Solving Equations on the Computer

Throughout this chapter we have used graphical and analytical methods to analyze first-order systems. Every budding dynamicist should master a third tool numerical methods. In the old days, numerical methods were impractical because they required enormous amounts of tedious hand-calculation. But all that has changed, thanks to the computer. Computers enable us to approximate the solutions to analytically intractable problems, and also to visualize those solutions. In this section we take our first look at dynamics on the computer, in the context of numerical integration of X = f (x). 

Numerical integration is a vast subject. We will barely scratch the surface. See Chapter 15 of Press et al. (1986) for an excellent treatment. 

The problem can be posed this way given the differential equation x = /(x), subject to the condition x = Xg at t =1, find a systematic way to approximate the solution x(t). 

This is the simplest possible numerical integration scheme. It is known as Euler s method. 

One problem with the Euler method is that it estimates the derivative only at the left end of the time interval between Z and. A more sensible approach would be to use the average derivative across this interval. This is the idea behind the improved Euler method. We first take a trial step across the interval, using the Euler method. This produces a trial value x + = jc, + the tilde above the 

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