NAG Library Function Document
nag_dsteqr (f08jec)
1
Purpose
nag_dsteqr (f08jec) computes all the eigenvalues and, optionally, all the eigenvectors of a real symmetric tridiagonal matrix, or of a real symmetric matrix which has been reduced to tridiagonal form.
2
Specification
#include <nag.h> |
#include <nagf08.h> |
void |
nag_dsteqr (Nag_OrderType order,
Nag_ComputeZType compz,
Integer n,
double d[],
double e[],
double z[],
Integer pdz,
NagError *fail) |
|
3
Description
nag_dsteqr (f08jec) 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
The function may also be used to compute all the eigenvalues and eigenvectors of a real symmetric matrix
which has been reduced to tridiagonal form
:
In this case, the matrix
must be formed explicitly and passed to
nag_dsteqr (f08jec), which must be called with
. The functions which must be called to perform the reduction to tridiagonal form and form
are:
nag_dsteqr (f08jec) uses the implicitly shifted
algorithm, switching between the
and
variants in order to handle graded matrices effectively (see
Greenbaum and Dongarra (1980)). The eigenvectors are normalized so that
, but are determined only to within a factor
.
If only the eigenvalues of
are required, it is more efficient to call
nag_dsterf (f08jfc) instead. If
is positive definite, small eigenvalues can be computed more accurately by
nag_dpteqr (f08jgc).
4
References
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Greenbaum A and Dongarra J J (1980) Experiments with QR/QL methods for the symmetric triangular eigenproblem
LAPACK Working Note No. 17 (Technical Report CS-89-92) University of Tennessee, Knoxville
http://www.netlib.org/lapack/lawnspdf/lawn17.pdf
Parlett B N (1998) The Symmetric Eigenvalue Problem SIAM, Philadelphia
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_ComputeZTypeInput
-
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 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 in ascending order, unless
NE_CONVERGENCE (in which case see
Section 6).
- 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 .
On exit:
e is overwritten.
- 6:
– doubleInput/Output
-
Note: the dimension,
dim, of the array
z
must be at least
when
or
.
The
th element of the matrix
is stored in
- when ;
- when .
On entry: if
,
z must contain the orthogonal matrix
from the reduction to tridiagonal form.
If
,
z must be allocated, but its contents need not be set.
If
,
z is not referenced and may be
NULL.
On exit: if
or
, the
required orthonormal eigenvectors stored as columns of
; the
th column corresponds to the
th eigenvalue, where
, unless
.
z is not changed if
.
- 7:
– IntegerInput
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
z.
Constraints:
- if or , ;
- if , z may be NULL.
- 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 has failed to find all the eigenvalues after a total of
iterations. In this case,
d and
e contain on exit the diagonal and off-diagonal elements, respectively, of a tridiagonal matrix orthogonally similar to
.
off-diagonal elements have not converged to zero.
- NE_ENUM_INT_2
-
On entry, , and .
Constraint: if or , .
On entry, , , .
Constraint: .
- 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_dsteqr (f08jec) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_dsteqr (f08jec) 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 typically about if and about if or , but depends on how rapidly the algorithm converges. When , the operations are all performed in scalar mode; the additional operations to compute the eigenvectors when or can be vectorized and on some machines may be performed much faster.
The complex analogue of this function is
nag_zsteqr (f08jsc).
10
Example
This example computes all the eigenvalues and eigenvectors of the symmetric tridiagonal matrix
, where
See also the examples for
nag_dorgtr (f08ffc),
nag_dopgtr (f08gfc) or
nag_dsbtrd (f08hec), which illustrate the use of this function to compute the eigenvalues and eigenvectors of a full or band symmetric matrix.
10.1
Program Text
Program Text (f08jece.c)
10.2
Program Data
Program Data (f08jece.d)
10.3
Program Results
Program Results (f08jece.r)