Sample Questions on Numerical Methods

1.

(a) Explain the meaning of the term ``local error''.     Solution

(b) The global error is not simply the sum of the local errors. Why not?     Solution

(c) Show that the local error is O(h³) for the Improved Euler method with step size h on the differential equation y' = 3 y.     Solution

2.

A student is using a numerical method to approximate the solution to a certain initial value problem. With a step size h = 0.2 she gets an answer of 1.752, while with h = 0.1 she gets 1.692. The true answer happens to be 1.671.

(a) Which method do you think she is using: Euler, Improved Euler, or some other method? Hint: what does p appear to be for this method?     Solution

(b) What answer will she get using Richardson extrapolation?     Solution

3.

Using the Improved Euler method to approximate the solution of an initial value problem, I obtain an answer of 3.7621 with step size h = 0.1 and 3.5521 with step size h = 0.05. Predict what would be obtained with step size h = 0.01.     Solution

4.

A mathematician is trying to use the Improved Euler method to solve numerically the initial value problem $$\frac{dy}{dx}=y^2-x$$, y(0)=0. She needs to approximate y(2) with an error of at most 0.00001.

(a) Perform the first step of the procedure for step size h=0.1 (obtaining the approximate answer for y(.1)).

(b) With step size h = 0.1 the approximation for y(2) is -1.192670233. With step size h= 0.05 the approximation is -1.193360798. Calculate and improved approximation using Richardson extrapolation, and suggest a step size that is likely to produce an error of approximately .00001.

(c) Another mathematician tried the same methods on the same equation, but with initial condition y(0)=1. His computer only gave him ``Overflow'' error messages. What do you think could be causing the trouble?     Solution