CHE/COSC/MATH 4340-01 Numerical Analysis
Computer Laboratory Assignment 01
Due Date: Thursday, 01/22/09
Problem 1. We have briefly discussed in class how to quantify an error associated with approximating
the derivative f′(x)of a given function f(x)by
D(x) = f(x+∆x)−f(x)
∆x.(0.1)
Using the Taylor’s Theorem we concluded that the error is |f′(x)−D(x)| ≈ C∆x, where ∆xis the step-size
parameter, and Cis a constant independent of ∆x. In fact, we have found that this Cdepends on f′′. This
assessment says that as ∆xis decreased, the approximation (0.1) is expected to be more accurate (hence
the error is smaller). Your task in this assignment is to present a numerical observation of this theoretical
finding. Write down the following MATLAB function and save and name it as fprime.m.
function fprime(p, func, exactdfunc, N)
%
%
for i=1:N
dx(i) = 4ˆ(-i); % step-size parameter.
approx = (func(p+dx(i)) - func(p))/dx(i); % approximate derivative.
exact = exactdfunc(p);
error(i) = abs(exact - approx); % relative error
end
loglog(dx,error); % A logarithmic plot of error against dx.
xlabel(’step-size’);
ylabel(’error’);
This MATLAB function takes four arguments: pis the point at which the derivative is computed,
func is a MATLAB inline function whose derivative is to be computed, exactdfunc is a MATLAB
inline function containing the exact derivative of the function, and Nis the number of step-size parameter.
The output of the MATLAB function above is a logarithmic plot of the error against ∆x(or dx).
To invoke an inline function, use the MATLAB command inline. For example, if you want to
create inline function f(x) = sin(x)and save it in a variable f, then from MATLAB prompt, you should
invoke f=inline(’sin(x)’).
Run the function for p=0.1, and N=10. Plot the error for computing derivative of f(x) = x2and
f(x) = sin(x). In your report show both error plots and discuss what you observe. Estimate the slope of
the line in the plot and compare it with the theoretical finding above.
Problem 2. Redo Problem 1 above for another approximation of derivative
D2(x) = f(x+∆x)−f(x−∆x)
2∆x.(0.2)
What do you observe compared to Problem 1?
1