非线性问题的牛顿法
2006-1
科学出版社
D.彼得
424
519000
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本书论述科学和工程中的非线性问题的数值解法，包括有限维系统(代数系统)和无限维系统(常微分方程和偏微分方程)。着重论述直接问题的局部和全局牛顿方法及反问题的高斯一牛顿方法。书中的“仿射不变性”是指所阐述的算法及其收敛性在四类仿射变换下是不变的。相对于传统教科书而言，利用“仿射不变性”这一独特方式来论述，令定理和证明更加简洁，使构造完全自适应算法成为可能。大量的数值示例、比较图表及练习使得本书非常适合计算数学课程；同时，本书开辟了未来研究的许多可能方向。
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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