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Title: Numerical methods for the unsymmetric tridiagonal eigenvalue problem

Abstract

This report summarizes the results of our project {open_quotes}Numerical Methods for the Unsymmetric Tridiagonal Eigenvalue Problem{close_quotes}. It was funded by both by a DOE grant (No. DE-FG02-92ER25122, 6/1/92-5/31/94, $100,000) and by an NSF Research Initiation Award (No. CCR-9109785, 7/1/91-6/30/93, $46,564.) The publications resulting from that project during the DOE funding period are listed below. Two other journal papers and two other conference papers were produced during the NSF funding period. Most of the listed conference papers are early or partial versions of the listed journal papers.

Authors:
Publication Date:
Research Org.:
Colorado Univ., Boulder, CO (United States). Dept. of Computer Science
Sponsoring Org.:
USDOE, Washington, DC (United States); National Science Foundation, Washington, DC (United States)
OSTI Identifier:
152659
Report Number(s):
DOE/ER/25122-T1
ON: DE96002721; CNN: Award CCR-9109785; TRN: 96:000032
DOE Contract Number:
FG02-92ER25122
Resource Type:
Technical Report
Resource Relation:
Other Information: PBD: [1996]
Country of Publication:
United States
Language:
English
Subject:
99 MATHEMATICS, COMPUTERS, INFORMATION SCIENCE, MANAGEMENT, LAW, MISCELLANEOUS; MATRICES; FACTORIZATION; EIGENVALUES; NUMERICAL SOLUTION; PARALLEL PROCESSING

Citation Formats

Jessup, E.R. Numerical methods for the unsymmetric tridiagonal eigenvalue problem. United States: N. p., 1996. Web. doi:10.2172/152659.
Jessup, E.R. Numerical methods for the unsymmetric tridiagonal eigenvalue problem. United States. doi:10.2172/152659.
Jessup, E.R. Tue . "Numerical methods for the unsymmetric tridiagonal eigenvalue problem". United States. doi:10.2172/152659. https://www.osti.gov/servlets/purl/152659.
@article{osti_152659,
title = {Numerical methods for the unsymmetric tridiagonal eigenvalue problem},
author = {Jessup, E.R.},
abstractNote = {This report summarizes the results of our project {open_quotes}Numerical Methods for the Unsymmetric Tridiagonal Eigenvalue Problem{close_quotes}. It was funded by both by a DOE grant (No. DE-FG02-92ER25122, 6/1/92-5/31/94, $100,000) and by an NSF Research Initiation Award (No. CCR-9109785, 7/1/91-6/30/93, $46,564.) The publications resulting from that project during the DOE funding period are listed below. Two other journal papers and two other conference papers were produced during the NSF funding period. Most of the listed conference papers are early or partial versions of the listed journal papers.},
doi = {10.2172/152659},
journal = {},
number = ,
volume = ,
place = {United States},
year = {Tue Dec 31 00:00:00 EST 1996},
month = {Tue Dec 31 00:00:00 EST 1996}
}

Technical Report:

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  • A multiprocessor algorithm for finding few or all eigenvalues and the corresponding eigenvectors of a symmetric tridiagonal matrix is presented. It is a pipelined variation of EISPACK routines - BISECT and TINVIT which consists of the three steps: isolation, extraction - inverse iteration, and partial orthogonalization. Multisections are performed for isolating the eigenvalues in a given interval, while bisection or the Zeroin method is used to extract these isolated eigenvalues. After the corresponding eigenvectors have been computed by inverse iteration, the modified Gram-Schmidt method is used to orthogonalize certain groups of these vectors. Experiments on the Alliant FX/8 and CRAYmore » X-MP/48 multiprocessors show that this algorithm achieves high speedup over BISECT and TINVIT; in fact it is much faster than TQL2 when all the eigenvalues and eigenvectors are required.« less
  • Two new numerical methods, the log derivative and the renormalized Numerov, are developed and applied to the calculation of bound-state solutions of the one-dimensional Schroedinger equation. They are efficient and stable; no convergence difficulties are encountered with double minimum potentials. A useful interpolation formula for calculating eigenfunctions at nongrid points is also derived. Results of example calculations are presented and discussed. (Author)
  • This project deals with the problem of solving nonlinear eigenvalue problems, namely, nonlinear systems with parameter dependence, primarily through the use of continuation methods. One of the major issues that we are interested in is in solving large and sparse problems. In particular, we put special emphasis on nonlinear elliptic eigenvalue problems, partly because these constitute an important class of applications, and partly because this class brings out some of the fundamental computational aspects of algorithms for solving general nonlinear eigenvalue problems. Specifically, we are interested in the nonlinear and linear algebraic techniques involved, which usually constitute the most timemore » consuming part of these algorithms. Another major objective is the accurate computation of singular points, which are often of great physical interest. A primary goal of our project is to develop a well-documented piece of mathematical software that incorporates results from our algorithmic studies (and others) and that can be used to trace solution curves of rather general, large and sparse nonlinear eigenvalue problems.« less
  • The solution of linear systems with a block-tridiagonal structure is a very common requirement in many applications. This report describes two Fortran packages for solving such systems. The first is for the case when all the relevant blocks in the coefficient matrix can be stored at once. The second is for the case when they cannot; instead the equations are generated, one block-row at a time, by a user-supplied subroutine. The blocks are assumed to be full, and partial pivoting is done within block-rows only.