In many of the following problems, it will be essential to have a calculator or comp...

In many of the following problems, it will be essential to have a calculator or computer available. You may use a software package† or write a program for solving initial value problems using the improved Euler’s method algorithms on pages 127 and 128. (Remember, all trigonometric calculations are done in radians.)

Local versus Global Error. In deriving formula (4) for Euler’s method, a rectangle was used to approximate the area under a curve (see Figure 3.14). With this approximation can be written as

(a) Show that if g has a continuous derivative that is bounded in absolute value by B, then the rectangle approximation has error O(h²); that is, for some constant M,

This is called the local truncation error of the scheme. [Hint: Write

Next, using the mean value theorem, show that

(b) In applying Euler’s method, local truncation errors occur in each step of the process and are propagated throughout the further computations. Show that the sum of the local truncation errors in part (a) that arise after n steps O(h) is . This is the global error, which is the same as the convergence rate of Euler’s method.