1.3. Stationary Points and Closed Orbits 19 amplified or attenuated in the same manner as truncation errors, we will often consolidate them and pretend that our computers do precise real number arithmetic, but there are situations where it is important to take it into consideration.) For Euler’s Method the stepping procedure is particularly simple and natural. It is defined by ΣE(f, yn,tn,h) := yn + h f(tn, yn). It is easy to see why this is a good choice. If as above we denote σ(f, yn,tn,t) by y(t), then by Taylor’s Theorem, y(tn + h) = y(tn) + h y (tn) + O(h2) = yn + h f(tn, yn) + O(h2) = ΣE(f, yn,tn,h) + O(h2), so that σ(f, yn,tn,tn + h) ΣE(f, yn,tn,h) , the local truncation error for Euler’s Method, does go to zero quadratically in h. When we partition [to,to + T ] into N equal parts, h = T N , each step in the recursion for computing yn will contribute a local truncation error that is O(h2) = O( 1 N2 ). Since there are N steps in the recursion and at each step we add O( 1 N2 ) to the error, this suggests that the global error will be O( 1 N ) and hence will go to zero as N tends to infinity. However, because of the potential amplification of prior errors, this is not a complete proof that Euler’s Method is convergent, and we will put off the details of the rigorous argument until the chapter on numerical methods. Exercise 1–15. Show that Euler’s Method applied to the initial value problem dy dt = y with y(0) = 1 gives limN→∞(1 + T N )N = eT . For T = 1 and N = 2, show that the global error is indeed greater than the sum of the two local truncation errors. 1.3. Stationary Points and Closed Orbits We next describe certain special types of solutions of a differential equation that play an important role in the description and analysis of the global behavior of its flow. For generality we will also consider the case of time-dependent vector fields, but these solutions are really most important in the study of autonomous equations.
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