CiNii Books - An introduction to numerical methods : a MATLAB approach

Numerical methods are a mainstay of researchers and professionals across the many mathematics, scientific, and engineering disciplines. The importance of these methods combined with the power and availability of today's computers virtually demand that students in these fields be well versed not only in the numerical techniques, but also in the use of a modern computational software package. Updated to reflect the latest version of MATLAB, the second edition of An Introduction to Numerical Methods continues to fulfill both these needs. It introduces the theory and applications of the most commonly used techniques for solving numerical problems on a computer. It covers a wide range of useful algorithms, each presented with full details so that readers can visualize and interpret each step. Highlights of the second edition: A new chapter on numerical optimization New sections on finite elements More exercises and applied problems in each chapter MATLAB incorporated as an integral part of the text Emphasis on understanding how the methods work, a simple, direct style, and thorough coverage make this book an outstanding initiation that allows students to see almost immediate results. It will boost their confidence in their ability to master the subject and give them valuable experience in the use of MATLAB.

Table of Contents

INTRODUCTION About the software MATLAB An Introduction to MATLAB Taylor Series NUMBER SYSTEM AND ERRORS Floating-Point Arithmetic Round-Off Errors Truncation Error Interval Arithmetic ROOTS OF EQUATIONS The Bisection Method The Method of False Position Fixed-Point Iteration The Secant method Newton's Method Convergence of the Newton and Secant Methods Multiple Roots and the Modified Newton Method Newton's Method for Nonlinear Systems SYSTEM OF LINEAR EQUATIONS Matrices and Matrix Operations Naive Gaussian Elimination Gaussian Elimination with Scaled Partial Pivoting LU Decomposition Iterative Methods INTERPOLATION Polynomial Interpolation Theory Newton's Divided Difference Interpolating Polynomial The Error of the Interpolating Polynomial Lagrange Interpolating Polynomial INTERPOLATION WITH SPLINE FUNCTIONS Piecewise Linear Interpolation Quadratic Spline Natural Cubic Splines THE METHOD OF LEAST SQUARES Linear Least Squares Least Squares Polynomial Nonlinear Least Squares Trigonometric Least Squares Polynomial NUMERICAL OPTIMIZATION Analysis of Single-Variable Functions Line Search Methods Minimization Using Derivatives NUMERICAL DIFFERENTIATION Numerical Differentiation Richardson's Formula NUMERICAL INTEGRATION Trapezoidal Rule Simpson's Rule Romberg Algorithm Gaussian Quadrature NUMERICAL METHODS FOR DIFFERENTIAL EQUATIONS Euler's Method Error Analysis Higher Order Taylor Series Methods Runge-Kutta Methods Multistep Methods Adams-Bashforth Method Predictor-Corrector Methods Adams-Moulton Method Numerical Stability Higher Order Equations and Systems of Differential Equations Implicit Methods and Stiff Systems Phase Plane Analysis: Chaotic Differential Equations BOUNDARY-VALUE PROBLEMS Finite-Difference Methods Shooting Methods EIGENVALUES AND EIGENVECTORS Basic Theory The Power Method The Quadratic Method Eigenvalues for Boundary-Value Problems Bifurcations in Differential Equations PARTIAL DIFFERENTIAL EQUATIONS Parabolic Equations Hyperbolic Equations Elliptic Equations Introduction to Finite Element Method Bibliography and References Appendices Calculus Review MATLAB Built-in Functions Text MATLAB Functions Answers to Selected Exercises Index Each chapter also contains a section of Applied Problems.

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An introduction to numerical methods : a MATLAB approach

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