
=
R
*
Q
,
B
=
Z
*
T
*
Q
, where
Q
is an
n
-by-
n
orthogonal/unitary matrix,
Z
is a
p
-by-
p
orthogonal/unitary matrix, and
R
and
T
assume one of the forms:
or
where
R
11
or
R
21
is upper triangular, and
or
where
T
11
is upper triangular.
In particular, if
B
is square and nonsingular, the
GRQ
factorization of
A
and
B
implicitly gives the
RQ
factorization of
A
*
B
-1
as:
A
*
B
-1
= (
R
*
T
-1
)*
Z
T
(for real flavors) or
A
*
B
-1
= (
R
*
T
-1
)*
Z
H
(for complex flavors).
Input Parameters
matrix_layout
Specifies whether matrix storage layout is row major (
LAPACK_ROW_MAJOR
) or column major (
LAPACK_COL_MAJOR
).
m
The number of rows of the matrix
A
(
m
0
).
p
The number of rows in
B
(
p
0
).
n
The number of columns of the matrices
A
and
B
(
n
0
).
a
,
b
Arrays:
a
(size max(1,
lda
*
n
) for column major layout and max(1,
lda
*
m
) for row major layout)
contains the
m
-by-
n
matrix
A
.
b
(size max(1,
ldb
*
n
) for column major layout and max(1,
ldb
*
p
) for row major layout)
contains the
p
-by-
n
matrix
B
.
lda
a
; at least max(1,
m
)
for column major layout and max(1,
n
) for row major layout
.
ldb
b
; at least max(1,
p
)
for column major layout and max(1,
n
) for row major layout
.
Output Parameters
a
,
b
Overwritten by the factorization data as follows:
on exit, if
m
n
, element
R
i j
(1<=
i
j
m
) of upper triangular matrix
R
is stored in
a
[(
i
- 1) + (
n
-
m
+
j
- 1)*
lda
]
for column major layout and in
a
[(
i
- 1)*
lda
+ (
n
-
m
+
j
- 1)]
for row major layout.
if
m
>
n
, the elements on and above the (
m
-
n
)th subdiagonal contain the
m
-by-
n
upper trapezoidal matrix
R
;
the remaining elements, with the array
taua
, represent the orthogonal/unitary matrix
Q
as a product of elementary reflectors.
The elements on and above the diagonal of the array
b
contain the min(
p
,
n
)-by-
n
upper trapezoidal matrix
T
(
T
is upper triangular if
p
n
); the elements below the diagonal, with the array
taub
, represent the orthogonal/unitary matrix
Z
as a product of elementary reflectors.
taua
,
taub
Arrays, size at least max (1, min(
m
,
n
)) for
taua
and at least max (1, min(
p
,
n
)) for
taub
.
The array
taua
contains the scalar factors of the elementary reflectors which represent the orthogonal/unitary matrix
Q
.
The array
taub
contains the scalar factors of the elementary reflectors which represent the orthogonal/unitary matrix
Z
.
Return Values
This function returns a value
info
.
If
info
=0
, the execution is successful.
If
info
=
-i
, the
i
-th parameter had an illegal value.
Application Notes
The matrix
Q
is represented as a product of elementary reflectors
Q
=
H
(1)
H
(2)...
H
(
k
)
, where
k
= min(m,n)
.
Each
H
(i) has the form
H
(i) =
I
-
taua
*
v
*
v
T
for real flavors, or
H
(i) =
I
-
taua
*
v
*
v
H
for complex flavors,
where
taua
is a real/complex scalar, and
v
is a real/complex vector with
v
n
-
k
+
i
= 1,
v
n
-
k
+
i
+ 1:
n
= 0.
On exit,
v
1:
n
-
k
+
i
- 1
is stored in
a
(m-k+i,1:n-k+i-1)
and
taua
is stored in
taua
[
i
- 1]
.
The matrix
Z
is represented as a product of elementary reflectors
Z
=
H
(1)
H
(2)...
H
(
k
)
, where
k
= min(p,n)
.
Each
H
(i) has the form
H
(i) =
I
-
taub
*
v
*
v
T
for real flavors, or
H
(i) =
I
-
taub
*
v
*
v
H
for complex flavors,
where
taub
is a real/complex scalar, and
v
is a real/complex vector with
v
1:
i
- 1
= 0,
v
i
= 1.
On exit,
v
i
+ 1:
p
is stored in
b
(i+1:p, i)
and
taub
is stored in
taub
[
i
- 1]
.

#### Product and Performance Information

1

Performance varies by use, configuration and other factors. Learn more at www.Intel.com/PerformanceIndex.