Developer Reference for Intel® oneAPI Math Kernel Library - C

Developer Reference

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

?trexc

Reorders the Schur factorization of a general matrix.

Syntax

lapack_int LAPACKE_strexc
(
int
matrix_layout
,
char
compq
,
lapack_int
n
,
float*
t
,
lapack_int
ldt
,
float*
q
,
lapack_int
ldq
,
lapack_int*
ifst
,
lapack_int*
ilst
);
lapack_int LAPACKE_dtrexc
(
int
matrix_layout
,
char
compq
,
lapack_int
n
,
double*
t
,
lapack_int
ldt
,
double*
q
,
lapack_int
ldq
,
lapack_int*
ifst
,
lapack_int*
ilst
);
lapack_int LAPACKE_ctrexc
(
int
matrix_layout
,
char
compq
,
lapack_int
n
,
lapack_complex_float*
t
,
lapack_int
ldt
,
lapack_complex_float*
q
,
lapack_int
ldq
,
lapack_int
ifst
,
lapack_int
ilst
);
lapack_int LAPACKE_ztrexc
(
int
matrix_layout
,
char
compq
,
lapack_int
n
,
lapack_complex_double*
t
,
lapack_int
ldt
,
lapack_complex_double*
q
,
lapack_int
ldq
,
lapack_int
ifst
,
lapack_int
ilst
);
Include Files
  • mkl.h
Description
The routine reorders the Schur factorization of a general matrix
A
 =
Q*T*Q
H
, so that the diagonal element or block of
T
 with row index
ifst
 is moved to row
ilst
.
The reordered Schur form
S
 is computed by an unitary (or, for real flavors, orthogonal) similarity transformation:
S
 =
Z
H
*T*Z
. Optionally the updated matrix
P
 of Schur vectors is computed as
P
 =
Q*Z
, giving
A
=
P*S*P
H
.
Input Parameters
matrix_layout
Specifies whether matrix storage layout is row major (
LAPACK_ROW_MAJOR
) or column major (
LAPACK_COL_MAJOR
).
compq
Must be
'V'
 or
'N'
.
If
compq
 =
'V'
, then the Schur vectors (
Q
) are updated.
If
compq
 =
'N'
, then no Schur vectors are updated.
n
The order of the matrix
T
 (
n
≥
 0
).
t
,
q
Arrays:
t
(size max(1,
ldt
*
n
))
 contains the
n
-by-
n
 matrix
T
.
q
(size max(1,
ldq
*
n
))
If
compq
 =
'V'
, then
q
 must contain
Q
 (Schur vectors).
If
compq
 =
'N'
, then
q
 is not referenced.
ldt
The leading dimension of
t
; at least max(1,
n
).
ldq
The leading dimension of
q
;
If
compq
 =
'N'
, then
ldq
≥
 1
.
If
compq
 =
'V'
, then
ldq
≥
 max(1,
n
)
.
ifst
,
ilst
1
≤
ifst
≤
n
; 1
≤
ilst
≤
n
.
Must specify the reordering of the diagonal elements (or blocks, which is possible for real flavors) of the matrix
T
. The element (or block) with row index
ifst
 is moved to row
ilst
 by a sequence of exchanges between adjacent elements (or blocks).
Output Parameters
t
Overwritten by the updated matrix
S
.
q
If
compq
 =
'V'
,
q
 contains the updated matrix of Schur vectors.
ifst
,
ilst
Overwritten for real flavors only.
If
ifst
 pointed to the second row of a 2 by 2 block on entry, it is changed to point to the first row;
ilst
 always points to the first row of the block in its final position (which may differ from its input value by
±
1).
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 computed matrix
S
 is exactly similar to a matrix
T
+
E
, where
||
E
||
2
 =
O
(
ε
)*||
T
||
2
, and
ε
 is the machine precision.
Note that if a 2 by 2 diagonal block is involved in the re-ordering, its off-diagonal elements are in general changed; the diagonal elements and the eigenvalues of the block are unchanged unless the block is sufficiently ill-conditioned, in which case they may be noticeably altered. It is possible for a 2 by 2 block to break into two 1 by 1 blocks, that is, for a pair of complex eigenvalues to become purely real.
The approximate number of floating-point operations is
for real flavors:
6
n
(
ifst
-
ilst
)
 if
compq
 =
'N'
;
12
n
(
ifst
-
ilst
)
 if
compq
 =
'V'
;
for complex flavors:
20
n
(
ifst
-
ilst
)
 if
compq
 =
'N'
;
40
n
(
ifst
-
ilst
)
 if
compq
 =
'V'
.

Product and Performance Information

1

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