NAG FL Interface
f08psf (zhseqr)
1
Purpose
f08psf computes all the eigenvalues and, optionally, the Schur factorization of a complex Hessenberg matrix or a complex general matrix which has been reduced to Hessenberg form.
2
Specification
Fortran Interface
Subroutine f08psf ( |
job, compz, n, ilo, ihi, h, ldh, w, z, ldz, work, lwork, info) |
Integer, Intent (In) |
:: |
n, ilo, ihi, ldh, ldz, lwork |
Integer, Intent (Out) |
:: |
info |
Complex (Kind=nag_wp), Intent (Inout) |
:: |
h(ldh,*), w(*), z(ldz,*) |
Complex (Kind=nag_wp), Intent (Out) |
:: |
work(max(1,lwork)) |
Character (1), Intent (In) |
:: |
job, compz |
|
C Header Interface
#include <nag.h>
void |
f08psf_ (const char *job, const char *compz, const Integer *n, const Integer *ilo, const Integer *ihi, Complex h[], const Integer *ldh, Complex w[], Complex z[], const Integer *ldz, Complex work[], const Integer *lwork, Integer *info, const Charlen length_job, const Charlen length_compz) |
|
C++ Header Interface
#include <nag.h> extern "C" {
void |
f08psf_ (const char *job, const char *compz, const Integer &n, const Integer &ilo, const Integer &ihi, Complex h[], const Integer &ldh, Complex w[], Complex z[], const Integer &ldz, Complex work[], const Integer &lwork, Integer &info, const Charlen length_job, const Charlen length_compz) |
}
|
The routine may be called by the names f08psf, nagf_lapackeig_zhseqr or its LAPACK name zhseqr.
3
Description
f08psf computes all the eigenvalues and, optionally, the Schur factorization of a complex upper Hessenberg matrix
:
where
is an upper triangular matrix (the Schur form of
), and
is the unitary matrix whose columns are the Schur vectors
. The diagonal elements of
are the eigenvalues of
.
The routine may also be used to compute the Schur factorization of a complex general matrix
which has been reduced to upper Hessenberg form
:
In this case, after
f08nsf has been called to reduce
to Hessenberg form,
f08ntf must be called to form
explicitly;
is then passed to
f08psf, which must be called with
.
The routine can also take advantage of a previous call to
f08nvf which may have balanced the original matrix before reducing it to Hessenberg form, so that the Hessenberg matrix
has the structure:
where
and
are upper triangular. If so, only the central diagonal block
(in rows and columns
to
) needs to be further reduced to Schur form (the blocks
and
are also affected). Therefore the values of
and
can be supplied to
f08psf directly. Also,
f08nwf must be called after this routine to permute the Schur vectors of the balanced matrix to those of the original matrix. If
f08nvf has not been called however, then
must be set to
and
to
. Note that if the Schur factorization of
is required,
f08nvf must
not be called with
or
, because the balancing transformation is not unitary.
f08psf uses a multishift form of the upper Hessenberg
algorithm, due to
Bai and Demmel (1989). The Schur vectors are normalized so that
, but are determined only to within a complex factor of absolute value
.
4
References
Bai Z and Demmel J W (1989) On a block implementation of Hessenberg multishift iteration Internat. J. High Speed Comput. 1 97–112
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
5
Arguments
-
1:
– Character(1)
Input
-
On entry: indicates whether eigenvalues only or the Schur form
is required.
- Eigenvalues only are required.
- The Schur form is required.
Constraint:
or .
-
2:
– Character(1)
Input
-
On entry: indicates whether the Schur vectors are to be computed.
- No Schur vectors are computed (and the array z is not referenced).
- The Schur vectors of are computed (and the array z must contain the matrix on entry).
- The Schur vectors of are computed (and the array z is initialized by the routine).
Constraint:
, or .
-
3:
– Integer
Input
-
On entry: , the order of the matrix .
Constraint:
.
-
4:
– Integer
Input
-
5:
– Integer
Input
-
On entry: if the matrix
has been balanced by
f08nvf,
ilo and
ihi must contain the values returned by that routine. Otherwise,
ilo must be set to
and
ihi to
n.
Constraint:
and .
-
6:
– Complex (Kind=nag_wp) array
Input/Output
-
Note: the second dimension of the array
h
must be at least
.
On entry: the
by
upper Hessenberg matrix
, as returned by
f08nsf.
On exit: if
, the array contains no useful information.
If
,
h is overwritten by the upper triangular matrix
from the Schur decomposition (the Schur form) unless
.
-
7:
– Integer
Input
-
On entry: the first dimension of the array
h as declared in the (sub)program from which
f08psf is called.
Constraint:
.
-
8:
– Complex (Kind=nag_wp) array
Output
-
Note: the dimension of the array
w
must be at least
.
On exit: the computed eigenvalues, unless
(in which case see
Section 6). The eigenvalues are stored in the same order as on the diagonal of the Schur form
(if computed).
-
9:
– Complex (Kind=nag_wp) array
Input/Output
-
Note: the second dimension of the array
z
must be at least
if
or
and at least
if
.
On entry: if
,
z must contain the unitary matrix
from the reduction to Hessenberg form.
If
,
z need not be set.
On exit: if
or
,
z contains the unitary matrix of the required Schur vectors, unless
.
If
,
z is not referenced.
-
10:
– Integer
Input
-
On entry: the first dimension of the array
z as declared in the (sub)program from which
f08psf is called.
Constraints:
- if or , ;
- if , .
-
11:
– Complex (Kind=nag_wp) array
Workspace
-
On exit: if
, the real part of
contains the minimum value of
lwork required for optimal performance.
-
12:
– Integer
Input
-
On entry: the dimension of the array
work as declared in the (sub)program from which
f08psf is called, unless
, in which case a workspace query is assumed and the routine only calculates the minimum dimension of
work.
Constraint:
or .
-
13:
– Integer
Output
On exit:
unless the routine detects an error (see
Section 6).
6
Error Indicators and Warnings
If , argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
-
Dynamic memory allocation failed.
See
Section 9 in the Introduction to the NAG Library FL Interface for further information.
An explanatory message is output, and execution of the program is terminated.
-
The algorithm has failed to find all the eigenvalues after a total of iterations.
7
Accuracy
The computed Schur factorization is the exact factorization of a nearby matrix
, where
and
is the
machine precision.
If
is an exact eigenvalue, and
is the corresponding computed value, then
where
is a modestly increasing function of
, and
is the reciprocal condition number of
. The condition numbers
may be computed by calling
f08qyf.
8
Parallelism and Performance
f08psf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08psf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the
X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the
Users' Note for your implementation for any additional implementation-specific information.
The total number of real floating-point operations depends on how rapidly the algorithm converges, but is typically about:
- if only eigenvalues are computed;
- if the Schur form is computed;
- if the full Schur factorization is computed.
The real analogue of this routine is
f08pef.
10
Example
This example computes all the eigenvalues and the Schur factorization of the upper Hessenberg matrix
, where
See also
Section 10 in
f08ntf, which illustrates the use of this routine to compute the Schur factorization of a general matrix.
10.1
Program Text
10.2
Program Data
10.3
Program Results