NAG CL Interface
f08jcc (dstevd)
1
Purpose
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
void |
f08jcc (Nag_OrderType order,
Nag_JobType job,
Integer n,
double d[],
double e[],
double z[],
Integer pdz,
NagError *fail) |
|
The function may be called by the names: f08jcc, nag_lapackeig_dstevd or nag_dstevd.
3
Description
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_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:
– Integer
Input
-
On entry: , the order of the matrix .
Constraint:
.
-
4:
– double
Input/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:
– double
Input/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:
– double
Output
-
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:
– Integer
Input
-
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 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
e did not converge to zero.
- 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 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.
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
f08jcc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
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
10.2
Program Data
10.3
Program Results