NAG Library Function Document
nag_dstevd (f08jcc)
1
Purpose
nag_dstevd (f08jcc) computes all the eigenvalues and, optionally, all the eigenvectors of a real symmetric tridiagonal 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_dstevd (Nag_OrderType order,
Nag_JobType job,
Integer n,
double d[],
double e[],
double z[],
Integer pdz,
NagError *fail) |
|
3
Description
nag_dstevd (f08jcc) computes all the eigenvalues and, optionally, all the eigenvectors of a real symmetric tridiagonal matrix
.
In other words, it can compute the spectral factorization of
as
where
is a diagonal matrix whose diagonal elements are the eigenvalues
, and
is the orthogonal matrix whose columns are the eigenvectors
. Thus
4
References
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: the eigenvalues of the matrix in ascending order.
- 5:
– doubleInput/Output
-
Note: the dimension,
dim, of the array
e
must be at least
.
On entry: the off-diagonal elements of the tridiagonal matrix . The th element of this array is used as workspace.
On exit:
e is overwritten with intermediate results.
- 6:
– doubleOutput
-
Note: the dimension,
dim, of the array
z
must be at least
- when
;
- when
.
The
th element of the matrix
is stored in
- when ;
- when .
On exit: if
,
z is overwritten by the orthogonal matrix
which contains the eigenvectors of
.
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 , ;
- if , .
- 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; eigenvectors did not converge.
- NE_ENUM_INT_2
-
On entry, , and .
Constraint: if , ;
if , .
- 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.
If
is an exact eigenvalue and
is the corresponding computed value, then
where
is a modestly increasing function of
.
If
is the corresponding exact eigenvector, and
is the corresponding computed eigenvector, then the angle
between them is bounded as follows:
Thus the accuracy of a computed eigenvector depends on the gap between its eigenvalue and all the other eigenvalues.
8
Parallelism and Performance
nag_dstevd (f08jcc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_dstevd (f08jcc) 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.
There is no complex analogue of this function.
10
Example
This example computes all the eigenvalues and eigenvectors of the symmetric tridiagonal matrix
, where
10.1
Program Text
Program Text (f08jcce.c)
10.2
Program Data
Program Data (f08jcce.d)
10.3
Program Results
Program Results (f08jcce.r)