Developer Reference for Intel® oneAPI Math Kernel Library - C

Developer Reference

  • 2021.1
  • 12/04/2020
  • Public Content
Contents

?gghrd

Reduces a pair of matrices to generalized upper Hessenberg form using orthogonal/unitary transformations.

Syntax

lapack_int
LAPACKE_sgghrd
(
int
matrix_layout
,
char
compq
,
char
compz
,
lapack_int
n
,
lapack_int
ilo
,
lapack_int
ihi
,
float
*
a
,
lapack_int
lda
,
float
*
b
,
lapack_int
ldb
,
float
*
q
,
lapack_int
ldq
,
float
*
z
,
lapack_int
ldz
);
lapack_int
LAPACKE_dgghrd
(
int
matrix_layout
,
char
compq
,
char
compz
,
lapack_int
n
,
lapack_int
ilo
,
lapack_int
ihi
,
double
*
a
,
lapack_int
lda
,
double
*
b
,
lapack_int
ldb
,
double
*
q
,
lapack_int
ldq
,
double
*
z
,
lapack_int
ldz
);
lapack_int
LAPACKE_cgghrd
(
int
matrix_layout
,
char
compq
,
char
compz
,
lapack_int
n
,
lapack_int
ilo
,
lapack_int
ihi
,
lapack_complex_float
*
a
,
lapack_int
lda
,
lapack_complex_float
*
b
,
lapack_int
ldb
,
lapack_complex_float
*
q
,
lapack_int
ldq
,
lapack_complex_float
*
z
,
lapack_int
ldz
);
lapack_int
LAPACKE_zgghrd
(
int
matrix_layout
,
char
compq
,
char
compz
,
lapack_int
n
,
lapack_int
ilo
,
lapack_int
ihi
,
lapack_complex_double
*
a
,
lapack_int
lda
,
lapack_complex_double
*
b
,
lapack_int
ldb
,
lapack_complex_double
*
q
,
lapack_int
ldq
,
lapack_complex_double
*
z
,
lapack_int
ldz
);
Include Files
  • mkl.h
Description
The routine reduces a pair of real/complex matrices (
A
,
B
) to generalized upper Hessenberg form using orthogonal/unitary transformations, where
A
 is a general matrix and
B
 is upper triangular. The form of the generalized eigenvalue problem is
A
*
x
 =
λ
*
B
*
x
, and
B
 is typically made upper triangular by computing its
QR
 factorization and moving the orthogonal matrix
Q
 to the left side of the equation.
This routine simultaneously reduces
A
 to a Hessenberg matrix
H
:
Q
H
*
A
*
Z
 =
H
and transforms
B
 to another upper triangular matrix
T
:
Q
H
*
B
*
Z
 =
T
in order to reduce the problem to its standard form
H
*
y
 =
λ
*
T
*
y
, where
y
 =
Z
H
*
x
.
The orthogonal/unitary matrices
Q
 and
Z
 are determined as products of Givens rotations. They may either be formed explicitly, or they may be postmultiplied into input matrices
Q
1
 and
Z
1
, so that
Q
1
*
A
*
Z
1
H
 = (
Q
1
*
Q
)*
H
*(
Z
1
*
Z
)
H
Q
1
*
B*
Z
1
H
 = (
Q
1
*
Q
)*
T*
(
Z
1
*
Z
)
H
If
Q
1
 is the orthogonal/unitary matrix from the
QR
 factorization of
B
 in the original equation
A
*
x
 =
λ
*
B
*
x
, then the routine
?gghrd
 reduces the original problem to generalized Hessenberg form.
Input Parameters
matrix_layout
Specifies whether matrix storage layout is row major (
LAPACK_ROW_MAJOR
) or column major (
LAPACK_COL_MAJOR
).
compq
Must be
'N'
,
'I'
, or
'V'
.
If
compq
 =
'N'
, matrix
Q
 is not computed.
If
compq
 =
'I'
,
Q
 is initialized to the unit matrix, and the orthogonal/unitary matrix
Q
 is returned;
If
compq
 =
'V'
,
Q
 must contain an orthogonal/unitary matrix
Q
1
 on entry, and the product
Q
1
*
Q
 is returned.
compz
Must be
'N'
,
'I'
, or
'V'
.
If
compz
 =
'N'
, matrix
Z
 is not computed.
If
compz
 =
'I'
,
Z
 is initialized to the unit matrix, and the orthogonal/unitary matrix
Z
 is returned;
If
compz
 =
'V'
,
Z
 must contain an orthogonal/unitary matrix
Z
1
 on entry, and the product
Z
1
*
Z
 is returned.
n
The order of the matrices
A
 and
B
 (
n
≥
 0).
ilo
,
ihi
ilo
 and
ihi
 mark the rows and columns of
A
 which are to be reduced. It is assumed that
A
 is already upper triangular in rows and columns 1:
ilo
-1 and
ihi
+1:
n
. Values of
ilo
 and
ihi
 are normally set by a previous call to ggbal; otherwise they should be set to 1 and
n
 respectively.
Constraint:
If
n
 > 0
, then 1
≤
ilo
≤
ihi
≤
n
;
if
n
 = 0
, then
ilo
 = 1
 and
ihi
 = 0
.
a
,
b
,
q
,
z
Arrays:
a
(size max(1,
lda
*
n
))
 contains the
n
-by-
n
 general matrix
A
.
b
(size max(1,
ldb
*
n
))
 contains the
n
-by-
n
 upper triangular matrix
B
.
q
(size max(1,
ldq
*
n
))
If
compq
 =
'N'
, then
q
 is not referenced.
If
compq
 =
'V'
, then
q
 must contain the orthogonal/unitary matrix
Q
1
, typically from the
QR
 factorization of
B
.
z
(size max(1,
ldz
*
n
))
If
compz
 =
'N'
, then
z
 is not referenced.
If
compz
 =
'V'
, then
z
 must contain the orthogonal/unitary matrix
Z
1
.
lda
The leading dimension of
a
; at least max(1,
n
).
ldb
The leading dimension of
b
; at least max(1,
n
).
ldq
The leading dimension of
q
;
If
compq
 =
'N'
, then
ldq
≥
 1.
If
compq
 =
'I'
or
'V'
, then
ldq
≥
 max(1,
n
).
ldz
The leading dimension of
z
;
If
compz
 =
'N'
, then
ldz
≥
 1.
If
compz
 =
'I'
or
'V'
, then
ldz
≥
 max(1,
n
).
Output Parameters
a
On exit, the upper triangle and the first subdiagonal of
A
 are overwritten with the upper Hessenberg matrix
H
, and the rest is set to zero.
b
On exit, overwritten by the upper triangular matrix
T
 =
Q
H
*
B*
Z
. The elements below the diagonal are set to zero.
q
If
compq
 =
'I'
, then
q
 contains the orthogonal/unitary matrix
Q
, ;
If
compq
 =
'V'
, then
q
 is overwritten by the product
Q
1
*
Q
.
z
If
compz
 =
'I'
, then
z
 contains the orthogonal/unitary matrix
Z
;
If
compz
 =
'V'
, then
z
 is overwritten by the product
Z
1
*
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.

Product and Performance Information

1

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