Applied Numerical Methods Using MATLAB
Applied Numerical Methods Using MATLAB
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DOI：
10.5860/choice.43-1615
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发表时间：
2005
期刊：
影响因子：
--
通讯作者：
Wŏn-yŏng Yang;Wenwu Cao;Tae-Sang Chung;John Morris
中科院分区：
文献类型：
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作者：
Wŏn-yŏng Yang;Wenwu Cao;Tae-Sang Chung;John Morris
前言。1。MATLAB的使用与计算误差。1.1 MATLAB的基本操作1.1.1 MATLAB命令窗口数据输入/输出1.1.2文件数据输入/输出1.1.3键盘数据输入/输出1.1.4二维图形输入/输出1.1.5三维图形输出、1.1.6数学函数、1.1.7向量和矩阵运算、1.1.8随机数生成器、1.1.9流量控制、1.2计算机错误与人为错误、1.2.1 IEEE 64位浮点数表示、1.2.2各种计算错误、1.2.3绝对/相对计算错误、1.2.4错误传播、1.2.5避免大错误的技巧、1.3走向好的程序、1.3.1提高计算效率的嵌套计算、1.3.2向量运算与循环迭代。1.3.3迭代例程与嵌套例程。1.3.4避免运行时错误。1.3.5通过全局变量共享参数。1.3.6通过Varargin传递参数。1.3.7自适应输入参数列表。问题。2。线性方程组。2.1线性方程组的解。2.1.1非奇异情况（M = N）. 2.1.2待定情况（M N）：最小二乘误差解。2.1.4 RLSE（递推最小二乘估计）2.2线性方程组的求解。2.2.1高斯消去。2.2.2偏轴化。2.2.3高斯-约当消去。2.3逆矩阵。2.4分解（因数分解）。2.4.1 LU分解（分解）：三角化。2.4.2其他分解（分解）：Cholesky、QR、SVD。2.5求解方程的迭代方法。2.5.1 Jacobi迭代。2.5.2 Gauss-Seidel迭代。2.5.3 Jacobi和Gauss-Seidel迭代的收敛性。问题。3。插补与曲线拟合3.1拉格朗日多项式插补3.2牛顿多项式插补3.3切比雪夫多项式逼近3.4有理函数Pade逼近3.5三次样条插补3.6 Hermite插值多项式3.7二维插补3.8曲线拟合3.8.1直线拟合一阶多项式函数3.8.2多项式曲线拟合一个更高次的多项式函数。3.8.3指数曲线拟合和其他函数。3.9傅立叶变换。3.9.1 FFT与DFT。3.9.2 DFT的物理含义。3.9.3使用DFS插值。问题。4。非线性方程。4.1不动点迭代法。4.2对分法。4.3假位置法或正则假法。4.4牛顿（-Raphson）法。4.5割线法。4.6非线性方程组的牛顿法。4.7方程的符号解。4.8一个现实问题。问题。5。数值微分/集成。5.1一阶导数的差分近似。5.2一阶导数的近似误差。5.3二阶导数和高阶导数的差分近似。5.4插值多项式和数值微分。5.5数值积分和正交。5.6梯形法和辛普森法。5.7递归规则和Romberg积分。5.8自适应正交。5.9高斯正交。5.9.1高斯-勒让德积分。5.9.2高斯-埃尔米特积分。5.9.3高斯-拉盖尔积分。5.9.4高斯-切比雪夫积分。5.10二重积分。问题。6。常微分方程。6.1欧拉法。6.2 Heun法：梯形法。6.3龙格-库塔法。6.4预测校正法。6.4.1 Adams-Bashforth-Moulton法。6.4.2 Hamming法。6.4.3方法比较。6.5矢量微分方程。6.5.1状态方程。6.5.2 LTI状态方程的离散化。6.5.3高阶微分方程到状态方程。6.5.4 Stiff方程。6.6边值问题（BVP）。6.6.1射击法。6.6.2有限差分法。问题。7。7.1无约束优化[L-2， Chapter 7]。7.1.1黄金搜索法。7.1.2二次逼近法。7.1.3 Nelder-Mead法[W-8]。7.1.4最速下降法。7.1.5牛顿法。7.1.6共轭梯度法。7.1.7模拟退火法[W-7]。7.1.8遗传算法[W-7]。7.2约束优化[L-2，第10章]。7.2.1拉格朗日乘子法。7.2.2罚函数法。7.3 MATLAB内置优化例程。7.3.1无约束优化。7.3.2约束优化。7.3.3线性规划（LP）。问题。8。矩阵与特征值。8.1特征值与特征向量。8.2相似变换与对角化。8.3幂次法。8.3.1比例幂次法。8.3.2幂次逆法。8.3.3移位幂次逆法。8.4 Jacobi法。8.5特征值/特征向量的物理意义。8.6特征值方程。问题。9。9.1椭圆偏微分方程。9.2抛物线PDE。9.2.1显式前向欧拉法9.2.2隐式后向欧拉法9.2.3 Crank-Nicholson法9.2.4二维抛物型偏微分方程9.3双曲PDE。9.3.1显式中心差分法。9.3.2二维双曲偏微分方程。9.4求解PDE的有限元法。9.5求解偏微分方程的MATLAB GUI: PDETOOL。9.5.1 PDETOOL可解析的基本pde。9.5.2 PDETOOL的使用说明9.5.3使用PDETOOL解析pde举例问题。附录A：中值定理。附录B：矩阵操作/属性。附录C：关于向量的微分。附录D：拉普拉斯变换。附录E：傅里叶变换。附录F：实用公式。附录G：符号计算。附录H：稀疏矩阵。附录一：MATLAB。引用。主题索引。索引MATLAB例程。表的索引。
Preface. 1. MATLAB Usage and Computational Errors. 1.1 Basic Operations of MATLAB. 1.1.1 Input/Output of Data from MATLAB Command Window. 1.1.2 Input/Output of Data Through Files. 1.1.3 Input/Output of Data Using Keyboard. 1.1.4 2-D Graphic Input/Output. 1.1.5 3-D Graphic Output. 1.1.6 Mathematical Functions. 1.1.7 Operations on Vectors and Matrices. 1.1.8 Random Number Generators. 1.1.9 Flow Control. 1.2 Computer Errors Versus Human Mistakes. 1.2.1 IEEE 64-bit Floating-Point Number Representation. 1.2.2 Various Kinds of Computing Errors. 1.2.3 Absolute/Relative Computing Errors. 1.2.4 Error Propagation. 1.2.5 Tips for Avoiding Large Errors. 1.3 Toward Good Program. 1.3.1 Nested Computing for Computational Efficiency. 1.3.2 Vector Operation Versus Loop Iteration. 1.3.3 Iterative Routine Versus Nested Routine. 1.3.4 To Avoid Runtime Error. 1.3.5 Parameter Sharing via Global Variables. 1.3.6 Parameter Passing Through Varargin. 1.3.7 Adaptive Input Argument List. Problems. 2. System of Linear Equations. 2.1 Solution for a System of Linear Equations. 2.1.1 The Nonsingular Case (M = N). 2.1.2 The Underdetermined Case (M N): Least-Squares Error Solution. 2.1.4 RLSE (Recursive Least-Squares Estimation). 2.2 Solving a System of Linear Equations. 2.2.1 Gauss Elimination. 2.2.2 Partial Pivoting. 2.2.3 Gauss-Jordan Elimination. 2.3 Inverse Matrix. 2.4 Decomposition (Factorization). 2.4.1 LU Decomposition (Factorization): Triangularization. 2.4.2 Other Decomposition (Factorization): Cholesky, QR, and SVD. 2.5 Iterative Methods to Solve Equations. 2.5.1 Jacobi Iteration. 2.5.2 Gauss-Seidel Iteration. 2.5.3 The Convergence of Jacobi and Gauss-Seidel Iterations. Problems. 3. Interpolation and Curve Fitting. 3.1 Interpolation by Lagrange Polynomial. 3.2 Interpolation by Newton Polynomial. 3.3 Approximation by Chebyshev Polynomial. 3.4 Pade Approximation by Rational Function. 3.5 Interpolation by Cubic Spline. 3.6 Hermite Interpolating Polynomial. 3.7 Two-dimensional Interpolation. 3.8 Curve Fitting. 3.8.1 Straight Line Fit: A Polynomial Function of First Degree. 3.8.2 Polynomial Curve Fit: A Polynomial Function of Higher Degree. 3.8.3 Exponential Curve Fit and Other Functions. 3.9 Fourier Transform. 3.9.1 FFT Versus DFT. 3.9.2 Physical Meaning of DFT. 3.9.3 Interpolation by Using DFS. Problems. 4. Nonlinear Equations. 4.1 Iterative Method Toward Fixed Point. 4.2 Bisection Method. 4.3 False Position or Regula Falsi Method. 4.4 Newton(-Raphson) Method. 4.5 Secant Method. 4.6 Newton Method for a System of Nonlinear Equations. 4.7 Symbolic Solution for Equations. 4.8 A Real-World Problem. Problems. 5. Numerical Differentiation/Integration. 5.1 Difference Approximation for First Derivative. 5.2 Approximation Error of First Derivative. 5.3 Difference Approximation for Second and Higher Derivative. 5.4 Interpolating Polynomial and Numerical Differential. 5.5 Numerical Integration and Quadrature. 5.6 Trapezoidal Method and Simpson Method. 5.7 Recursive Rule and Romberg Integration. 5.8 Adaptive Quadrature. 5.9 Gauss Quadrature. 5.9.1 Gauss-Legendre Integration. 5.9.2 Gauss-Hermite Integration. 5.9.3 Gauss-Laguerre Integration. 5.9.4 Gauss-Chebyshev Integration. 5.10 Double Integral. Problems. 6. Ordinary Differential Equations. 6.1 Euler's Method. 6.2 Heun's Method: Trapezoidal Method. 6.3 Runge-Kutta Method. 6.4 Predictor-Corrector Method. 6.4.1 Adams-Bashforth-Moulton Method. 6.4.2 Hamming Method. 6.4.3 Comparison of Methods. 6.5 Vector Differential Equations. 6.5.1 State Equation. 6.5.2 Discretization of LTI State Equation. 6.5.3 High-Order Differential Equation to State Equation. 6.5.4 Stiff Equation. 6.6 Boundary Value Problem (BVP). 6.6.1 Shooting Method. 6.6.2 Finite Difference Method. Problems. 7. Optimization. 7.1 Unconstrained Optimization [L-2, Chapter 7]. 7.1.1 Golden Search Method. 7.1.2 Quadratic Approximation Method. 7.1.3 Nelder-Mead Method [W-8]. 7.1.4 Steepest Descent Method. 7.1.5 Newton Method. 7.1.6 Conjugate Gradient Method. 7.1.7 Simulated Annealing Method [W-7]. 7.1.8 Genetic Algorithm [W-7]. 7.2 Constrained Optimization [L-2, Chapter 10]. 7.2.1 Lagrange Multiplier Method. 7.2.2 Penalty Function Method. 7.3 MATLAB Built-In Routines for Optimization. 7.3.1 Unconstrained Optimization. 7.3.2 Constrained Optimization. 7.3.3 Linear Programming (LP). Problems. 8. Matrices and Eigenvalues. 8.1 Eigenvalues and Eigenvectors. 8.2 Similarity Transformation and Diagonalization. 8.3 Power Method. 8.3.1 Scaled Power Method. 8.3.2 Inverse Power Method. 8.3.3 Shifted Inverse Power Method. 8.4 Jacobi Method. 8.5 Physical Meaning of Eigenvalues/Eigenvectors. 8.6 Eigenvalue Equations. Problems. 9. Partial Differential Equations. 9.1 Elliptic PDE. 9.2 Parabolic PDE. 9.2.1 The Explicit Forward Euler Method. 9.2.2 The Implicit Backward Euler Method. 9.2.3 The Crank-Nicholson Method. 9.2.4 Two-Dimensional Parabolic PDE. 9.3 Hyperbolic PDE. 9.3.1 The Explicit Central Difference Method. 9.3.2 Two-Dimensional Hyperbolic PDE. 9.4 Finite Element Method (FEM) for solving PDE. 9.5 GUI of MATLAB for Solving PDEs: PDETOOL. 9.5.1 Basic PDEs Solvable by PDETOOL. 9.5.2 The Usage of PDETOOL. 9.5.3 Examples of Using PDETOOL to Solve PDEs. Problems. Appendix A: Mean Value Theorem. Appendix B: Matrix Operations/Properties. Appendix C: Differentiation with Respect to a Vector. Appendix D: Laplace Transform. Appendix E: Fourier Transform. Appendix F: Useful Formulas. Appendix G: Symbolic Computation. Appendix H: Sparse Matrices. Appendix I: MATLAB. References. Subject Index. Index for MATLAB Routines. Index for Tables.