The routine may be called by the names f08jlf, nagf_lapackeig_dstegr or its LAPACK name dstegr.
3Description
f08jlf computes selected eigenvalues and, optionally, the corresponding eigenvectors, of a real symmetric tridiagonal matrix . That is, the routine 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 routine 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 routines which must be called to perform the reduction to tridiagonal form and apply are:
This routine 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. f08jlf 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 routines in this chapter when all the eigenvectors are required, particularly so compared to those routines that are based on the algorithm.
4References
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 https://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
5Arguments
1: – Character(1)Input
On entry: indicates whether eigenvectors are computed.
Only eigenvalues are computed.
Eigenvalues and eigenvectors are computed.
Constraint:
or .
2: – Character(1)Input
On entry: indicates which eigenvalues should be returned.
All eigenvalues will be found.
All eigenvalues in the half-open interval will be found.
On entry: in earlier versions, this argument was the absolute error tolerance for the eigenvalues/eigenvectors. It is now deprecated, and only included for backwards-compatibility.
11: – IntegerOutput
On exit: the total number of eigenvalues found. .
If , .
If , .
12: – Real (Kind=nag_wp) arrayOutput
Note: the dimension of the array w
must be at least
.
On exit: the eigenvalues in ascending order.
13: – Real (Kind=nag_wp) arrayOutput
Note: the second dimension of the array z
must be at least
if , and at least otherwise.
On exit: if , then if , the columns of z contain the orthonormal eigenvectors of the matrix , with the th column of holding the eigenvector associated with .
Note: you must ensure that at least columns are supplied in the array z; if , the exact value of m is not known in advance and an upper bound of at least n must be used.
14: – IntegerInput
On entry: the first dimension of the array z as declared in the (sub)program from which f08jlf is called.
Constraints:
if , ;
otherwise .
15: – Integer arrayOutput
Note: the dimension 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 .
On entry: the dimension of the array work as declared in the (sub)program from which f08jlf is called.
If , a workspace query is assumed; the routine only calculates the minimum sizes of the work and iwork arrays, returns these values as the first entries of the work and iwork arrays, and no error message related to lwork or liwork is issued.
On entry: the dimension of the array iwork as declared in the (sub)program from which f08jlf is called.
If , a workspace query is assumed; the routine only calculates the minimum sizes of the work and iwork arrays, returns these values as the first entries of the work and iwork arrays, and no error message related to lwork or liwork is issued.
Constraint:
or .
20: – IntegerOutput
On exit: unless the routine detects an error (see Section 6).
6Error Indicators and Warnings
If , argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
Background information to multithreading can be found in the Multithreading documentation.
f08jlf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08jlf 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 routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
9Further Comments
The total number of floating-point operations required to compute all the eigenvalues and eigenvectors is approximately proportional to .