The Symmetric Tridiagonal Eigenproblem has been the topic of some recent work. Many methods have been advanced for the computation of the eigenvalues of such a matrix. In this paper, we present a divide-and-conquer approach to the computation of the eigenvalues of a symmetric tridiagonal matrix via the evaluation of the characteristic polynomial. The problem of evaluation of the characteristic polynomial is partitioned into smaller parts which are solved and these solutions are then combined to form the solution to the original problem. We give the update equations for the characteristic polynomial and certain auxiliary polynomials used in the computation. Furthermore, this set of recursions can be implemented on a regulartree structure. If the concurrency exhibited by this algorithm is exploited, it can be shown that thetime for computation of all the eigenvalues becomesO(nlogn) instead ofO(n2) as is the case for the approach where the order is increased by only one at every step. We address the numerical problems associated with the use of the characteristic polynomial and present a numerically stable technique for the eigenvalue computation.

Eigenvalues of a symmetric tridiagonal matrix: A divide-and-conquer approach

This paper presents a short survey of recent work on parallel implementations of Numerical Linear Algebra algorithms with emphasis on those relating to the solution of the symmetric eigenvalue problem on loosely coupled multiprocessor architectures. The vital operations in the formulation of most eigenvalue algorithms are matrix vector multiplication, matrix transposition, and linear system solution. Their implementations on several representative multiprocessor systems will be described, as well as parallel implementations of the following classes of eigenvalue methods : QR, bisection, divide-and-conquer, and Lanczos algorithm.

The Impact of Parallel Architectures on The Solution of Eigenvalue Problems

In this paper, we present a divide-and-conquer approach to the evaluation of the characteristic polynomial of a symmetric tridiagonal matrix for a real argument. Here, the problem is partitioned into smaller parts which are solved and these solutions are then combined to form the solution to the original problem. We give the update equations for the characteristic polynomial and certain auxiliary polynomials used in the computation. We show that the three-term recurrence is a specific case of our equations. Furthermore, this set of recursions can be implemented on a regular tree structure. If the concurrency exhibited by order is increased by one at every step.

A divide-and-conquer approach to the evaluation of the characteristic polynomial of a symmetric tridiagonal matrix

The author describes an algorithm for computing the characteristic polynomial of a tridiagonal matrix. It is quite general and consists in the application of the divide-and-conquer technique to the evaluation of a three-term recurrence relation. The algorithm developed requires O(n log/sup 2/n) arithmetic operations as compared to the classical algorithm that requires O(n/sup 2/) arithmetic operations. >

https://ieeexplore.ieee.org/iel1/5/585/00016343.pdf

An asymptotically superior algorithm for computing the characteristic polynomial of a tridiagonal matrix

The eigenvalues of a symmetric tridiagonal matrix can be computed by bisection using Sturm sequences and their corresponding eigenvectors found by inverse iteration. The independent computing tasks comprising this combination are especially appropriate for implementation on a distributed memory multiprocessor, but the parallel efficiency of bisection and inverse iteration is not enough to recommend their use in general. Bisection produces eigenvalues to high accuracy. Inverse iteration results in eigenvectors with quality dependent on the spacing of the eigenvalues and on the starting vector. This paper examines the factors influencing the accuracy of inverse iteration. The effects of random starting vectors are discussed, and some uses of statistical analysis in the design of an inverse iteration algorithm are presented.

A Statistical Evaluation of Inverse Iteration

In this paper we present a tree architecture foreigendecomposition based on a Divide -and-Conquer approach. A comparison of this method with the previously proposed systolic array architecture is made. Strategies for serial -parallel computation and fault -tolerantdesign are indicated. The suitability of this architecture for VLSI implementation is pointed out. 1. Introduction 1.1. Problem Description Many signal and image processing algo-rithms require the eigendecomposition of large matrices for efficient application [1,2,3]. Often, these problems have to be solved in a distri- buted fashion, e.g. in a Distributed Sensor Net- work involving the combination of sub -spaces. Another notable example is the so- called 'super- radar'. There have been many solutions to the eigendecomposition problem. We will be concerned mainly with the symmetric eigen- problem. The quantities of interest are the eigenvalues (all or some of them) and the corresponding eigenvectors. The specializationto symmetric matrices is not overly restrictive,

A tree architecture for the symmetric eigenproblem

Handle everyday research tasks with reliable, citation-backed results

Your personal Research Agent to handle research tasks with citation-backed results

Popular Tasks used by Researchers

How can I help with your research?

Meet SciSpace

Get more enhanced response by uploading the PDFs you want me to reference.

No relevant PDFs in your library

SciSpace is the AI research assistant for academics. Run systematic literature reviews on 280M+ papers, and write papers with cited sources. Trusted by 1M+ students, PhDs & researchers.

SciSpace | AI for Research

Analyze PDFs

Code & Manuscripts

Funding & Grants

Literature & Patents

Medical & Clinical Data

Systematic Review

Visualize & Present

Web & Data

Build a Google Scholar-like website for your research.

Build a website

Create charts and images for your research

Create a Chart

Write a paper for submission to a journal

Draft a manuscript

Patent Search

Design eye-catching scientific posters in minutes.

Scientific Poster Generation

Systematic Literature Review

One task is running at the moment. Your messages will be shown right after.

Drag and drop or click here to browse

Loved by <highlight>1 million+</highlight> researchers

Extract a list of specific topics and their sources from unstructured text

Topics

Compare and analyze relevant papers that matches with your search

Papers

Get insights from PDFs and bookmarked papers from your library

My library

Recent searches

Try searching for:

Catch AI-generated content in scholarly and non-scholarly content

{ai} Detector

Ai Writer

Get PDF Summaries, highlighted text explanations 

Chat with PDF

Effortlessly create in-text citations and bibliographies in APA and 2,500 other formats

Citation generator

Get explanations, summaries, and answers on academic papers

Ease up your research workflow with {scispace}'s cohort of exciting AI tools

Elevate your academic writing skills and convey your ideas the way you want

Paraphraser

Explore our range of reading and writing tools

Your file is being prepared and should be ready in a few minutes. If it's a large file, it might take a bit longer. You can close this window, and we'll email you the file when it's done.

You have reached a maximum limit of <strong>{limit}</strong> columns in the table. Remove at least <strong>1</strong> column to add or create another one.

A. S. Krishnakumar

Author Tools

Chat about Author

Papers

A tree architecture for the symmetric eigenproblem

A divide-and-conquer approach to the evaluation of the characteristic polynomial of a symmetric tridiagonal matrix