NAG FL Interface
f08fqf (zheevd)
1
Purpose
f08fqf 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
Fortran Interface
Subroutine f08fqf ( |
job, uplo, n, a, lda, w, work, lwork, rwork, lrwork, iwork, liwork, info) |
Integer, Intent (In) |
:: |
n, lda, lwork, lrwork, liwork |
Integer, Intent (Out) |
:: |
iwork(max(1,liwork)), info |
Real (Kind=nag_wp), Intent (Inout) |
:: |
w(*) |
Real (Kind=nag_wp), Intent (Out) |
:: |
rwork(max(1,lrwork)) |
Complex (Kind=nag_wp), Intent (Inout) |
:: |
a(lda,*) |
Complex (Kind=nag_wp), Intent (Out) |
:: |
work(max(1,lwork)) |
Character (1), Intent (In) |
:: |
job, uplo |
|
C Header Interface
#include <nag.h>
void |
f08fqf_ (const char *job, const char *uplo, const Integer *n, Complex a[], const Integer *lda, double w[], Complex work[], const Integer *lwork, double rwork[], const Integer *lrwork, Integer iwork[], const Integer *liwork, Integer *info, const Charlen length_job, const Charlen length_uplo) |
|
C++ Header Interface
#include <nag.h> extern "C" {
void |
f08fqf_ (const char *job, const char *uplo, const Integer &n, Complex a[], const Integer &lda, double w[], Complex work[], const Integer &lwork, double rwork[], const Integer &lrwork, Integer iwork[], const Integer &liwork, Integer &info, const Charlen length_job, const Charlen length_uplo) |
}
|
The routine may be called by the names f08fqf, nagf_lapackeig_zheevd or its LAPACK name zheevd.
3
Description
f08fqf 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
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: 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 .
-
3:
– Integer
Input
-
On entry: , the order of the matrix .
Constraint:
.
-
4:
– Complex (Kind=nag_wp) array
Input/Output
-
Note: the second dimension of the array
a
must be at least
.
On entry: the
by
Hermitian matrix
.
- 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
.
-
5:
– Integer
Input
-
On entry: the first dimension of the array
a as declared in the (sub)program from which
f08fqf is called.
Constraint:
.
-
6:
– Real (Kind=nag_wp) array
Output
-
Note: the dimension of the array
w
must be at least
.
On exit: the eigenvalues of the matrix in ascending order.
-
7:
– Complex (Kind=nag_wp) array
Workspace
-
On exit: if
, the real part of
contains the required minimal size of
lwork.
-
8:
– Integer
Input
-
On entry: the dimension of the array
work as declared in the (sub)program from which
f08fqf is called.
If
, a workspace query is assumed; the routine only calculates the minimum dimension of the
work array, returns this value as the first entry of the
work array, and no error message related to
lwork is issued.
Constraints:
- if , or ;
- if and , or ;
- if and , or .
-
9:
– Real (Kind=nag_wp) array
Workspace
-
On exit: if , contains the required minimal size of .
-
10:
– Integer
Input
-
On entry: the dimension of the array
rwork as declared in the (sub)program from which
f08fqf is called.
If
, a workspace query is assumed; the routine only calculates the minimum dimension of the
rwork array, returns this value as the first entry of the
rwork array, and no error message related to
lrwork is issued.
Constraints:
- if , or ;
- if and , or ;
- if and , or .
-
11:
– Integer array
Workspace
-
On exit: if
,
contains the required minimal size of
liwork.
-
12:
– Integer
Input
-
On entry: the dimension of the array
iwork as declared in the (sub)program from which
f08fqf is called.
If
, a workspace query is assumed; the routine only calculates the minimum dimension of the
iwork array, returns this value as the first entry of the
iwork array, and no error message related to
liwork is issued.
Constraints:
- if , or ;
- if and , or ;
- if and , or .
-
13:
– 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.
-
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 .
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
f08fqf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08fqf 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 real analogue of this routine is
f08fcf.
10
Example
This example computes all the eigenvalues and eigenvectors of the Hermitian matrix
, where
The example program for f08fqf illustrates the computation of error bounds for the eigenvalues and eigenvectors.
10.1
Program Text
10.2
Program Data
10.3
Program Results