NAG Library Function Document
nag_zhpev (f08gnc)
1
Purpose
nag_zhpev (f08gnc) computes all the eigenvalues and, optionally, all the eigenvectors of a complex by Hermitian matrix in packed storage.
2
Specification
#include <nag.h> |
#include <nagf08.h> |
void |
nag_zhpev (Nag_OrderType order,
Nag_JobType job,
Nag_UploType uplo,
Integer n,
Complex ap[],
double w[],
Complex z[],
Integer pdz,
NagError *fail) |
|
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
http://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:
– Nag_OrderTypeInput
-
On entry: the
order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by
. See
Section 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint:
or .
- 2:
– Nag_JobTypeInput
-
On entry: indicates whether eigenvectors are computed.
- Only eigenvalues are computed.
- Eigenvalues and eigenvectors are computed.
Constraint:
or .
- 3:
– Nag_UploTypeInput
-
On entry: if
, the upper triangular part of
is stored.
If , the lower triangular part of is stored.
Constraint:
or .
- 4:
– IntegerInput
-
On entry: , the order of the matrix .
Constraint:
.
- 5:
– ComplexInput/Output
-
Note: the dimension,
dim, of the array
ap
must be at least
.
On entry: the upper or lower triangle of the
by
Hermitian matrix
, packed by rows or columns.
The storage of elements
depends on the
order and
uplo arguments as follows:
- if and ,
is stored in , for ; - if and ,
is stored in , for ; - if and ,
is stored in , for ; - if and ,
is stored 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
.
- 6:
– doubleOutput
-
On exit: the eigenvalues in ascending order.
- 7:
– ComplexOutput
-
Note: the dimension,
dim, of the array
z
must be at least
- when
;
- otherwise.
The
th element of the matrix
is stored in
- when ;
- when .
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.
- 8:
– IntegerInput
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
z.
Constraints:
- if , ;
- otherwise .
- 9:
– NagError *Input/Output
-
The NAG error argument (see
Section 3.7 in How to Use the NAG Library and its Documentation).
6
Error Indicators and Warnings
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
See
Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
- NE_BAD_PARAM
-
On entry, argument had an illegal value.
- NE_CONVERGENCE
-
The algorithm failed to converge; off-diagonal elements of an intermediate tridiagonal form did not converge to zero.
- NE_ENUM_INT_2
-
On entry, , and .
Constraint: if , ;
otherwise .
- NE_INT
-
On entry, .
Constraint: .
On entry, .
Constraint: .
- NE_INTERNAL_ERROR
-
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact
NAG for assistance.
See
Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
- NE_NO_LICENCE
-
Your licence key may have expired or may not have been installed correctly.
See
Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
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
nag_zhpev (f08gnc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_zhpev (f08gnc) 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 function. 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 function is
nag_dspev (f08gac).
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
Program Text (f08gnce.c)
10.2
Program Data
Program Data (f08gnce.d)
10.3
Program Results
Program Results (f08gnce.r)