The function may be called by the names: f08jyc, nag_lapackeig_zstegr or nag_zstegr.
f08jyc 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 eigenvectors computed.
The function stores the real orthogonal matrix in a complex array, so that it may also be used to compute selected eigenvalues and the corresponding eigenvectors of a complex Hermitian 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. f08jyc 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.
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
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 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
2: – Nag_JobTypeInput
On entry: indicates whether eigenvectors are computed.
Only eigenvalues are computed.
Eigenvalues and eigenvectors are computed.
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.
On entry: the stride separating row or column elements (depending on the value of order) in the array z.
if , ;
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 7 in the Introduction to the NAG Library CL Interface).
6Error Indicators and Warnings
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument had an illegal value.
Inverse iteration failed to converge.
The algorithm failed to converge.
On entry, , and .
Constraint: if , ;
On entry, , , and .
Constraint: if and , ;
if and , and .
On entry, , and .
Constraint: if , .
On entry, .
On entry, . Constraint: .
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 7.5 in the Introduction to the NAG Library CL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
Background information to multithreading can be found in the Multithreading documentation.
f08jyc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08jyc 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 .