稀疏線性系統(tǒng)的迭代方法(第二版)

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?！　某霭娣矫鎭碇v，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?？茖W(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?，都是由該領(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?！　‘?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。　　總之，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

Iterative Methods for Sparse Linear Systems, Second Edition gives an in-depth, up-to-date view of practical algorithms for solving large-scale linear systems of equations. These equations can number in the millions and are sparse in the sense that each involves only a small number of unknowns. The methods described are iterative, i.e., they provide sequences of approximations that will converge to the solution.    This new edition includes a wide range of the best methods available today. The author has added a new chapter on multigrid techniques and has updated material throughout the text, particularly the chapters on sparse matrices, Krylov subspace methods, preconditioning techniques, and parallel preconditioners. Material on older topics has been removed or shortened, numerous exercises have been added, and many typographical errors have been corrected. The updated and expanded bibliography now includes more recent works emphasizing new and important research topics in this field.    This book can be used to teach graduate-level courses on iterative methods for linear systems. Engineers and mathematicians will find its contents easily accessible, and practitioners and educators will value it as a helpful resource. The preface includes syllabi that can be used for either a semester- or quarter-length course in both mathematics and computer science.

書籍目錄

Preface to the Second EditionPreface to the First Edition1  Background in Linear Algebra  1.1  Matrices  1.2  Square Matrices and Eigenvalues  1.3  Types of Matrices  1.4  Vector Inner Products and Norms  1.5  Matrix Norms  1.6  Subspaces, Range, and Kernel  1.7  Orthogonal Vectors and Subspaces  1.8  Canonical Forms of Matrices    1.8.1  Reduction to the Diagonal Form    1.8.2  The Jordan Canonical Form    1.8.3  The Schur Canonical Form    1.8.4  Application to Powers of Matrices  1.9  Normal and Hermitian Matrices     1.9.1  Normal Matrices    1.9.2  Hermitian Matrices  1.10  Nonnegative Matrices, M-Matrices  1.11  Positive Definite Matrices  1.12  Projection Operators    1.12.1  Range and Null Space of a Projector    1.12.2  Matrix Representations    1.12.3  Orthogonal and Oblique Projectors    1.12.4  Properties of Orthogonal Projectors  1.13  Basic Concepts in Linear Systems    1.13.1  Existence of a Solution    1.13.2  Perturbation Analysis  Exercises  Notes and References2  Discretization of Partial Differential Equations  2.1  Partial Differential Equations    2.1.1  Elliptic Operators    2.1.2  The Convection Diffusion Equation  2.2  Finite Difference Methods    2.2.1  Basic Approximations    2.2.2  Difference Schemes for the Laplacian Operator     2.2.3  Finite Differences for One-Dimensional Problems     2.2.4  Upwind Schemes    2.2.5  Finite Differences for Two-Dimensional Problems     2.2.6  Fast Poisson Solvers  2.3  The Finite Element Method  2.4  Mesh Generation and Refinement  2.5  Finite Volume Method  Exercises  Notes and References3  Sparse Matrices  3.1  Introduction  ……4  Basic Iterative Methods5  Projection Methods6  Krylov Subspace Methods, Part Ⅰ7  Krylov Subspace Methods, Part Ⅱ8  Methods Related to the Normal Equations9  Preconditioned Iterations10  Preconditioning Techniques11  Parallel Implementations12  Parallel Preconditioners13  Multigrid Methods14  Domain Decomposition MethodsBibliographyIndex

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　　《國(guó)外數(shù)學(xué)名著系列（續(xù)1）（影印版）39：稀疏線性系統(tǒng)的迭代方法（第2版）》由科學(xué)出版社出版。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。 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，《《國(guó)外數(shù)學(xué)名著系列（續(xù)1）（影印版）39：稀疏線性系統(tǒng)的迭代方法（第2版）》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，其中有些也具有交叉性質(zhì)。 這些書可以使讀者較快地了解數(shù)學(xué)某方面的前沿，對(duì)從事這方面研究的數(shù)學(xué)家了解該領(lǐng)域的前沿與全貌也很有幫助。

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