The Probability That a Numerical, Analysis Problem Is Difficult (Classic Reprint) - Softcover

Demmel, James W.

Softcover

ISBN 10: 1330257162 ISBN 13: 9781330257166

Verlag: Forgotten Books, 2018

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Excerpt from The Probability That a Numerical, Analysis Problem Is Difficult

The main justification for using a uniform distribution is that it appears to be fair: each problem is as likely as any other. However, it does not appear to apply in many practical cases for a variety of reasons, including the fact that any set of problems which can be represented in a computer is necessarily discrete rather than continuous. We will discuss the validity of our choice of uniform distribution as well as alternatives at length in section 6 below.

Finally, given this distribution, we must compute the induced probability distribution of the condition number. It turns out that all the problems we consider here have a common geometric structure which lets us compute the distributions of their condition numbers with a single analysis, which goes as follows.

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Excerpt from The Probability That a Numerical, Analysis Problem Is Difficult

Numerous problems in numerical analysis, including matrix inversion, eigenvalue calculations and polynomial zero finding, share the following property: the difficulty of solving a given problem is large when the distance from that problem to the nearest "ill-posed" one is small. For example, the closer a matrix is to the set of noninvertible matrices, the larger its condition number with respect to inversion. We show that the sets of ill-posed problems for matrix inversion, eigenproblems, and polynomial zero finding all have a common algebraic and geometric structure which lets us compute the probability distribution of the distance from a "random" problem to the set. From this probability distribution we derive, for example, the distribution of the condition number of a random matrix. We examine the relevance of this theory to the analysis and construction of numerical algorithms destined to be run in finite precision arithmetic.

To investigate the probability that a numerical analysis problem is difficult, we need to do three things:
1) Choose a measure of difficulty,
2) Choose a probability distribution on the set of problems,
3) Compute the distribution of the measure of difficulty induced by the distribution on the set of problems.

The measure of difficulty we shall use in this paper is the condition number, which measures the sensitivity of the solution to small changes in the problem. For the problems we consider in this paper (matrix inversion, polynomial zero finding and eigenvalue calculation), there are well known condition numbers in the literature of which we shall use slightly modified versions to be discussed more fully later. The condition number is an appropriate measure of difficulty because it can be used to measure the expected loss of accuracy in the computed solution, or even the number of iterations required for an iterative algorithm to converge to a solution.

The probability distribution on the set of problems for which we will attain most of our results will be the "uniform distribution" which we define as follows.

About the Publisher

Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com

This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works.

Details zur Ausgabe

• Verlag: Forgotten Books
• Erscheinungsdatum: 2018
• ISBN 10: 1330257162
• ISBN 13: 9781330257166
• Einband: Tapa blanda
• Sprache: Englisch
• Anzahl der Seiten: 40
• Kontakt zum Hersteller: Nicht verfügbar