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Unformatted text preview: CHAPTER 1 Numerical Methods for Ordinary Differential Equations
In this chapter we discuss numerical method for ODE . We will discuss the two basic methods, Euler's Method and RungeKutta Method. 1. Numerical Algorithm and Programming in Mathcad 1.1. Numerical Algorithm. If you look at dictionary, you will the following definition for algorithm, 1. a set of rules for solving a problem in a finite number of steps; 2. a sequence of steps designed for programming a computer to solve a specific problem. A numerical algorithm is a set of rules for solving a problem in finite number of steps that can be easily implemented in computer using any programming language. The following is an algorithm for compute the root of f (x) = 0, Input f , a, N and tol . Output: the approximate solution to f (x) = 0 with initial guess a or failure message. Step One: Set x = a Step Two: For i=0 to N do Step Three  Four f Step Three: Compute x = x  f (x) (x) Step Four: If f (x) tol return x Step Five return "failure". In analogy, a numerical algorithm is like a cook recipe that specify the input  cooking material, the outputthe cooking product, and steps of carrying computation  cooking steps. In an algorithm, you will see loops (for, while), decision making statements(if, then, else(otherwise)) and return statements. for loop: Typically used when specific number of steps need to be carried out. You can break a for loop with return or break statement.
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 Spring '11
 Dr.Han

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