NAG CL Interface
f08hnc (zhbev)
1
Purpose
f08hnc computes all the eigenvalues and, optionally, all the eigenvectors of a complex by Hermitian band matrix of bandwidth .
2
Specification
void |
f08hnc (Nag_OrderType order,
Nag_JobType job,
Nag_UploType uplo,
Integer n,
Integer kd,
Complex ab[],
Integer pdab,
double w[],
Complex z[],
Integer pdz,
NagError *fail) |
|
The function may be called by the names: f08hnc, nag_lapackeig_zhbev or nag_zhbev.
3
Description
The Hermitian band 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:
– Nag_OrderType
Input
-
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 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint:
or .
-
2:
– Nag_JobType
Input
-
On entry: indicates whether eigenvectors are computed.
- Only eigenvalues are computed.
- Eigenvalues and eigenvectors are computed.
Constraint:
or .
-
3:
– Nag_UploType
Input
-
On entry: if
, the upper triangular part of
is stored.
If , the lower triangular part of is stored.
Constraint:
or .
-
4:
– Integer
Input
-
On entry: , the order of the matrix .
Constraint:
.
-
5:
– Integer
Input
-
On entry: if
, the number of superdiagonals,
, of the matrix
.
If , the number of subdiagonals, , of the matrix .
Constraint:
.
-
6:
– Complex
Input/Output
-
Note: the dimension,
dim, of the array
ab
must be at least
.
On entry: the upper or lower triangle of the
by
Hermitian band matrix
.
This is stored as a notional two-dimensional array with row elements or column elements stored contiguously. The storage of elements of
, depends on the
order and
uplo arguments as follows:
- if and ,
- is stored in , for and ;
- if and ,
- is stored in , for and ;
- if and ,
- is stored in , for and ;
- if and ,
- is stored in , for and .
On exit:
ab is overwritten by values generated during the reduction to tridiagonal form.
The first superdiagonal or subdiagonal and the diagonal of the tridiagonal matrix
are returned in
ab using the same storage format as described above.
-
7:
– Integer
Input
On entry: the stride separating row or column elements (depending on the value of
order) of the matrix
in the array
ab.
Constraint:
.
-
8:
– double
Output
-
On exit: the eigenvalues in ascending order.
-
9:
– Complex
Output
-
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.
-
10:
– Integer
Input
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
z.
Constraints:
- if , ;
- otherwise .
-
11:
– NagError *
Input/Output
-
The NAG error argument (see
Section 7 in the Introduction to the NAG Library CL Interface).
6
Error Indicators and Warnings
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
See
Section 3.1.2 in the Introduction to the NAG Library CL Interface 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: .
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 7.5 in the Introduction to the NAG Library CL Interface for further information.
- NE_NO_LICENCE
-
Your licence key may have expired or may not have been installed correctly.
See
Section 8 in the Introduction to the NAG Library CL Interface 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
f08hnc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08hnc 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 total number of floating-point operations is proportional to if and is proportional to otherwise.
The real analogue of this function is
f08hac.
10
Example
This example finds all the eigenvalues and eigenvectors of the Hermitian band matrix
together with approximate error bounds for the computed eigenvalues and eigenvectors.
10.1
Program Text
10.2
Program Data
10.3
Program Results