MACM 316 - Computing Assignment 5
Read the Guidelines for Assignments rst.
Submit a one-page PDF report to Canvas and upload your Matlab scripts (as m- les). Do
not use any other le formats.
Keep in mind that Canvas discussions are open forums.
You must acknowledge any collaborations/assistance from colleagues, TAs, instructors etc.
Runge’s phenomenon and aliasing
Two major concerns in polynomial interpolation are Runge’s phenomenon and aliasing. Runge’s
phenomenon describes oscillations of the interpolant near the boundaries. Aliasing refers to lower
degree polynomials pretending to be higher degree polynomials. In this report you will investigate
the e ects of both issues.
First, de ne a set of equispaced points on the interval [ 1;1]:
xi = 1 + 2in; i = 0;:::;n:
Next, de ne the set of Chebyshev points on the same interval:
xi = cos(i =n); i = 0;:::;n:
These two sets will act as your interpolation points. In addition, you’ll need a set of points to test
your interpolants; this set should also span the interval [ 1;1] but have several more points than
your interpolation points.
Download the le baryinterp.m, which uses barycentric Lagrange interpolation to construct
the Lagrange interpolating polynomial. All inputs are column vectors. Let x be the interpolation
points, f the values of the function to interpolate and grid the points to evaluate the interpolant
at. The syntax to use baryinterp() is then:
u = baryinterp(x,f,grid);
Measure the error of interpolation for the function
f1(x) = 12 +x; f1 = @(x) 1./(2 + x);
for n between 1 and 100. How does Runge’s phenomenon a ect the accuracy? Which set of
interpolation points deals with it better?
Next, measure the error of interpolation for the function
f2(x) = cos(10 x); f2 = @(x) cos(10*pi*x);
for n between 1 and 100. How many points do you need to overcome aliasing? Were there any
di erences between the types of interpolation points?
TipsThe following lines of code will be useful when de ning the interpolation points:
x_equi = 0:n; x_equi = -1 + 2*x_equi’/n;
x_cheb = 0:n; x_cheb = cos(pi*x_cheb’/n);
You may use any set of points as the grid on which to test your interpolants. Use the same
grid for all tests and make sure the number of points on the grid is at least 10 times as many
interpolation points you use. The easiest choice is 1000 equispaced points.
You are free to use any measure of error. The most common way to measure error is to use
the norm() function in Matlab:
error = norm(f1(grid) - u, p);
Here, the p refers to which p{norm the function will use. You can use any whole number (not
zero) and you can also use inf for the max norm. The most popular choices of p are 1, 2 and
inf.
If you run into any kind of error message when using the function baryinterp() make sure
that your inputs are all column vectors. You can do this by checking size(x) or by nding
their size in the workspace section of the console.