NAG Library Function Document
nag_dstev (f08jac)
1
Purpose
nag_dstev (f08jac) computes all the eigenvalues and, optionally, all the eigenvectors of a real by symmetric tridiagonal matrix .
2
Specification
#include <nag.h> |
#include <nagf08.h> |
void |
nag_dstev (Nag_OrderType order,
Nag_JobType job,
Integer n,
double d[],
double e[],
double z[],
Integer pdz,
NagError *fail) |
|
3
Description
nag_dstev (f08jac) computes all the eigenvalues and, optionally, all the eigenvectors of using a combination of the and algorithms, with an implicit shift.
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:
– IntegerInput
-
On entry: , the order of the matrix.
Constraint:
.
- 4:
– doubleInput/Output
-
Note: the dimension,
dim, of the array
d
must be at least
.
On entry: the diagonal elements of the tridiagonal matrix .
On exit: if NE_NOERROR, the eigenvalues in ascending order.
- 5:
– doubleInput/Output
-
Note: the dimension,
dim, of the array
e
must be at least
.
On entry: the subdiagonal elements of the tridiagonal matrix .
On exit: the contents of
e are destroyed.
- 6:
– doubleOutput
-
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
, then if
NE_NOERROR,
z contains the orthonormal eigenvectors of the matrix
, with the
th column of
holding the eigenvector associated with
.
If
,
z is not referenced.
- 7:
– IntegerInput
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
z.
Constraints:
- if , ;
- otherwise .
- 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
-
The algorithm failed to converge;
off-diagonal elements of
e 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_dstev (f08jac) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_dstev (f08jac) 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 if .
10
Example
This example finds all the eigenvalues and eigenvectors of the symmetric tridiagonal matrix
together with approximate error bounds for the computed eigenvalues and eigenvectors.
10.1
Program Text
Program Text (f08jace.c)
10.2
Program Data
Program Data (f08jace.d)
10.3
Program Results
Program Results (f08jace.r)