NAG Library Function Document
nag_zstedc (f08jvc)
1
Purpose
nag_zstedc (f08jvc) computes all the eigenvalues and, optionally, all the eigenvectors of a real by symmetric tridiagonal matrix, or of a complex full or banded Hermitian matrix which has been reduced to tridiagonal form.
2
Specification
#include <nag.h> |
#include <nagf08.h> |
void |
nag_zstedc (Nag_OrderType order,
Nag_ComputeEigVecsType compz,
Integer n,
double d[],
double e[],
Complex z[],
Integer pdz,
NagError *fail) |
|
3
Description
nag_zstedc (f08jvc) computes all the eigenvalues and, optionally, the eigenvectors of a real symmetric tridiagonal matrix
. That is, the function computes the spectral factorization of
given by
where
is a diagonal matrix whose diagonal elements are the eigenvalues,
, of
and
is an orthogonal matrix whose columns are the eigenvectors,
, of
. Thus
The function may also be used to compute all the eigenvalues and eigenvectors of a complex full, or banded, Hermitian matrix
which has been reduced to real tridiagonal form
as
where
is unitary. The spectral factorization of
is then given by
In this case
must be formed explicitly and passed to
nag_zstedc (f08jvc) in the array
z, and the function called with
. Functions which may be called to form
and
are
When only eigenvalues are required then this function calls
nag_dsterf (f08jfc) to compute the eigenvalues of the tridiagonal matrix
, but when eigenvectors of
are also required and the matrix is not too small, then a divide and conquer method is used, which can be much faster than
nag_zsteqr (f08jsc), although more storage is required.
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_ComputeEigVecsTypeInput
-
On entry: indicates whether the eigenvectors are to be computed.
- Only the eigenvalues are computed (and the array z is not referenced).
- The eigenvalues and eigenvectors of are computed (and the array z must contain the matrix on entry).
- The eigenvalues and eigenvectors of are computed (and the array z is initialized by the function).
Constraint:
, or .
- 3:
– IntegerInput
-
On entry: , the order of the symmetric tridiagonal 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:
e is overwritten.
- 6:
– ComplexInput/Output
-
Note: the dimension,
dim, of the array
z
must be at least
- when
or ;
- otherwise.
If
then the
th element of the matrix
is stored in
- when ;
- when .
On entry: if
,
z must contain the unitary matrix
used in the reduction to tridiagonal form.
On exit: if
,
z contains the orthonormal eigenvectors of the original Hermitian matrix
, and if
,
z contains the orthonormal eigenvectors of the symmetric tridiagonal matrix
.
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 or , ;
- 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 compute an eigenvalue while working on the submatrix lying in rows and columns through .
- NE_ENUM_INT_2
-
On entry, , and .
Constraint: if or , ;
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.
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.
See Section 4.7 of
Anderson et al. (1999) for further details. See also
nag_ddisna (f08flc).
8
Parallelism and Performance
nag_zstedc (f08jvc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_zstedc (f08jvc) 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.
If only eigenvalues are required, the total number of floating-point operations is approximately proportional to
. When eigenvectors are required the number of operations is bounded above by approximately the same number of operations as
nag_zsteqr (f08jsc), but for large matrices
nag_zstedc (f08jvc) is usually much faster.
The real analogue of this function is
nag_dstedc (f08jhc).
10
Example
This example finds all the eigenvalues and eigenvectors of the Hermitian band matrix
is first reduced to tridiagonal form by a call to
nag_zhbtrd (f08hsc).
10.1
Program Text
Program Text (f08jvce.c)
10.2
Program Data
Program Data (f08jvce.d)
10.3
Program Results
Program Results (f08jvce.r)