An Introduction to Numerical Methods and Analysis (Revised Edition)(Hardcover)
暫譯: 數值方法與分析導論（修訂版）（精裝本）

James F. Epperson

An Introduction to Numerical Methods and Analysis (Revised Edition)(Hardcover)
  • 出版商: Wiley
  • 出版日期: 2007-09-17
  • 售價: $5,370
  • 貴賓價: 9.5 折 $5,102
  • 語言: 英文
  • 頁數: 590
  • 裝訂: Hardcover
  • ISBN: 0470049634
  • ISBN-13: 9780470049631
  • 相關分類: 數值分析 Numerical-analysis、程式語言

    • 海外代購書籍(需單獨結帳)

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The objective of this book is for the reader to learn where approximation methods come from, why they work, why they sometimes don't work, and when to use which of many techniques that are available, and to do all this in a way that emphasizes readability and usefulness to the numerical methods novice. Each chapter and each section begins with the basic, elementary material and gradually builds up to more advanced topics. The text begins with a review of the important calculus results, and why and where these ideas play an important role throughout the book. Some of the concepts required for the study of computational mathematics are introduced, and simple approximations using Taylor's Theorem are treated in some depth. The exposition is intended to be lively and "student friendly". Exercises run the gamut from simple hand computations that might be characterized as "starter exercises", to challenging derivations and minor proofs, to programming exercises.

Table of Contents

Preface.
1. Introductory Concepts and Calculus Review.

1.1 Basic Tools of Calculus.

1.1.1 Taylor's Theorem.

1.1.2 Mean Value and Extreme Value Theorems.

1.2 Error, Approximate Equality, and Asymptotic Order Notation.

1.2.1 Error.

1.2.2 Notation: Approximate Equality.

1.2.3 Notation: Asymptotic Order.

1.3 A Primer on Computer Arithmetic.

1.4 A Word on Computer Languages and Software.

1.5 Simple Approximations.

1.6 Application: Approximating the Natural Logarithm.

References.

2. A Survey of Simple Methods and Tools.

2.1 Horner’s Rule and Nested Multiplication.

2.2 Difference Approximations to the Derivative.

2.3 Application: Euler’s Method for Initial Value Problems.

2.4 Linear Interpolation.

2.5 Application - The Trapezoid Rule.

2.6 Solution of Tridiagonal Linear Systems.

2.7 Application: Simple Two-Point Boundary Value Problems.

3. Root-Finding.

3.1 The Bisection Method.

3.2 Newton's Method: Derivation and Examples.

3.3 How to Stop Newton’s Method.

3.4 Application: Division Using Newton’s Method.

3.5 The Newton Error Formula.

3.6 Newton's Method: Theory and Convergence.

3.7 Application: Computation of the Square Root.

3.8 The Secant Method: Derivation and Examples.

3.9 Fixed Point Iteration.

3.10 Special Topics in Root-finding Methods.

3.10.1 Extrapolation and Acceleration.

3.10.2 Variants of Newton's Method.

3.10.3 The Secant Method: Theory and Convergence.

3.10.4 Multiple Roots.

3.10.5 In Search of Fast Global Convergence: Hybrid Algorithms.

3.11 Literature and Software Discussion 156.

References.

4. Interpolation and Approximation.

4.1 Lagrange Interpolation.

4.2 Interpolation and Divided Differences.

4.3 Interpolation Error.

4.4 Application: Muller’s Method and Inverse Quadratic Interpolation.

4.5 Application: More Approximations to the Derivative.

4.6 Hermite Interpolation.

4.7 Piecewise Polynomial Interpolation.

4.8 An Introduction to Splines.

4.8.1 Definition of the Problem.

4.8.2 Cubic B-Splines.

4.9 Application: Solution of Boundary Value Problems.

4.10 Least Squares Concepts in Approximation.

4.10.1 An Introduction to Data Fitting.

4.10.2 Least Squares Approximation and Orthogonal Polynomials.

4.11 Advanced Topics in Interpolation Error.

4.11.1 Stability of Polynomial Interpolation.

4.11.2 The Runge Example.

4.11.3 The Chebyshev nodes.

4.12 Literature and Software Discussion.

References.

5. Numerical Integration.

5.1 A Review of the Definite Integral.

5.2 Improving the Trapezoid Rule.

5.3 Simpson’s Rule and Degree of Precision.

5.4 The Midpoint Rule.

5.5 Application: Stirling's Formula.

5.6 Gaussian Quadrature.

5.7 Extrapolation Methods.

5.8 Special Topics in Numerical Integration.

5.8.1 Romberg Integration.

5.8.2 Quadrature with Non-Smooth Integrands.

5.8.3 Adaptive Integration.

5.8.4 Peano Estimates for the Trapezoid Rule.

5.9 Literature and Software Discussion.

References.

6. Numerical Methods for Ordinary Differential Equations.

6.1 The Initial Value Problem - Background.

6.2 Euler’s Method.

6.3 Analysis of Euler’s Method.

6.4 Variants of Euler’s Method.

6.4.1 The Residual and Truncation Error.

6.4.2 Implicit Methods and Predictor-Corrector Schemes.

6.4.3 Starting Values and Multistep Methods.

6.4.4 The Midpoint Method and Weak Stability.

6.5 Single Step Methods? Runge-Kutta.

6.6 Multi-step Methods.

6.6.1 The Adams Families.

6.6.2 The BDF Family.

6.7 Stability Issues.

6.7.1 Stability Theory for Multistep Methods.

6.7.2 Stability Regions.

6.8 Application to Systems of Equations.

6.8.1 Implementation Issues and Examples.

6.8.2 Stiff Equations.

6.8.3 A-Stability.

6.9 Adaptive Solvers.

6.10 Boundary Value Problems.

6.10.1 Simple Difference Methods.

6.10.2 Shooting Methods.

6.11 Literature and Software Discussion.

References.

7. Numerical Methods for the Solution of Systems of Equations.

7.1 Linear Algebra Review.

7.2 Linear Systems and Gaussian Elimination.

7.3 Operation Counts.

7.4 The LU Factorization.

7.5 Perturbation, Conditioning and Stability.

7.5.1 Vector and Matrix Norms.

7.5.2 The Condition Number and Perturbations.

7.5.3 Estimating the Condition Number.

7.5.4 Iterative Refinement.

7.6 SPD Matrices and the Cholesky Decomposition.

7.7 Iterative Methods for Linear Systems - A Brief Survey.

7.8 Nonlinear Systems: Newton's Method and Related Ideas.

7.8.1 Newton's Method.

7.8.2 Fixed Point Methods.

7.9 Application: Numerical Solution of Nonlinear BVP’s.

7.10 Literature and Software Discussion.

References.

8. Approximate Solution of the Algebraic Eigenvalue Problem.

8.1 Eigenvalue Review.

8.2 Reduction to Hessenberg Form.

8.3 Power Methods.

8.4 An Overview of the QR Iteration.

8.5 Literature and Software Discussion.

References.

9. A Survey of Finite Difference Methods for Partial Differential Equations.

9.1 Difference Methods for the Diffusion Equation.

9.1.1 The Basic Problem.

9.1.2 The Explicit Method and Stability.

9.1.3 Implicit Methods and the Crank-Nicolson Method.

9.2 Difference Methods for Poisson Equations.

9.2.1 Discretization.

9.2.2 Banded Cholesky Solvers.

9.2.3 Iteration and the Method of Conjugate Gradients.

9.3 Literature and Software Discussion.

References.

Appendix A: Proofs of Selected Theorems, and Other Additional Material.

A.1 Proofs of the Interpolation Error Theorems.

A.2 Proof of Stability.

A.3 Stiff Systems of Differential Equations and Eigenvalues.

A.4 The Matrix Perturbation Theorem.

Index.

商品描述(中文翻譯)

描述

本書的目標是讓讀者了解近似方法的來源、為什麼它們有效、為什麼有時無效，以及在眾多可用技術中何時使用哪一種，並以強調可讀性和對數值方法初學者的實用性為重。每一章和每一節都從基本的、初步的材料開始，逐漸深入到更高級的主題。文本首先回顧重要的微積分結果，以及這些概念在整本書中扮演的重要角色。介紹了一些計算數學研究所需的概念，並對使用泰勒定理的簡單近似進行了深入探討。該書的表述旨在生動且「對學生友好」。練習題涵蓋從簡單的手動計算（可視為「入門練習」）到具有挑戰性的推導和小型證明，再到程式設計練習。

目錄

前言。
1. 介紹概念與微積分回顧。

1.1 微積分的基本工具。

1.1.1 泰勒定理。

1.1.2 平均值定理與極值定理。

1.2 誤差、近似相等與漸近階數表示法。

1.2.1 誤差。

1.2.2 表示法：近似相等。

1.2.3 表示法：漸近階數。

1.3 電腦算術入門。

1.4 關於電腦語言與軟體的說明。

1.5 簡單的近似。

1.6 應用：近似自然對數。

參考文獻。

2. 簡單方法與工具的調查。

2.1 霍納法則與巢狀乘法。

2.2 對導數的差分近似。

2.3 應用：初值問題的歐拉法。

2.4 線性插值。

2.5 應用 - 梯形法則。

2.6 三對角線性系統的解。

2.7 應用：簡單的兩點邊值問題。

3. 根尋找。

3.1 二分法。

3.2 牛頓法：推導與範例。

3.3 如何停止牛頓法。

3.4 應用：使用牛頓法進行除法。

3.5 牛頓誤差公式。

3.6 牛頓法：理論與收斂性。

3.7 應用：平方根的計算。

3.8 割線法：推導與範例。

3.9 不動點迭代。

3.10 根尋找方法中的特殊主題。

3.10.1 外推與加速。

3.10.2 牛頓法的變體。

3.10.3 割線法：理論與收斂性。

3.10.4 多重根。

3.10.5 尋找快速全局收斂：混合算法。

3.11 文獻與軟體討論。

參考文獻。

4. 插值與近似。

4.1 拉格朗日插值。

4.2 插值與劃分差。

4.3 插值誤差。

4.4 應用：穆勒法與反二次插值。

4.5 應用：導數的更多近似。

4.6 赫米特插值。

4.7 分段多項式插值。

4.8 介紹樣條。

4.8.1 問題的定義。

4.8.2 三次B樣條。

4.9 應用：邊值問題的解。

4.10 近似中的最小二乘概念。

4.10.1 數據擬合入門。

4.10.2 最小二乘近似與正交多項式。

4.11 插值誤差中的高級主題。

4.11.1 多項式插值的穩定性。

4.11.2 隆格例子。

4.11.3 切比雪夫節點。

4.12 文獻與軟體討論。

參考文獻。

5. 數值積分。

5.1 定積分回顧。

5.2 改進梯形法則。

5.3 辛普森法則與精度等級。

5.4 中點法則。

5.5 應用：斯特林公式。

5.6 高斯求積法。

5.7 外推方法。

5.8 數值積分中的特殊主題。

5.8.1 隆伯格積分。

5.8.2 非光滑積分的求積法。

5.8.3 自適應積分。

5.8.4 梯形法則的佩阿諾估計。

5.9 文獻與軟體討論。

參考文獻。

6. 常微分方程的數值方法。

6.1 初值問題 - 背景。

6.2 歐拉法。

6.3 歐拉法的分析。

6.4 歐拉法的變體。

6.4.1 殘差與截斷誤差。

6.4.2 隱式方法與預測-修正方案。

6.4.3 起始值與多步驟方法。

6.4.4 中點法與弱穩定性。

6.5 單步驟方法？龍格-庫塔法。

6.6 多步驟方法。

6.6.1 亞當斯家族。

6.6.2 BDF家族。

6.7 穩定性問題。

6.7.1 多步驟方法的穩定性理論。

6.7.2 穩定區域。

6.8 應用於方程組。

6.8.1 實現問題與範例。

6.8.2 剛性方程。

6.8.3 A-穩定性。

6.9 自適應求解器。

6.10 邊值問題。

6.10.1 簡單差分方法。

6.10.2 射擊方法。

6.11 文獻與軟體討論。

參考文獻。

7. 方程組解的數值方法。

7.1 線性代數回顧。

7.2 線性系統與高斯消去法。

7.3 操作計數。

7.4 LU分解。

7.5 擾動、條件與穩定性。

7.5.1 向量與矩陣範數。

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