NAG Library Function Document
nag_dstegr (f08jlc)
1
Purpose
nag_dstegr (f08jlc) computes selected eigenvalues and, optionally, the corresponding eigenvectors of a real by symmetric tridiagonal matrix.
2
Specification
#include <nag.h> |
#include <nagf08.h> |
void |
nag_dstegr (Nag_OrderType order,
Nag_JobType job,
Nag_RangeType range,
Integer n,
double d[],
double e[],
double vl,
double vu,
Integer il,
Integer iu,
Integer *m,
double w[],
double z[],
Integer pdz,
Integer isuppz[],
NagError *fail) |
|
3
Description
nag_dstegr (f08jlc) computes selected eigenvalues and, optionally, the corresponding eigenvectors, of a real symmetric tridiagonal matrix
. That is, the function computes the (partial) spectral factorization of
given by
where
is a diagonal matrix whose diagonal elements are the selected eigenvalues,
, of
and
is an orthogonal matrix whose columns are the corresponding eigenvectors,
, of
. Thus
where
is the number of selected eigenvalues computed.
The function may also be used to compute selected eigenvalues and eigenvectors of a real symmetric matrix
which has been reduced to tridiagonal form
:
In this case, the matrix
must be explicitly applied to the output matrix
. The functions which must be called to perform the reduction to tridiagonal form and apply
are:
This function uses the dqds and the Relatively Robust Representation algorithms to compute the eigenvalues and eigenvectors respectively; see for example
Parlett and Dhillon (2000) and
Dhillon and Parlett (2004) for further details.
nag_dstegr (f08jlc) can usually compute all the eigenvalues and eigenvectors in
floating-point operations and so, for large matrices, is often considerably faster than the other symmetric tridiagonal functions in this chapter when all the eigenvectors are required, particularly so compared to those functions that are based on the
algorithm.
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
Barlow J and Demmel J W (1990) Computing accurate eigensystems of scaled diagonally dominant matrices SIAM J. Numer. Anal. 27 762–791
Dhillon I S and Parlett B N (2004) Orthogonal eigenvectors and relative gaps. SIAM J. Appl. Math. 25 858–899
Parlett B N and Dhillon I S (2000) Relatively robust representations of symmetric tridiagonals Linear Algebra Appl. 309 121–151
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:
– Nag_RangeTypeInput
-
On entry: indicates which eigenvalues should be returned.
- All eigenvalues will be found.
- All eigenvalues in the half-open interval will be found.
- The ilth through iuth eigenvectors will be found.
Constraint:
, or .
- 4:
– IntegerInput
-
On entry: , the order of the matrix .
Constraint:
.
- 5:
– 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:
d is overwritten.
- 6:
– doubleInput/Output
-
Note: the dimension,
dim, of the array
e
must be at least
.
On entry: to are the subdiagonal elements of the tridiagonal matrix . need not be set.
On exit:
e is overwritten.
- 7:
– doubleInput
- 8:
– doubleInput
-
On entry: if
,
vl and
vu contain the lower and upper bounds respectively of the interval to be searched for eigenvalues.
If
or
,
vl and
vu are not referenced.
Constraint:
if , .
- 9:
– IntegerInput
- 10:
– IntegerInput
-
On entry: if
,
il and
iu contains the indices (in ascending order) of the smallest and largest eigenvalues to be returned, respectively.
If
or
,
il and
iu are not referenced.
Constraints:
- if and , ;
- if and , and .
- 11:
– Integer *Output
-
On exit: the total number of eigenvalues found.
.
If , .
If , .
- 12:
– doubleOutput
-
Note: the dimension,
dim, of the array
w
must be at least
.
On exit: the eigenvalues in ascending order.
- 13:
– 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, the columns of
z contain the orthonormal eigenvectors of the matrix
, with the
th column of
holding the eigenvector associated with
.
If
,
z is not referenced.
- 14:
– IntegerInput
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
z.
Constraints:
- if , ;
- otherwise .
- 15:
– IntegerOutput
-
Note: the dimension,
dim, of the array
isuppz
must be at least
.
On exit: the support of the eigenvectors in , i.e., the indices indicating the nonzero elements in . The th eigenvector is nonzero only in elements through .
- 16:
– 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
-
Inverse iteration failed to converge.
The algorithm failed to converge.
- NE_ENUM_INT_2
-
On entry, , and .
Constraint: if , ;
otherwise .
- NE_ENUM_INT_3
-
On entry, , , and .
Constraint: if and , ;
if and , and .
- NE_ENUM_REAL_2
-
On entry, , and .
Constraint: 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
8
Parallelism and Performance
nag_dstegr (f08jlc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_dstegr (f08jlc) 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 required to compute all the eigenvalues and eigenvectors is approximately proportional to .
The complex analogue of this function is
nag_zstegr (f08jyc).
10
Example
This example finds all the eigenvalues and eigenvectors of the symmetric tridiagonal matrix
10.1
Program Text
Program Text (f08jlce.c)
10.2
Program Data
Program Data (f08jlce.d)
10.3
Program Results
Program Results (f08jlce.r)