实例介绍
【实例简介】
《逼近理论和方法》是一部本科生和研究生学习数值逼近的教程，将逼近理论经典结果和当前发展巧妙衔接起来。由于在计算机计算中许多数学函数不能直接被应用，然而它们可以被多项式和分段多项式这些易于处理的函数逼近。这个科目的一般理论以及在多项式逼近中的应用已经十分经典，但分段多项式在过去的二十年中得到了大力应用，并且发现了众多重要的理论性质，以及许多描述逼近精确度的技巧，书中全面透彻，系统地讲述了当前逼近方法的基础。
Approximation Theory and Methods M.J.D. POWELL John Humphrey Plummer Professor of Applied Numerical Analysis University of cambridge Approximation theory and methods EO CAMBRIDGe 动沙 UNIVERSITY PRESS 图书在版编目(CIP)数据 逼近理论和方法= Approximation theory and methods:英文/(英)鲍威尔 ( Powell. m.J.D.)著.一影印本.一北京:世界图书出版公司北京公司,2014.9 ISBN978-7-5100-8625-0 I.①逼…Ⅱ.①鲍…Ⅲ.①逼近论一教材一英文Ⅳ.①O174.41 中国版本图书馆CP数据核字(2014)第211071号 书名: Approximation Theory and Methods 作者:M.J.D. Powell 中译名:逼近理论和方法 责任编辑:高蓉刘慧 出版者:世界图书出版公司北京公司 印刷者:三河市国英印务有限公司 发行:世界图书出版公司北京公司(北京朝内大街137号100010) 联系电话:010-64021602,010-64015659 电子信箱:kjb@wpcbj.com.cn 开本:24开 印张:15 版次:2015年1月 版权登记:图字:01-2013-8236 书号:978-7-5100-8625-0 定价:49.00元 Approximation theory and methods Approximation Theory and Methods(978-0-521-29514-7) by M J.D Powell, first published by Cambridge University Press 2001 All rights reserved. This reprint edition for the People's Republic of China is published by arrangement with the Press Syndicate of the University of Cam- bridge, Cambridge, United Kingdom. o Cambridge University Press Beijing World Publishing Corpora tion 2014This book is in copyright. No reproduction of any part may take place without the written permission of Cambridge University Press or Beijing World Publishing Corporation. This edition is for sale in the mainland of China only, excluder Hong Kong SAR, Macao SAR and Taiwan, and may not be bought for export therefrom. 此版本仅限中华人民共和国境内销售,不包括香港、澳门特别 行政区及中国台湾。不得出口。 PREFACE There are several reasons for studying approximation theory and methods, ranging from a need to represent functions in computer cal- culations to an interest in the mathematics of the subject. Although approximation algorithms are used throughout the sciences and in many industrial and commercial fields, some of the theory has become highl specialized and abstract Work in numerical analysis and in mathematical software is one of the main links between these two extremes for its purpose is to provide computer users with efficient programs for general approximation calculations, in order that useful advances in the subject can be applied This book presents the view of a numerical analyst, who enjoys the theory, and who is keenly interested in its importance to practical computer calculations. It is based on a course of twenty-four lectures, given to third-year mathematics undergraduates at the uni- versity of Cambridge. There is really far too much material for such a course, but it is possible to speak coherently on each chapter for about one hour, and to include proofs of most of the main theorems. The pre requisites are an introduction to linear spaces and operators and an inter mediate course on analysis, but complex variable theory is not required Spline functions have transformed approximation techniques and theory during the last fifteen years. Not only are they convenient and suitable for computer calculations, but also they provide optimal theoretical solutions to the estimation of functions from limited data Therefore seven chapters are given to spline approximations. The classi- cal theory of best approximations from linear spaces with respect to the minimax, least squares and Li-norms is also studied and algorithms are described and analysed for the calculation of these approximations Interpolation is considered also, and the accuracy of interpolation and Preface X other linear operators is related to the accuracy of optimal algorithms Special attention is given to polynomial functions, and there is one chapter on rational functions but, due to the constraints of twenty-four lectures, the approximation of functions of several variables is not included. Also there are no computer listings, and little attention is given to the consequences of the rounding errors of computer arithmetic. All theorems are proved, and the reader will find that the subject provides a wide range of techniques of proof. Some material is included in order to demonstrate these techniques, for example the analysis of the con- vergence of the exchange algorithm for calculating the best minimax approximation to a continuous function. Several of the proofs are new. In particular, the uniform boundedness theorem is established in a way that does not require any ideas that are more advanced than cauchy sequences and completeness Less functional analysis is used than in other books on approximation theory, and normally functions are assumed to be continuous, in order to simplify the presentation Exercises are included with each chapter which support and extend the text. all references to related work are given in an appendix. It is a pleasure to acknowledge the excellent opportunities I have received for research and study in the department of Applied mathema tics and Theoretical Physics at the University of Cambridge since 1976, and before that at the atomic Energy research Establishment, Harwell My interest in approximation theory began at Harwell, stimulated by the enthusiasm of Alan Curtis, and strengthened by Pat Gaffney, who developed some of the theory that is reported in Chapter 24. I began to write this book in the summer of 1978 at the university of victoria Canada, and I am grateful for the facilities of their department of Mathematics, for the encouragement of lan Barrodale and frank Roberts, and for financial support from grants A5251 and A7143 of the National Research Council of Canada. At Cambridge David Carter of Kings College kindly studied drafts of the chapters and offered helpful comments. The manuscript was typed most expertly by judy roberts, Hazel Felton, Margaret Harrison and Paula Lister. I wish to express special thanks to Hazel for her assistance and patience when I was redrafting the text. My wife, Caroline, not only showed sympathetic understanding at home during the time when I worked long hours to complete the manuscript, but also she assisted with the figures. This work is dedicated to Caroline Pembroke College, Cambridge M.J.D. POWELL January 1980 CONTENTS Preface 1 The approximation problem and existence of best approximations 1. 1 Examples of approximation problems 1.2 Approximation in a metric space 1.3 Approximation in a normed linear space 1. 4 The L-norms 113569 1. 5 a geometric view of best approximations 2 The uniqueness of best approximations 2. 1 Convexity conditions 13 2.2 Conditions for the uniqueness of the best approximation 2.3 The continuity of best approximation operators 2.4 The 1-2-and oo-norms 17 3 Approximation operators and some approximating functions 22 3. 1 Approximation operators 22 3.2 Lebesgue constants 24 3.3 Polynomial approximations to differentiable functions 25 3. 4 Piecewise polynomial approximations 28 4 Polynomial interpolation 33 4. 1 The Lagrange interpolation formula 33 4.2 The error in polynomial interpolation 35 4.3 The Chebyshev interpolation points 37 4.4 The norm of the Lagrange interpolation operator 41 5 Divided difierences 46 5.1 Basic properties of divided differences 46 5.2 Newton's interpolation method 48 【实例截图】
【核心代码】
《逼近理论和方法》是一部本科生和研究生学习数值逼近的教程，将逼近理论经典结果和当前发展巧妙衔接起来。由于在计算机计算中许多数学函数不能直接被应用，然而它们可以被多项式和分段多项式这些易于处理的函数逼近。这个科目的一般理论以及在多项式逼近中的应用已经十分经典，但分段多项式在过去的二十年中得到了大力应用，并且发现了众多重要的理论性质，以及许多描述逼近精确度的技巧，书中全面透彻，系统地讲述了当前逼近方法的基础。
Approximation Theory and Methods M.J.D. POWELL John Humphrey Plummer Professor of Applied Numerical Analysis University of cambridge Approximation theory and methods EO CAMBRIDGe 动沙 UNIVERSITY PRESS 图书在版编目(CIP)数据 逼近理论和方法= Approximation theory and methods:英文/(英)鲍威尔 ( Powell. m.J.D.)著.一影印本.一北京:世界图书出版公司北京公司,2014.9 ISBN978-7-5100-8625-0 I.①逼…Ⅱ.①鲍…Ⅲ.①逼近论一教材一英文Ⅳ.①O174.41 中国版本图书馆CP数据核字(2014)第211071号 书名: Approximation Theory and Methods 作者:M.J.D. Powell 中译名:逼近理论和方法 责任编辑:高蓉刘慧 出版者:世界图书出版公司北京公司 印刷者:三河市国英印务有限公司 发行:世界图书出版公司北京公司(北京朝内大街137号100010) 联系电话:010-64021602,010-64015659 电子信箱:kjb@wpcbj.com.cn 开本:24开 印张:15 版次:2015年1月 版权登记:图字:01-2013-8236 书号:978-7-5100-8625-0 定价:49.00元 Approximation theory and methods Approximation Theory and Methods(978-0-521-29514-7) by M J.D Powell, first published by Cambridge University Press 2001 All rights reserved. This reprint edition for the People's Republic of China is published by arrangement with the Press Syndicate of the University of Cam- bridge, Cambridge, United Kingdom. o Cambridge University Press Beijing World Publishing Corpora tion 2014This book is in copyright. No reproduction of any part may take place without the written permission of Cambridge University Press or Beijing World Publishing Corporation. This edition is for sale in the mainland of China only, excluder Hong Kong SAR, Macao SAR and Taiwan, and may not be bought for export therefrom. 此版本仅限中华人民共和国境内销售,不包括香港、澳门特别 行政区及中国台湾。不得出口。 PREFACE There are several reasons for studying approximation theory and methods, ranging from a need to represent functions in computer cal- culations to an interest in the mathematics of the subject. Although approximation algorithms are used throughout the sciences and in many industrial and commercial fields, some of the theory has become highl specialized and abstract Work in numerical analysis and in mathematical software is one of the main links between these two extremes for its purpose is to provide computer users with efficient programs for general approximation calculations, in order that useful advances in the subject can be applied This book presents the view of a numerical analyst, who enjoys the theory, and who is keenly interested in its importance to practical computer calculations. It is based on a course of twenty-four lectures, given to third-year mathematics undergraduates at the uni- versity of Cambridge. There is really far too much material for such a course, but it is possible to speak coherently on each chapter for about one hour, and to include proofs of most of the main theorems. The pre requisites are an introduction to linear spaces and operators and an inter mediate course on analysis, but complex variable theory is not required Spline functions have transformed approximation techniques and theory during the last fifteen years. Not only are they convenient and suitable for computer calculations, but also they provide optimal theoretical solutions to the estimation of functions from limited data Therefore seven chapters are given to spline approximations. The classi- cal theory of best approximations from linear spaces with respect to the minimax, least squares and Li-norms is also studied and algorithms are described and analysed for the calculation of these approximations Interpolation is considered also, and the accuracy of interpolation and Preface X other linear operators is related to the accuracy of optimal algorithms Special attention is given to polynomial functions, and there is one chapter on rational functions but, due to the constraints of twenty-four lectures, the approximation of functions of several variables is not included. Also there are no computer listings, and little attention is given to the consequences of the rounding errors of computer arithmetic. All theorems are proved, and the reader will find that the subject provides a wide range of techniques of proof. Some material is included in order to demonstrate these techniques, for example the analysis of the con- vergence of the exchange algorithm for calculating the best minimax approximation to a continuous function. Several of the proofs are new. In particular, the uniform boundedness theorem is established in a way that does not require any ideas that are more advanced than cauchy sequences and completeness Less functional analysis is used than in other books on approximation theory, and normally functions are assumed to be continuous, in order to simplify the presentation Exercises are included with each chapter which support and extend the text. all references to related work are given in an appendix. It is a pleasure to acknowledge the excellent opportunities I have received for research and study in the department of Applied mathema tics and Theoretical Physics at the University of Cambridge since 1976, and before that at the atomic Energy research Establishment, Harwell My interest in approximation theory began at Harwell, stimulated by the enthusiasm of Alan Curtis, and strengthened by Pat Gaffney, who developed some of the theory that is reported in Chapter 24. I began to write this book in the summer of 1978 at the university of victoria Canada, and I am grateful for the facilities of their department of Mathematics, for the encouragement of lan Barrodale and frank Roberts, and for financial support from grants A5251 and A7143 of the National Research Council of Canada. At Cambridge David Carter of Kings College kindly studied drafts of the chapters and offered helpful comments. The manuscript was typed most expertly by judy roberts, Hazel Felton, Margaret Harrison and Paula Lister. I wish to express special thanks to Hazel for her assistance and patience when I was redrafting the text. My wife, Caroline, not only showed sympathetic understanding at home during the time when I worked long hours to complete the manuscript, but also she assisted with the figures. This work is dedicated to Caroline Pembroke College, Cambridge M.J.D. POWELL January 1980 CONTENTS Preface 1 The approximation problem and existence of best approximations 1. 1 Examples of approximation problems 1.2 Approximation in a metric space 1.3 Approximation in a normed linear space 1. 4 The L-norms 113569 1. 5 a geometric view of best approximations 2 The uniqueness of best approximations 2. 1 Convexity conditions 13 2.2 Conditions for the uniqueness of the best approximation 2.3 The continuity of best approximation operators 2.4 The 1-2-and oo-norms 17 3 Approximation operators and some approximating functions 22 3. 1 Approximation operators 22 3.2 Lebesgue constants 24 3.3 Polynomial approximations to differentiable functions 25 3. 4 Piecewise polynomial approximations 28 4 Polynomial interpolation 33 4. 1 The Lagrange interpolation formula 33 4.2 The error in polynomial interpolation 35 4.3 The Chebyshev interpolation points 37 4.4 The norm of the Lagrange interpolation operator 41 5 Divided difierences 46 5.1 Basic properties of divided differences 46 5.2 Newton's interpolation method 48 【实例截图】
【核心代码】
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