無約束最優(yōu)化與非線性方程的數(shù)值方法

前言

要使我國(guó)的數(shù)學(xué)事業(yè)更好地發(fā)展起來，需要數(shù)學(xué)家淡泊名利并付出更艱苦地努力。另一方面，我們也要從客觀上為數(shù)學(xué)家創(chuàng)造更有利的發(fā)展數(shù)學(xué)事業(yè)的外部環(huán)境，這主要是加強(qiáng)對(duì)數(shù)學(xué)事業(yè)的支持與投資力度，使數(shù)學(xué)家有較好的工作與生活條件，其中也包括改善與加強(qiáng)數(shù)學(xué)的出版工作。從出版方面來講，除了較好較快地出版我們自己的成果外，引進(jìn)國(guó)外的先進(jìn)出版物無疑也是十分重要與必不可少的。從數(shù)學(xué)來說，施普林格（springer）出版社至今仍然是世界上最具權(quán)威的出版社。科學(xué)出版社影印一批他們出版的好的新書，使我國(guó)廣大數(shù)學(xué)家能以較低的價(jià)格購(gòu)買，特別是在邊遠(yuǎn)地區(qū)工作的數(shù)學(xué)家能普遍見到這些書，無疑是對(duì)推動(dòng)我國(guó)數(shù)學(xué)的科研與教學(xué)十分有益的事。這次科學(xué)出版社購(gòu)買了版權(quán)，一次影印了23本施普林格出版社出版的數(shù)學(xué)書，就是一件好事，也是值得繼續(xù)做下去的事情。大體上分一下，這23本書中，包括基礎(chǔ)數(shù)學(xué)書5本，應(yīng)用數(shù)學(xué)書6本與計(jì)算數(shù)學(xué)書12本，其中有些書也具有交叉性質(zhì)。這些書都是很新的，2000年以后出版的占絕大部分，共計(jì)16本，其余的也是1990年以后出版的。這些書可以使讀者較快地了解數(shù)學(xué)某方面的前沿，例如基礎(chǔ)數(shù)學(xué)中的數(shù)論、代數(shù)與拓?fù)淙荆际怯稍擃I(lǐng)域大數(shù)學(xué)家編著的“數(shù)學(xué)百科全書”的分冊(cè)。對(duì)從事這方面研究的數(shù)學(xué)家了解該領(lǐng)域的前沿與全貌很有幫助。按照學(xué)科的特點(diǎn)，基礎(chǔ)數(shù)學(xué)類的書以“經(jīng)典”為主，應(yīng)用和計(jì)算數(shù)學(xué)類的書以“前沿”為主。這些書的作者多數(shù)是國(guó)際知名的大數(shù)學(xué)家，例如《拓?fù)鋵W(xué)》一書的作者諾維科夫是俄羅斯科學(xué)院的院士，曾獲“菲爾茲獎(jiǎng)”和“沃爾夫數(shù)學(xué)獎(jiǎng)”。這些大數(shù)學(xué)家的著作無疑將會(huì)對(duì)我國(guó)的科研人員起到非常好的指導(dǎo)作用。當(dāng)然，23本書只能涵蓋數(shù)學(xué)的一部分，所以，這項(xiàng)工作還應(yīng)該繼續(xù)做下去。更進(jìn)一步，有些讀者面較廣的好書還應(yīng)該翻譯成中文出版，使之有更大的讀者群?？傊?，我對(duì)科學(xué)出版社影印施普林格出版社的部分?jǐn)?shù)學(xué)著作這一舉措表示熱烈的支持，并盼望這一工作取得更大的成績(jī)。

內(nèi)容概要

This book is a standard for a complete description of the methods for unconstrained optimization and the solution ofnonlinear equations....this republication is most welcome and this volume should be in every library. Of course, there exist more recent books on the topics and somebody interested in the subject cannot be satiated by looking only at this book. However, it contains much quite-well-presented material and I recommend reading it before going ,to other.publications.

作者簡(jiǎn)介

作者：(美國(guó))丹尼斯 (J.E.Dennis Jr.) (美國(guó))Robert B.Schnabel

書籍目錄

PREFACE TO THE CLASSICS EDITION     PREFACE    1   INTRODUCTION    1.1 Problems to be considered   1.2 Characteristics of"real-world" problems   1.3 Finite-precision arithmetic and measurement of error 　1.4 Exercises 2   NONLINEAR PROBLEMS IN ONE VARIABLE        　2.1 What is not possible 　2.2 Newton's method for solving one equation in one unknown 　2.3 Convergence of sequences of real numbers 　2.4 Convergence of Newton's method 　2.5 Globally convergent methods for solving one equation in one unknown 　2.6 Methods when derivatives are unavailable 　2.7 Minimization of a function of one variable 　2.8 Exercises 3   NUMERICAL LINEAR  ALGEBRA BACKGROUND     　3.1 Vector and matrix norms and orthogonality 　3.2 Solving systems of linear equations--matrix factorizations 　3.3 Errors in solving linear systems 　3.4 Updating matrix factorizations 　3.5 Eigenvalues and positive definiteness 　3.6 Linear least squares 　3.7 Exercises 4   MULTIVARIABLE  CALCULUS  BACKGROUND         　4.1 Derivatives and multivariable models 　4.2 Multivariable finite-difference derivatives 　4.3 Necessary and sufficient conditions for unconstrained minimization 　4.4 Exercises 835   NEWTON'S METHOD FOR NONLINEAR EQUATIONS AND UNCONSTRAINED MINIMIZATION     　5.1 Newton's method for systems of nonlinear equations 　5.2 Local convergence of Newton's method 　5.3 The Kantorovich and contractive mapping theorems 　5.4 Finite-difference derivative methods for systems of nonlinear equations 　5.5 Newton's method for unconstrained minimization 　5.6 Finite-difference derivative methods for unconstrained minimization 　5.7 Exercises 6   GLOBALLY  CONVERGENT  MODIFICATIONS OF NEWTON'S METHOD               　6.1 The quasi-Newton framework 　6.2 Descent directions 　6.3 Line searches     6.3.1 Convergence results for properly chosen steps      6.3.2 Step selection by backtracking 　6.4 The model-trust region approach     6.4.1 The locally constrained optimal ("hook") step     6.4.2 The double dogleg step     6.4.3 Updating the trust region 　6.5 Global methods for systems of nonlinear equations 　6.6 Exercises 7   STOPPING,  SCALING,  AND  TESTING          　7.1 Scaling 　7.2 Stopping criteria 　7.3 Testing 　　7.4 Exercises 8   SECANT  METHODS  FOR  SYSTEMS OF NONLINEAR EQUATIONS        　8.1 Broyden's method 　8.2 Local convergence analysis of Broyden's method 　8.3 Implementation of quasi-Newton algorithms using Broyden's update 　8.4 Other secant updates for nonlinear equations 　8.5 Exercises 9   SECANT METHODS FOR UNCONSTRAINED MINIMIZATION     　9.1 The symmetric secant update of Powell 　9.2 Symmetric positive definite secant updates 　9.3 Local convergence of positive definite secant methods 　9.4 Implementation of quasi-Newton algorithms using the positive definite secant update 　9.5 Another convergence result for the positive definite secant method 　9.6 Other secant updates for unconstrained minimization 　9.7 Exercises 10   NONLINEAR  LEAST  SQUARES         　10.1 The nonlinear least-squares problem 　10.2 Gauss-Newton-type methods 　10.3 Full Newton-type methods 　10.4 Other considerations in solving nonlinear least-squares problems 　10.5 Exercises 11   METHODS FOR PROBLEMS WITH SPECIAL STRUCTURE     　11.1 The sparse finite-difference Newton method 　11.2 Sparse secant methods 　11.3 Deriving least-change secant updates 　11.4 Analyzing least-change secant methods 　11.5 Exercises A  APPENDIX: A MODULAR SYSTEM OF ALGORITHMS FOR UNCONSTRAINED MINIMIZATION AND NONLINEAR EQUATIONS  (by Robert Schnabel)B   APPENDIX: TEST PROBLEMS  (by Robert SchnabeI)  REFERENCES       AUTHOR INDEX      SUBJECT INDEX

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