Homework 0: Interpolation examples in Matlab¶

Developed by Randy LeVeque for a course on Approximation Theory and Spectral Methods at the University of Washington.

See http://faculty.washington.edu/rjl/classes/am590a2013/codes.html for more IPython Notebook examples.

This is a demonstration of using an IPython notebook to combine code with descriptions.

It also demonstrates how to use Matlab and Chebfun from within the notebook.

Sample problem¶

Determine the quadratic polynomial $p(x)$ that interpolates the data

$(-1,3),~~(0,-1),~~(1,2)$ first using the Vandermonde matrix and then using the Lagrange form of interpolating polynomial.

To use Matlab from within the IPython notebook, first you have to install pymatbridge, see https://github.com/arokem/python-matlab-bridge.

Then the next set of commands gets Matlab running in the background (assuming it's installed on the computer you are using).

After this, any cell starting with %%matlab will be evaluated by Matlab.

In [1]:

import pymatbridge
ip = get_ipython()
pymatbridge.load_ipython_extension(ip)

Starting MATLAB on http://localhost:54956
 visit http://localhost:54956/exit.m to shut down same
...MATLAB started and connected!

Interpolation data for all the examples:¶

In [2]:

%%matlab
xj = [-1.; 0.; 1.]
yj = [3.; -1.; 2.]

Using monomials¶

For a quadratic interpolation we use basis functions $1,~x,$ and $x^2$ .

Define the Vandermonde matrix: The columns are the basis functions evaluated at the interpolation points.

In [3]:

%%matlab
A = [xj.^0, xj, xj.^2]

A =
     1    -1     1
     1     0     0
     1     1     1

Solve the system for the monomial coefficients:

In [4]:

%%matlab
c = A\yj

c =
  -1.000000000000000
  -0.500000000000000
   3.500000000000000

plot the resulting polynomial on a fine grid:

In [5]:

%%matlab
x = linspace(-1, 1, 1001);
p = c(1) + c(2)*x + c(3)*x.^2;
plot(x,p)
hold on
plot(xj,yj,'.','markersize',20)

Lagrange form¶

Evaluate the Lagrange basis functions on the fine grid for plotting:

$\begin{align} L_1(x) &= \frac{(x-x_2)(x-x_3)}{(x_1-x_2)(x_1-x_3)},\\\\ L_2(x) &= \frac{(x-x_1)(x-x_3)}{(x_2-x_1)(x_2-x_3)},\\\\ L_3(x) &= \frac{(x-x_1)(x-x_2)}{(x_3-x_1)(x_3-x_2)} \end{align}$

In [6]:

%%matlab
L1 = (x-xj(2)).*(x-xj(3)) / ((xj(1)-xj(2))*(xj(1)-xj(3)));
L2 = (x-xj(1)).*(x-xj(3)) / ((xj(2)-xj(1))*(xj(2)-xj(3)));
L3 = (x-xj(1)).*(x-xj(2)) / ((xj(3)-xj(1))*(xj(3)-xj(2)));

In this form, the data values are the coefficients

In [7]:

%%matlab
p = yj(1)*L1 + yj(2)*L2 + yj(3)*L3;

Plot this version (should look the same as before!)

In [8]:

%%matlab
clf
plot(x,p)
hold on
plot(xj,yj,'.','markersize',20)

Using Chebfun¶

To use Chebfun, you must download and add the location to the matlab path:

In [9]:

%%matlab
path(path,'/Users/rjl/chebfun/chebfun_v4.2.2889/chebfun')
which chebfun

/Users/rjl/chebfun/chebfun_v4.2.2889/chebfun/@chebfun/chebfun.m  % chebfun constructor

Use Chebfun to plot the function $f(x) = \left(\frac{1}{1 + 25x^2}\right)^{10}$ and the function $g(x) = 5 \int_0^x f(s)\, ds.$

What degree polynomial approximation is used for each?

In [10]:

%%matlab
x = chebfun('x');
f = (1./(1+25*x.^2)).^10;
g = 5*cumsum(f);
clf
plot(f,'b')
hold on
plot(g,'r')
legend('f','g')

disp(sprintf('The degree of f is %i', length(f)-1))
disp(sprintf('The degree of g is %i', length(g)-1))

The degree of f is 272
The degree of g is 215

In [10]: