NAG Library Function Document
nag_dstevr (f08jdc)
1
Purpose
nag_dstevr (f08jdc) computes selected eigenvalues and, optionally, eigenvectors of a real by symmetric tridiagonal matrix . Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues.
2
Specification
#include <nag.h> |
#include <nagf08.h> |
void |
nag_dstevr (Nag_OrderType order,
Nag_JobType job,
Nag_RangeType range,
Integer n,
double d[],
double e[],
double vl,
double vu,
Integer il,
Integer iu,
double abstol,
Integer *m,
double w[],
double z[],
Integer pdz,
Integer isuppz[],
NagError *fail) |
|
3
Description
Whenever possible
nag_dstevr (f08jdc) computes the eigenspectrum using Relatively Robust Representations.
nag_dstevr (f08jdc) computes eigenvalues by the dqds algorithm, while orthogonal eigenvectors are computed from various ‘good’
representations (also known as Relatively Robust Representations). Gram–Schmidt orthogonalization is avoided as far as possible. More specifically, the various steps of the algorithm are as follows. For the
th unreduced block of
:
(a) |
compute
, such that
is a relatively robust representation, |
(b) |
compute the eigenvalues, , of
to high relative accuracy by the dqds algorithm, |
(c) |
if there is a cluster of close eigenvalues, ‘choose’ close to the cluster, and go to (a), |
(d) |
given the approximate eigenvalue of
, compute the corresponding eigenvector by forming a rank-revealing twisted factorization. |
The desired accuracy of the output can be specified by the argument
abstol. For more details, see
Dhillon (1997) and
Parlett and Dhillon (2000).
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
Demmel J W and Kahan W (1990) Accurate singular values of bidiagonal matrices SIAM J. Sci. Statist. Comput. 11 873–912
Dhillon I (1997) A new algorithm for the symmetric tridiagonal eigenvalue/eigenvector problem Computer Science Division Technical Report No. UCB//CSD-97-971 UC Berkeley
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
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: if
, all eigenvalues will be found.
If , all eigenvalues in the half-open interval will be found.
If
, the
ilth to
iuth eigenvalues 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: may be multiplied by a constant factor chosen to avoid over/underflow in computing the eigenvalues.
- 6:
– 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: may be multiplied by a constant factor chosen to avoid over/underflow in computing the eigenvalues.
- 7:
– doubleInput
- 8:
– doubleInput
-
On entry: if
, the lower and upper bounds 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
, the indices (in ascending order) of the smallest and largest eigenvalues to be returned.
If
or
,
il and
iu are not referenced.
Constraints:
- if and , and ;
- if and , .
- 11:
– doubleInput
-
On entry: the absolute error tolerance for the eigenvalues. An approximate eigenvalue is accepted as converged when it is determined to lie in an interval
of width less than or equal to
where
is the
machine precision. If
abstol is less than or equal to zero, then
will be used in its place. See
Demmel and Kahan (1990).
If high relative accuracy is important, set
abstol to
, although doing so does not currently guarantee that eigenvalues are computed to high relative accuracy. See
Barlow and Demmel (1990) for a discussion of which matrices can define their eigenvalues to high relative accuracy.
- 12:
– Integer *Output
-
On exit: the total number of eigenvalues found.
.
If , .
If , .
- 13:
– doubleOutput
-
Note: the dimension,
dim, of the array
w
must be at least
.
On exit: the first
m elements contain the selected eigenvalues in ascending order.
- 14:
– 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
, the first
m columns of
contain the orthonormal eigenvectors of the matrix
corresponding to the selected eigenvalues, with the
th column of
holding the eigenvector associated with
.
If
,
z is not referenced.
- 15:
– IntegerInput
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
z.
Constraints:
- if , ;
- otherwise .
- 16:
– IntegerOutput
-
Note: the dimension,
dim, of the array
isuppz
must be at least
.
On exit: the support of the eigenvectors in
z, i.e., the indices indicating the nonzero elements in
z. The
th eigenvector is nonzero only in elements
through
. Implemented only for
or
and
.
- 17:
– 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_ENUM_INT_2
-
On entry, , and .
Constraint: if , ;
otherwise .
- NE_ENUM_INT_3
-
On entry, , , and .
Constraint: if and , and ;
if 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.
An internal error has occurred in this function. Please refer to
fail in
nag_dstebz (f08jjc).
- 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. See Section 4.7 of
Anderson et al. (1999) for further details.
8
Parallelism and Performance
nag_dstevr (f08jdc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_dstevr (f08jdc) 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 proportional to if and is proportional to if and , otherwise the number of floating-point operations will depend upon the number of computed eigenvectors.
10
Example
This example finds the eigenvalues with indices in the range
, and the corresponding eigenvectors, of the symmetric tridiagonal matrix
10.1
Program Text
Program Text (f08jdce.c)
10.2
Program Data
Program Data (f08jdce.d)
10.3
Program Results
Program Results (f08jdce.r)