非線性問題的牛頓法

內(nèi)容概要

本書論述科學(xué)和工程中的非線性問題的數(shù)值解法，包括有限維系統(tǒng)(代數(shù)系統(tǒng))和無(wú)限維系統(tǒng)(常微分方程和偏微分方程)。著重論述直接問題的局部和全局牛頓方法及反問題的高斯一牛頓方法。書中的“仿射不變性”是指所闡述的算法及其收斂性在四類仿射變換下是不變的。相對(duì)于傳統(tǒng)教科書而言，利用“仿射不變性”這一獨(dú)特方式來(lái)論述，令定理和證明更加簡(jiǎn)潔，使構(gòu)造完全自適應(yīng)算法成為可能。大量的數(shù)值示例、比較圖表及練習(xí)使得本書非常適合計(jì)算數(shù)學(xué)課程；同時(shí)，本書開辟了未來(lái)研究的許多可能方向。

書籍目錄

Outline of Contents1 Introduction  1.1 Newton-Raphson Method for Scalar Equations  1.2 Newton's Method for General Nonlinear Problems  1.3 A Roadmap of Newton-type Methods  1.4 Adaptive Inner Solvers for Inexact Newton Methods  ExercisesPart I ALGEBRAIC EQUATIONS  2 Systems of Equations: Local Newton Methods    2.1 Error Oriented Algorithms    2.2 Residual Based Algorithms    2.3 Convex Optimization  Exercises  3 Systmes of Equations: Global Newton Methods    3.1 Globalization Concepts    3.2 Residual Based Descent    3.3 Error Oriented Descent    3.4 Convex Functional Descent  Exercises  4 Least Squares Problems: Gauss-Newton Methods    4.1 Linear Least Squares Problems    4.2 Residual Based Algorithms    4.3 Error Oriented Algorithms    4.4 Underdetermined Systmes of Equations  Exercises  5 Parameter Dependent Systems: Continuation Methods    5.1 Newton Continuation Methods    5.2 Gauss-Newton Continuation Method    5.3 Computation of Simple bifurcations  ExercisesPart II DIFFERENTIAL EQUATIONS  6 Stiff ODE Initial Value Problems    6.1 Affine Similar Linear Contractivity    6.2 Nonstiff versus Stiff Initial Value Problems    6.3 Uniqueness Theorems for Implicit One-step Methods    6.4 Pseudo-transient Continuation for Steady State Problems  Exercises  7. ODE Boundary Value Problems    7.1 Multiple Shooting for Timelike BVPs    7.2 Parameter Identification in ODEs    7.3 Periodic Orbit Computation    7.4 Polynomial Collocation for Spacelike BVPs  Exercises  8 PDE Boundary Value Problems    8.1 Asymptotic Mesh Independence    8.2 global Discrete Newton Methods    8.3 Inexact Newton Multilevel FEM for Elliptic PDEs  ExercisesReferencesSoftwareIndex

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