algos.m

Matlab numerical methods,examples of mathematical procedures

%---------------------------------------------------------------------------
%ALGOS
%
% NUMERICAL METHODS: MATLAB Programs, (c) John H. Mathews 1995
% To accompany the text:
% NUMERICAL METHODS for Mathematics, Science and Engineering, 2nd Ed, 1992
% Prentice Hall, Englewood Cliffs, New Jersey, 07632, U.S.A.
% Prentice Hall, Inc.; USA, Canada, Mexico ISBN 0-13-624990-6
% Prentice Hall, International Editions:   ISBN 0-13-625047-5
% This free software is compliments of the author.
% E-mail address:       in%"[e-mail]"


CONTENTS

Chapter 1. Preliminaries

  Theorem 1.1   Limits and Continuous Functions
  Theorem 1.2   Intermediate Value Theorem
  Theorem 1.3   Extreme Value Theorem for a Continuous Function
  Theorem 1.4   Differentiable function implies continuous function
  Theorem 1.5   Rolle's Theorem
  Theorem 1.6   Mean Value Theorem
  Theorem 1.7   Extreme Value Theorem for a Differentiable Function
  Theorem 1.8   Generalized Rolle's Theorem
  Theorem 1.9   First Fundamental Theorem
  Theorem 1.10  Second Fundamental Theorem
  Theorem 1.11  Mean Value Theorem for Integrals
  Theorem 1.12  Weighted Integral Mean Value Theorem
  Theorem 1.13  Taylor's Theorem
  Theorem 1.14  Horner's Method for Polynomial Evaluation
  Theorem 1.15  Geometric Series
  Theorem 1.16  Big "O" remainders for Taylor's Theorem
  Theorem 1.17  Remainder term for Taylor's Theorem

Chapter 2. The Solution of Nonlinear Equations f(x) = 0

  Algorithm 2.1   Fixed Point Iteration
  Algorithm 2.2   Bisection Method
  Algorithm 2.3   False position or Regula Falsi Method
  Algorithm 2.4   Approximate Location of Roots
  Algorithm 2.5   Newton-Raphson Iteration
  Algorithm 2.6   Secant Method
  Algorithm 2.7   Steffensen's Acceleration
  Algorithm 2.8   Muller's Method
  Algorithm 2.9   Nonlinear Seidel Iteration
  Algorithm 2.10  Newton-Raphson Method in 2-Dimensions

Chapter 3.  The Solution of Linear Systems  AX = B

  Algorithm 3.1   Back Substitution
  Algorithm 3.2   Upper-Triangularization Followed by Back Substitution
  Algorithm 3.3   PA = LU Factorization with Pivoting
  Algorithm 3.4   Jacobi Iteration
  Algorithm 3.5   Gauss-Seidel Iteration

Chapter 4.  Interpolation and Polynomial Approximation

  Algorithm 4.1   Evaluation of a Taylor Series
  Algorithm 4.2   Polynomial Calculus
  Algorithm 4.3   Lagrange Approximation
  Algorithm 4.4   Nested Multiplication with Multiple Centers
  Algorithm 4.5   Newton Interpolation Polynomial
  Algorithm 4.6   Chebyshev Approximation

Chapter 5.  Curve Fitting

  Algorithm 5.1   Least Squares Line
  Algorithm 5.2   Least Squares Polynomial
  Algorithm 5.3   Non-linear Curve Fitting
  Algorithm 5.4   Cubic Splines
  Algorithm 5.5   Trigonometric Polynomials

Chapter 6.  Numerical Differentiation

  Algorithm 6.1   Differentiation Using Limits
  Algorithm 6.2   Differentiation Using Extrapolation
  Algorithm 6.3   Differentiation Based on N+1 Nodes

Chapter 7.  Numerical Integration

  Algorithm 7.1   Composite Trapezoidal Rule
  Algorithm 7.2   Composite Simpson Rule
  Algorithm 7.3   Recursive Trapezoidal Rule
  Algorithm 7.4   Romberg Integration
  Algorithm 7.5   Adaptive Quadrature Using Simpson's Rule
  Algorithm 7.6   Gauss-Legendre Quadrature

Chapter 8.  Numerical Optimization

  Algorithm 8.1   Golden Search for a Minimum
  Algorithm 8.2   Nelder-Mead's Minimization Method
  Algorithm 8.3   Local Minimum Search Using Quadratic Interpolation
  Algorithm 8.4   Steepest Descent or Gradient Method

Chapter 9.  Solution of Differential Equations

  Algorithm 9.1   Euler's Method
  Algorithm 9.2   Heun's Method
  Algorithm 9.3   Taylor's Method of Order 4
  Algorithm 9.4   Runge-Kutta Method of Order 4
  Algorithm 9.5   Runge-Kutta-Fehlberg Method RKF45
  Algorithm 9.6   Adams-Bashforth-Moulton Method
  Algorithm 9.7   Milne-Simpson Method
  Algorithm 9.8   The Hamming Method
  Algorithm 9.9   Linear Shooting Method
  Algorithm 9.10  Finite-Difference Method

Chapter 10.  Solution of Partial Differential Equations

  Algorithm 10.1  Finite-Difference Solution for the Wave Equation
  Algorithm 10.2  Forward-Difference Method for the Heat Equation
  Algorithm 10.3  Crank-Nicholson Method for the Heat Equation
  Algorithm 10.4  Dirichlet Method for Laplace's Equation

Chapter 11.  Eigenvalues and Eigenvectors

  Algorithm 11.1  Power Method
  Algorithm 11.2  Shifted Inverse Power Method
  Algorithm 11.3  Jacobi Iteration for Eigenvalues and Eigenvectors
  Algorithm 11.4  Reduction to Tridiagonal Form
  Algorithm 11.5  The QL Method with Shifts