An Introduction to the Quality of Computed Solutions Hammarling, Sven 2005 MIMS EPrint: 2005.29 Manchester Institute for Mathematical Sciences School of Mathematics The University of Manchester Reports available from: http://eprints.maths.manchester.ac.uk/ And by contacting: The MIMS Secretary School of Mathematics The University of Manchester Manchester, M13 9PL, UK ISSN 1749-9097 An Introduction to the Quality of Computed Solutions1 Sven Hammarling The Numerical Algorithms Group Ltd Wilkinson House Jordan Hill Road Oxford, OX2 8DR, UK
[email protected] December 5, 2005 1Based on [Hammarling, 2005], Chapter 4 of [Einarsson, 2005], which is available for pur- chase from SIAM at http://ec-securehost.com/SIAM/SE18.html. The royalties go to the SIAM Student Travel Fund. CONTENTS 1 Contents 1 Introduction 2 2 Floating Point Numbers and IEEE Arithmetic 2 3 Why Worry about Computed Solutions? 5 4 Condition, Stability and Error Analysis 9 4.1 Condition ....................................... 9 4.2 Stability . 14 4.3 ErrorAnalysis..................................... 19 5 Floating Point Error Analysis 23 6 Posing the Mathematical Problem 29 7 Error Bounds and Software 30 8 Other Approaches 34 9 Summary 34 Bibliography 35 1 INTRODUCTION 2 1 Introduction This report is concerned with the quality of the computed numerical solutions of mathematical problems. For example, suppose we wish to solve the system of linear equations Ax = b using a numerical software package. The package will return a computed solution, say x˜, and we wish to judge whether or not x˜ is a reasonable solution to the equations. Sadly, all too often software packages return poor, or even incorrect, numerical results and give the user no means by which to judge the quality of the numerical results.