NAG FL Interface
f08gnf (zhpev)
1
Purpose
f08gnf computes all the eigenvalues and, optionally, all the eigenvectors of a complex by Hermitian matrix in packed storage.
2
Specification
Fortran Interface
Integer, Intent (In) |
:: |
n, ldz |
Integer, Intent (Out) |
:: |
info |
Real (Kind=nag_wp), Intent (Out) |
:: |
w(n), rwork(3*n-2) |
Complex (Kind=nag_wp), Intent (Inout) |
:: |
ap(*), z(ldz,*) |
Complex (Kind=nag_wp), Intent (Out) |
:: |
work(2*n-1) |
Character (1), Intent (In) |
:: |
jobz, uplo |
|
C Header Interface
#include <nag.h>
void |
f08gnf_ (const char *jobz, const char *uplo, const Integer *n, Complex ap[], double w[], Complex z[], const Integer *ldz, Complex work[], double rwork[], Integer *info, const Charlen length_jobz, const Charlen length_uplo) |
|
C++ Header Interface
#include <nag.h> extern "C" {
void |
f08gnf_ (const char *jobz, const char *uplo, const Integer &n, Complex ap[], double w[], Complex z[], const Integer &ldz, Complex work[], double rwork[], Integer &info, const Charlen length_jobz, const Charlen length_uplo) |
}
|
The routine may be called by the names f08gnf, nagf_lapackeig_zhpev or its LAPACK name zhpev.
3
Description
The Hermitian matrix is first reduced to real tridiagonal form, using unitary similarity transformations, and then the algorithm is applied to the tridiagonal matrix to compute the eigenvalues and (optionally) the eigenvectors.
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
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 eigenvectors are computed.
- Only eigenvalues are computed.
- Eigenvalues and eigenvectors are computed.
Constraint:
or .
-
2:
– Character(1)
Input
-
On entry: if
, the upper triangular part of
is stored.
If , the lower triangular part of is stored.
Constraint:
or .
-
3:
– Integer
Input
-
On entry: , the order of the matrix .
Constraint:
.
-
4:
– Complex (Kind=nag_wp) array
Input/Output
-
Note: the dimension of the array
ap
must be at least
.
On entry: the upper or lower triangle of the
by
Hermitian matrix
, packed by columns.
More precisely,
- if , the upper triangle of must be stored with element in for ;
- if , the lower triangle of must be stored with element in for .
On exit:
ap is overwritten by the values generated during the reduction to tridiagonal form. The elements of the diagonal and the off-diagonal of the tridiagonal matrix overwrite the corresponding elements of
.
-
5:
– Real (Kind=nag_wp) array
Output
-
On exit: the eigenvalues in ascending order.
-
6:
– 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
,
z contains the orthonormal eigenvectors of the matrix
, with the
th column of
holding the eigenvector associated with
.
If
,
z is not referenced.
-
7:
– Integer
Input
-
On entry: the first dimension of the array
z as declared in the (sub)program from which
f08gnf is called.
Constraints:
- if , ;
- otherwise .
-
8:
– Complex (Kind=nag_wp) array
Workspace
-
-
9:
– Real (Kind=nag_wp) array
Workspace
-
-
10:
– 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.
-
The algorithm failed to converge; off-diagonal elements of an intermediate tridiagonal form did not converge to zero.
7
Accuracy
The computed eigenvalues and eigenvectors are exact for a nearby matrix
, where
and
is the
machine precision. See Section 4.7 of
Anderson et al. (1999) for further details.
8
Parallelism and Performance
f08gnf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08gnf 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.
Each eigenvector is normalized so that the element of largest absolute value is real.
The total number of floating-point operations is proportional to .
The real analogue of this routine is
f08gaf.
10
Example
This example finds all the eigenvalues of the Hermitian matrix
together with approximate error bounds for the computed eigenvalues.
10.1
Program Text
10.2
Program Data
10.3
Program Results