![]() 21 Section 3: Numerical Integration.21 2.1 - Rectangular Rule.24 2.2 - Trapezoid Rule.31 2.3 - Simpson's Rule.31 2.4 - Gaussian Quadrature.31 2.5 - Monte Carlo Integration.31 Section 3: Initial Value Problems (Ordinary Differential Equations).34 3.1 - Euler's Method.34 3.2 - Runge-Kutta Methods.39 3.3 - Adams-Bashforth Methods.43 3.4 - Systems of Differential Equations.43 Section 4: Boundary Value Problems.44 4.1 - Linear Shooting Method.44 4.2 - Shooting Method for Nonlinear Problems.44 4.3 - Finite-Difference Methods.44 Section 5: Partial Differential Equations and the Finite Element Method.44īasic Numerical Methods and FreeMat Numerical methods provide a way to solve problems quickly and easily compared to analytic solutions. Table of Contents Basic Numerical Methods and FreeMat.1 About This Document.3 Basic Numerical Methods and FreeMat.5 Section 1: Root Finding.5 1.1 - The Bisection Method.5 1.2 - The Newton-Raphson Method.8 1.3 - The Secant Method.10 1.4 - Nonlinear Systems of Equations.16 The Newton-Raphson Method for Systems of Equations.16 Section 2: Numerical Differentiation. We just ask that you give FreeMat a try, and you might fall in love with it like we have. This text is free to copy, distribute, download and disseminate – we encourage exactly this. This text is not for commercial sale, and may not be printed, copied or modified for the purpose of selling it. ![]() This text is a basic introduction to the idiosyncrasies of the FreeMat programming environment, and does provide downloadable examples of every type of problem covered. This text very lightly addresses error and uncertainty, but substantially addresses examples of basic problems engineers may face. This text, as you'll see, doesn't present much of anything in the way of analytical solution of differential equations. It is by no means a full-on graduate-level text on the intricacies of numerics and solution of differential equations. If there are pictures, asides, explanations or examples that can make learning better, you can add them and republish the document (citing the original work, obviously)! There are certain things that this text is and isn't. Probably the most significant aspect of this document is that it is free (as in freedom). Also, it's a bit of an introduction to the language of FreeMat, which is easily interchangeable with commercial packages such as MATLAB, or is at the least readable pseudo-code for examples to use in other languages (such as C++ or FORTRAN). This is a basic introduction to numerical methods, showing how the commonly used methods work by some simple examples. This text is a response to that phenomenon. My personal experience with college texts has been that the ones where explanation would have greatly augmented learning lacked such explanation, but certainly didn't hold back on cost. Especially with respect to explanations of commercial software packages such as MATLAB, and introductions/explanations of numerical methods, texts tend to under-explain and overcharge. ![]() This document is free as in freedom, and as in beer.Ī lot of basic ideas put forth in most Engineering/Mathematics texts are rather old, and are generally common knowledge, at least to the educated. Engineering/Mathematics texts are expensive. FreeMat is free, as in freedom, and as in beer. Commercial software packages such as MATLAB are expensive. “We demand guaranteed, rigidly-defined areas of doubt and uncertainty.” - Douglas Adams ![]() “Essentially, all models are wrong, but some are useful.” - George Box
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