Homework 4
Due on Thursday, February 14
There are five problems in this assignment. The first two problems require MATLAB implementation, and the last
three problems are the usual exercise problems. You should hand in the programming assignment online: go to the
course webpage and click on the CourseTool link on the left. You should submit a zip file containing two MATLAB
m-files with names q1.m, q2.m. Obviously, q1.m is the code for problem 1 and q2.m is the code for Problem 2. The
written solutions to the last three problems should be handed in as before (i.e., in class).
1 (Problem 10 on page 47) Use Euler’s method to compute approximate solutions to each of the initial value problems
below, using h−1= 2,4,8,16.Compute the maximum error over the interval [0 1] for each value of h. Plot your
approximate solutions for the h=1
16 case.
1. y0=−yln y, y(0) = 3; y(t) = e(ln 3)e−t
2. y0=t−y, y(0) = 2; y(t) = 3e−t+t−1
3. y0+ 4y= 1, y(0) = 1; y(t) = 1
4(3e−4t+ 1).
(For this problem, you need to implement the Euler’s method using MATLAB. You code should plot the three approx-
imate solutions in one figure with different colors (say, red, blue and green).)
2We know that for approximating derivatives, the one-sided difference and centered difference have accuracies that
are linear and quadratic, respectively:
f0(x) = f(x+h)−f(x)
h+O(h)
f0(x) = f(x+h)−f(x−h)
2h+O(h2)
Therefore, if you halve the step size h, the error will decrease by a factor of 2and 4, respectively. However, since the
computer uses floating point numbers in its computations, rounding errors will eventually become significant for small
hsuch that the expected decrease in error will no longer be observed when his sufficiently small. For this problem,
you will run a simple experiment to witness this.
Let f(x) = ex, and of course f0(x) = ex. We will numerically compute the derivative of f(x)at x= 1 using
both one-sided and centered differences. Take hk= (1
2)k, for k= 0,1,2,3,· · · ,65. Let Dhkdenote the derivative
numerically computed (using either method) with h=hk. Denote E(hk)the error |f0(1) −Dhk|and E R(k)the ratio
ER(k) = E(hk−1)
E(hk).
For each approximation method, your program should produce a figure that contains (a) the plot of ER(k)(in blue)
for k= 1,2,3,· · · ,65 and (b) the plot of Dhk(in red). See Figure 1. For centered difference, you should see that
ER(k)is approximately 4for k < 20 and its values become somewhat random for k > 20. This happens because the
error due to rounding is now roughly in the same order as the error due to approximation (see page 46). The computed
derivative Dhkbecomes zero when k > 53.
1
