NAG FL Interface
f08jyf (zstegr)
1
Purpose
f08jyf computes selected eigenvalues and, optionally, the corresponding eigenvectors of a real by symmetric tridiagonal matrix.
2
Specification
Fortran Interface
Subroutine f08jyf ( |
jobz, range, n, d, e, vl, vu, il, iu, abstol, m, w, z, ldz, isuppz, work, lwork, iwork, liwork, info) |
Integer, Intent (In) |
:: |
n, il, iu, ldz, lwork, liwork |
Integer, Intent (Inout) |
:: |
isuppz(*) |
Integer, Intent (Out) |
:: |
m, iwork(max(1,liwork)), info |
Real (Kind=nag_wp), Intent (In) |
:: |
vl, vu, abstol |
Real (Kind=nag_wp), Intent (Inout) |
:: |
d(*), e(*), w(*) |
Real (Kind=nag_wp), Intent (Out) |
:: |
work(max(1,lwork)) |
Complex (Kind=nag_wp), Intent (Inout) |
:: |
z(ldz,*) |
Character (1), Intent (In) |
:: |
jobz, range |
|
C Header Interface
#include <nag.h>
void |
f08jyf_ (const char *jobz, const char *range, const Integer *n, double d[], double e[], const double *vl, const double *vu, const Integer *il, const Integer *iu, const double *abstol, Integer *m, double w[], Complex z[], const Integer *ldz, Integer isuppz[], double work[], const Integer *lwork, Integer iwork[], const Integer *liwork, Integer *info, const Charlen length_jobz, const Charlen length_range) |
|
C++ Header Interface
#include <nag.h> extern "C" {
void |
f08jyf_ (const char *jobz, const char *range, const Integer &n, double d[], double e[], const double &vl, const double &vu, const Integer &il, const Integer &iu, const double &abstol, Integer &m, double w[], Complex z[], const Integer &ldz, Integer isuppz[], double work[], const Integer &lwork, Integer iwork[], const Integer &liwork, Integer &info, const Charlen length_jobz, const Charlen length_range) |
}
|
The routine may be called by the names f08jyf, nagf_lapackeig_zstegr or its LAPACK name zstegr.
3
Description
f08jyf computes selected eigenvalues and, optionally, the corresponding eigenvectors, of a real symmetric tridiagonal matrix
. That is, the routine computes the (partial) spectral factorization of
given by
where
is a diagonal matrix whose diagonal elements are the selected eigenvalues,
, of
and
is an orthogonal matrix whose columns are the corresponding eigenvectors,
, of
. Thus
where
is the number of selected eigenvectors computed.
The routine stores the real orthogonal matrix
in a complex array, so that it may also be used to compute selected eigenvalues and the corresponding eigenvectors of a complex Hermitian matrix
which has been reduced to tridiagonal form
:
In this case, the matrix
must be explicitly applied to the output matrix
. The routines which must be called to perform the reduction to tridiagonal form and apply
are:
This routine uses the dqds and the Relatively Robust Representation algorithms to compute the eigenvalues and eigenvectors respectively; see for example
Parlett and Dhillon (2000) and
Dhillon and Parlett (2004) for further details.
f08jyf can usually compute all the eigenvalues and eigenvectors in
floating-point operations and so, for large matrices, is often considerably faster than the other symmetric tridiagonal routines in this chapter when all the eigenvectors are required, particularly so compared to those routines that are based on the
algorithm.
4
References
Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999)
LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia
https://www.netlib.org/lapack/lug
Barlow J and Demmel J W (1990) Computing accurate eigensystems of scaled diagonally dominant matrices SIAM J. Numer. Anal. 27 762–791
Dhillon I S and Parlett B N (2004) Orthogonal eigenvectors and relative gaps SIAM J. Appl. Math. 25 858–899
Parlett B N and Dhillon I S (2000) Relatively robust representations of symmetric tridiagonals Linear Algebra Appl. 309 121–151
5
Arguments
-
1:
– Character(1)
Input
-
On entry: indicates whether eigenvectors are computed.
- Only eigenvalues are computed.
- Eigenvalues and eigenvectors are computed.
Constraint:
or .
-
2:
– Character(1)
Input
-
On entry: indicates which eigenvalues should be returned.
- All eigenvalues will be found.
- All eigenvalues in the half-open interval will be found.
- The ilth through iuth eigenvectors will be found.
Constraint:
, or .
-
3:
– Integer
Input
-
On entry: , the order of the matrix .
Constraint:
.
-
4:
– Real (Kind=nag_wp) array
Input/Output
-
Note: the dimension of the array
d
must be at least
.
On entry: the diagonal elements of the tridiagonal matrix .
On exit:
d is overwritten.
-
5:
– Real (Kind=nag_wp) array
Input/Output
-
Note: the dimension of the array
e
must be at least
.
On entry: contains the subdiagonal elements of the tridiagonal matrix . need not be set.
On exit:
e is overwritten.
-
6:
– Real (Kind=nag_wp)
Input
-
7:
– Real (Kind=nag_wp)
Input
-
On entry: if
,
vl and
vu contain the lower and upper bounds respectively of the interval to be searched for eigenvalues.
If
or
,
vl and
vu are not referenced.
Constraint:
if , .
-
8:
– Integer
Input
-
9:
– Integer
Input
-
On entry: if
,
il and
iu specify the indices (in ascending order) of the smallest and largest eigenvalues to be returned, respectively.
If
or
,
il and
iu are not referenced.
Constraints:
- if and , ;
- if and , and .
-
10:
– Real (Kind=nag_wp)
Input
-
On entry: in earlier versions, this argument was the absolute error tolerance for the eigenvalues/eigenvectors. It is now deprecated, and only included for backwards-compatibility.
-
11:
– Integer
Output
-
On exit: the total number of eigenvalues found.
.
If , .
If , .
-
12:
– Real (Kind=nag_wp) array
Output
-
Note: the dimension of the array
w
must be at least
.
On exit: the eigenvalues in ascending order.
-
13:
– Complex (Kind=nag_wp) array
Output
-
Note: the second dimension of the array
z
must be at least
if
, and at least
otherwise.
On exit: if
, then if
, the columns of
z contain the orthonormal eigenvectors of the matrix
, with the
th column of
holding the eigenvector associated with
.
If
,
z is not referenced.
Note: you must ensure that at least
columns are supplied in the array
z; if
, the exact value of
m is not known in advance and an upper bound of at least
n must be used.
-
14:
– Integer
Input
-
On entry: the first dimension of the array
z as declared in the (sub)program from which
f08jyf is called.
Constraints:
- if , ;
- otherwise .
-
15:
– Integer array
Output
-
Note: the dimension of the array
isuppz
must be at least
.
On exit: the support of the eigenvectors in , i.e., the indices indicating the nonzero elements in . The th eigenvector is nonzero only in elements through .
-
16:
– Real (Kind=nag_wp) array
Workspace
-
On exit: if
,
returns the minimum
lwork.
-
17:
– Integer
Input
-
On entry: the dimension of the array
work as declared in the (sub)program from which
f08jyf is called.
If
, a workspace query is assumed; the routine only calculates the minimum sizes of the
work and
iwork arrays, returns these values as the first entries of the
work and
iwork arrays, and no error message related to
lwork or
liwork is issued.
Constraint:
or .
-
18:
– Integer array
Workspace
-
On exit: if
,
returns the minimum
liwork.
-
19:
– Integer
Input
-
On entry: the dimension of the array
iwork as declared in the (sub)program from which
f08jyf is called.
If
, a workspace query is assumed; the routine only calculates the minimum sizes of the
work and
iwork arrays, returns these values as the first entries of the
work and
iwork arrays, and no error message related to
lwork or
liwork is issued.
Constraint:
or .
-
20:
– 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.
-
Inverse iteration failed to converge.
The algorithm failed to converge.
7
Accuracy
See the description for
abstol. See also Section 4.7 of
Anderson et al. (1999) and
Barlow and Demmel (1990) for further details.
8
Parallelism and Performance
f08jyf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08jyf 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 floating-point operations required to compute all the eigenvalues and eigenvectors is approximately proportional to .
The real analogue of this routine is
f08jlf.
10
Example
This example finds all the eigenvalues and eigenvectors of the symmetric tridiagonal matrix
abstol is set to zero so that the default tolerance of
is used.
10.1
Program Text
10.2
Program Data
10.3
Program Results