On the computation of probabilities and eigenvalues for random and non-random matrices

Can you imagine doing hundreds of millions of operations on non-integers and not obtaining a single round-off error? For n < 12, the algorithm used in this thesis does exactly that. We took advantage of a floating point property that we have not seen used before. If only we had quad precision we could have gone even further and extended the algorithm without round-off error for higher values of 'n'. The problem in question concerns whether the eigenvalues are real or complex. The eigenvalues of an n-by-n real random matrix whose elements are independent random variables with standard normal are examined. An exact expression to determine the probability Pn,k that exactly k eigenvalues are real are derived in [1]. This expression was used to compute the probabilities Pn,k, but the computation was achieved only up to n = 9. For higher values of n, the symbolic expressions generated during the course of an algorithm to compute an exact probability as expressed in Mathematica code requires large amounts of memory. In this thesis, we target development of a more efficient algorithm. The symbolic algorithm implemented in Mathematica is converted into an equivalent numerical version and is implemented using MATLAB. After implementing the serial code in MATLAB, the code is parallelized using a client-server parallel computing platform named Star-p. This modified code implementation along with superior hardware in terms of better processor speeds and larger memory, has enabled the probability evaluation for all values of k up to n= 11, and for certain k values for n = 12 and 13.

Department dc:contributor.department

Massachusetts Institute of Technology. Computation for Design and Optimization Program.

Grantor dc:publisher

Massachusetts Institute of Technology

Year dc:date.issued

2009

Author dc:creator

• Peruvamba Sundaresh, Vignesh

Advisor dc:contributor.advisor

• Alan Edelman.

• Computation for Design and Optimization Program.

Statement dc:rights

• M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission.

Licence dc:rights.uri

• http://dspace.mit.edu/handle/1721.1/7582

Language dc:language.iso

eng

Handle dc:identifier.uri

http://hdl.handle.net/1721.1/54840

OAI identifier oai:identifier

oai:dspace.mit.edu:1721.1/54840