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The Runge phenomenon


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Introduction

In this lab, we will investigate the Runge Phenomenon and see how we might be able to fix it by choosing interpolating points wisely.

As a classic example of the Runge Phenomenon, we try to interpolate the function f(x) = 1/(1 + 25x^2) at equally spaced points.

clear all;
close all;
f = @(x) 1./(1 + 25*x.^2);

Plot the original function f(x) at set of equally spaced points

figure(1);
clf;

x = linspace(-1,1,500);
y_true = f(x);
plot(x,y_true,'r','linewidth',2);
hold on;

Construct an interpolating polynomial of degree N at N+1 points. We will use polyfit although the problem occurs with Lagrange polynomial interpolation or the Barycentric form as well.

% Equally spaced points
N = 10;    % Degree of the polynomial we try to fit
xdata = linspace(-1,1,N+1)';
ydata = f(xdata);

p = polyfit(xdata,ydata,N);

Plot the interpolating polynomial

y_fit = polyval(p,x);

poly_10 = plot(x,y_fit,'b','linewidth',2);

plot(xdata,ydata,'k.','markersize',30);
snapnow;

We see that a single interpolating polynomial does a particularly poor job of interpolating near the endpoints of the interval [-1,1]. Perhaps we should try increasing the order of accuracy.

N = 20;  % Increase the degree to 20 from 10.
xdata = linspace(-1,1,N+1)';
ydata = f(xdata);

% Ignore the warning about the matrix being ill-conditioned.  If we used
% the barycentric form, we do not have these ill-conditioning issues!
p = polyfit(xdata,ydata,N);

Plot the interpolating polynomial

y_fit = polyval(p,x);

poly_20 = plot(x,y_fit,'g','linewidth',2);

plot(xdata,ydata,'k.','markersize',30);

axis([-1 1 -5 5]);
snapnow;

Let's increase one more time to see that pattern

N = 40;  % Increase the degree to 40 from 20.
xdata = linspace(-1,1,N+1)';
ydata = f(xdata);

% plot(xdata,ydata,'k.','markersize',30);

% Ignore the warning about the matrix being ill-conditioned.  If we used
% the barycentric form, we do not have these ill-conditioning issues!
p = polyfit(xdata,ydata,N);
Warning: Polynomial is badly conditioned. Add points with distinct X values,
reduce the degree of the polynomial, or try centering and scaling as described
in HELP POLYFIT. 

Plot the interpolating polynomial

y_fit = polyval(p,x);

poly_40 = plot(x,y_fit,'m','linewidth',2);

plot(xdata,ydata,'k.','markersize',30);

axis([-1 1 -10 10]);
snapnow;

But are we doing at least a better job in the middle? Yes!

axis([-1 1 0 1]);
lh = legend([poly_10, poly_20, poly_40],{'N = 10','N = 20','N = 40'});
set(lh,'fontsize',18);
snapnow;

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Avoiding the Runge phenomenon (part I)

In the previous examples, our interpolating points were equally spaced points. Is there a better choice of points at which to interpolate our function?

We can try using "Chebychev nodes". These are special nodes which allow for a better approximation of the polynomial.

clf;
y_true = f(x);
plot(x,y_true,'r','linewidth',2);
hold on;

% Fit a 10th degree polynomial at Chebyshev nodes.
N = 10;    % Degree of the polynomial we try to fit
t = linspace(0,pi,N+1);
xdata = -cos(t);    % Chebyshev nodes
ydata = f(xdata);

p = polyfit(xdata,ydata,N);

Plot the interpolating polynomial

y_fit = polyval(p,x);

poly_10 = plot(x,y_fit,'b','linewidth',2);
hold on;

plot(xdata,ydata,'k.','markersize',30);
snapnow;

% Fit a 20th degree polynomial at Chebyshev nodes.
N = 20;    % Degree of the polynomial we try to fit
t = linspace(0,pi,N+1);
xdata = -cos(t);    % Chebyshev nodes
ydata = f(xdata);

p = polyfit(xdata,ydata,N);

Plot the interpolating polynomial

y_fit = polyval(p,x);

poly_20 = plot(x,y_fit,'g','linewidth',2);
hold on;

plot(xdata,ydata,'k.','markersize',30);
snapnow;

% Fit a 40th degree polynomial at Chebyshev nodes.
N = 40;    % Degree of the polynomial we try to fit
t = linspace(0,pi,N+1);
xdata = -cos(t);    % Chebyshev nodes
ydata = f(xdata);

% Ignore the warning about the matrix being ill-conditioned.  If we used
% the barycentric form, we do not have these ill-conditioning issues!
p = polyfit(xdata,ydata,N);
Warning: Polynomial is badly conditioned. Add points with distinct X values,
reduce the degree of the polynomial, or try centering and scaling as described
in HELP POLYFIT. 

Plot the interpolating polynomial

y_fit = polyval(p,x);

poly_40 = plot(x,y_fit,'m','linewidth',2);
hold on;

plot(xdata,ydata,'k.','markersize',30);
snapnow;

And how does the middle look?

axis([-1 1 0 1]);
lh = legend([poly_10, poly_20, poly_40],{'N = 10','N = 20','N = 40'});
set(lh,'fontsize',18);
snapnow;

We have completely avoided the problem at the endpoints by chosing our points carefully! But what if we don't have the flexibility to chose our interpolating points? That leads us to the idea of using piecewise polynomials.

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