Developer Reference

  • 0.9
  • 09/09/2020
  • Public Content
Contents

?ggrqf

Computes the generalized RQ factorization of two matrices.

Syntax

lapack_int
LAPACKE_sggrqf
(
int
matrix_layout
,
lapack_int
m
,
lapack_int
p
,
lapack_int
n
,
float
*
a
,
lapack_int
lda
,
float
*
taua
,
float
*
b
,
lapack_int
ldb
,
float
*
taub
);
lapack_int
LAPACKE_dggrqf
(
int
matrix_layout
,
lapack_int
m
,
lapack_int
p
,
lapack_int
n
,
double
*
a
,
lapack_int
lda
,
double
*
taua
,
double
*
b
,
lapack_int
ldb
,
double
*
taub
);
lapack_int
LAPACKE_cggrqf
(
int
matrix_layout
,
lapack_int
m
,
lapack_int
p
,
lapack_int
n
,
lapack_complex_float
*
a
,
lapack_int
lda
,
lapack_complex_float
*
taua
,
lapack_complex_float
*
b
,
lapack_int
ldb
,
lapack_complex_float
*
taub
);
lapack_int
LAPACKE_zggrqf
(
int
matrix_layout
,
lapack_int
m
,
lapack_int
p
,
lapack_int
n
,
lapack_complex_double
*
a
,
lapack_int
lda
,
lapack_complex_double
*
taua
,
lapack_complex_double
*
b
,
lapack_int
ldb
,
lapack_complex_double
*
taub
);
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:
Equation
or
Equation
where
R
11
or
R
21
is upper triangular, and
Equation
or
Equation
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
The leading dimension of
a
; at least max(1,
m
)
for column major layout and max(1,
n
) for row major layout
.
ldb
The leading dimension of
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

Intel's compilers may or may not optimize to the same degree for non-Intel microprocessors for optimizations that are not unique to Intel microprocessors. These optimizations include SSE2, SSE3, and SSSE3 instruction sets and other optimizations. Intel does not guarantee the availability, functionality, or effectiveness of any optimization on microprocessors not manufactured by Intel. Microprocessor-dependent optimizations in this product are intended for use with Intel microprocessors. Certain optimizations not specific to Intel microarchitecture are reserved for Intel microprocessors. Please refer to the applicable product User and Reference Guides for more information regarding the specific instruction sets covered by this notice.

Notice revision #20110804