NAG Library Function Document
nag_zheevd (f08fqc)
1
Purpose
nag_zheevd (f08fqc) computes all the eigenvalues and, optionally, all the eigenvectors of a complex Hermitian matrix.
If the eigenvectors are requested, then it uses a divide-and-conquer algorithm to compute eigenvalues and eigenvectors. However, if only eigenvalues are required, then it uses the Pal–Walker–Kahan variant of the or algorithm.
2
Specification
#include <nag.h> |
#include <nagf08.h> |
void |
nag_zheevd (Nag_OrderType order,
Nag_JobType job,
Nag_UploType uplo,
Integer n,
Complex a[],
Integer pda,
double w[],
NagError *fail) |
|
3
Description
nag_zheevd (f08fqc) computes all the eigenvalues and, optionally, all the eigenvectors of a complex Hermitian matrix
.
In other words, it can compute the spectral factorization of
as
where
is a real diagonal matrix whose diagonal elements are the eigenvalues
, and
is the (complex) unitary matrix whose columns are the eigenvectors
. Thus
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: indicates whether the upper or lower triangular part of
is stored.
- The upper triangular part of is stored.
- 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
a
must be at least
.
On entry: the
by
Hermitian matrix
.
If , is stored in .
If , is stored in .
If , the upper triangular part of must be stored and the elements of the array below the diagonal are not referenced.
If , the lower triangular part of must be stored and the elements of the array above the diagonal are not referenced.
On exit: if
,
a is overwritten by the unitary matrix
which contains the eigenvectors of
.
- 6:
– IntegerInput
On entry: the stride separating row or column elements (depending on the value of
order) of the matrix
in the array
a.
Constraint:
.
- 7:
– doubleOutput
-
Note: the dimension,
dim, of the array
w
must be at least
.
On exit: the eigenvalues of the matrix in ascending order.
- 8:
– 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
-
If and , the algorithm failed to converge; elements of an intermediate tridiagonal form did not converge to zero; if and , then the algorithm failed to compute an eigenvalue while working on the submatrix lying in rows and column through .
- NE_INT
-
On entry, .
Constraint: .
On entry, .
Constraint: .
- NE_INT_2
-
On entry, and .
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_zheevd (f08fqc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_zheevd (f08fqc) 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.
The real analogue of this function is
nag_dsyevd (f08fcc).
10
Example
This example computes all the eigenvalues and eigenvectors of the Hermitian matrix
, where
The example program for nag_zheevd (f08fqc) illustrates the computation of error bounds for the eigenvalues and eigenvectors.
10.1
Program Text
Program Text (f08fqce.c)
10.2
Program Data
Program Data (f08fqce.d)
10.3
Program Results
Program Results (f08fqce.r)