On Condition Numbers and the Distance to the Nearest Ill-Posed Problem

James W. Demmel

ISBN 10: 1334018197 / ISBN 13: 9781334018190

Verlag: Forgotten Books, 2024

Beschreibung

Print on Demand. This book explores the relationship between the condition number of a problem, a measure of how much its solution changes in response to small changes in its data, and the distance to the nearest ill-posed problem, a problem with an infinite condition number. The author shows that for many important problems, including matrix inversion, eigenvalue computation, and polynomial root finding, the condition number can be used to estimate how close a problem is to being ill-posed. This estimate can be used to determine how much effort should be put into solving a problem, and to guide the development of algorithms for solving ill-posed problems. This book is a reproduction of an important historical work, digitally reconstructed using state-of-the-art technology to preserve the original format. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in the book.

Bestandsnummer des Verkäufers 9781334018190_0

Bibliografische Details

• Titel: On Condition Numbers and the Distance to the Nearest Ill-Posed Problem
• Verlag: Forgotten Books
• Veröffentlichungsdatum: 2024
• Sprache: Englisch
• ISBN-10: 1334018197
• ISBN-13: 9781334018190
• Bindung: Paperback
• Zustand: New
• Buchtyp: print-on-demand item
• Verkäuferkataloge:
  • Calculus

Über diesen Titel

Inhaltsangabe

Excerpt from On Condition Numbers and the Distance to the Nearest Ill-Posed Problem

The condition number of a problem measures the sensitivity of the answer to small changes in the input. We call the problem ill-posed if its condition number is infinite. The ill-posed problems typically form a lower dimensional surface in the space of problems. It turns out that for many problems of numerical analysis, there is a simple relationship between the condition number of a problem and the shortest distance from that problem to the surface of ill-posed ones: the shortest distance is proportional to the reciprocal of the condition number, or bounded by the reciprocal of the condition number. Sometimes, the distance is bounded below by the reciprocal of the condition number squared. This is true for matrix inversion, computing eigenvalues and eigenvectors, finding zeros of polynomials.

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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.