Cubic Spline Program

Question

I\'m trying to write a cubic spline interpolation program. I have written the program but, the graph is not coming out correctly. The spline uses natural boundary conditions

Answer 1

I wrote a cubic spline package in Mathematica a long time ago. Here is my translation of that package into Matlab. Note I haven't looked at cubic splines in about 7 years, so I'm basing this off my own documentation. You should check everything I say.

The basic problem is we are given n data points (x(1), y(1)) , ... , (x(n), y(n)) and we wish to calculate a piecewise cubic interpolant. The interpolant is defined as

   S(x) = {  Sk(x)   when x(k) <= x <= x(k+1)
          {  0       otherwise 

Here Sk(x) is a cubic polynomial of the form

  Sk(x) = sk0 + sk1*(x-x(k)) + sk2*(x-x(k))^2 + sk3*(x-x(k))^3

The properties of the spline are:

1. The spline pass through the data point Sk(x(k)) = y(k)
2. The spline is continuous at the end-points and thus continuous everywhere in the interpolation interval Sk(x(k+1)) = Sk+1(x(k+1))
3. The spline has continuous first derivative Sk'(x(k+1)) = Sk+1'(x(k+1))
4. The spline has continuous second derivative Sk''(x(k+1)) = Sk+1''(x(k+1))

To construct a cubic spline from a set of data point we need to solve for the coefficients sk0, sk1, sk2 and sk3 for each of the n-1 cubic polynomials. That is a total of 4*(n-1) = 4*n - 4 unknowns. Property 1 supplies n constraints, and properties 2,3,4 each supply an additional n-2 constraints. Thus we have n + 3*(n-2) = 4*n - 6 constraints and 4*n - 4 unknowns. This leaves two degrees of freedom. We fix these degrees of freedom by setting the second derivative equal to zero at the start and end nodes.

Let m(k) = Sk''(x(k)) , h(k) = x(k+1) - x(k) and d(k) = (y(k+1) - y(k))/h(k). The following three-term recurrence relation holds

  h(k-1)*m(k-1) + 2*(h(k-1) + h(k))*m(k) + h(k)*m(k+1) = 6*(d(k) - d(k-1))

The m(k) are unknowns we wish to solve for. The h(k) and d(k) are defined by the input data. This three-term recurrence relation defines a tridiagonal linear system. Once the m(k) are determined the coefficients for Sk are given by

   sk0 = y(k)
   sk1 = d(k) - h(k)*(2*m(k) + m(k-1))/6
   sk2 = m(k)/2
   sk3 = m(k+1) - m(k)/(6*h(k))

Okay that is all the math you need to know to completely define the algorithm to compute a cubic spline. Here it is in Matlab:

function [s0,s1,s2,s3]=cubic_spline(x,y)

if any(size(x) ~= size(y)) || size(x,2) ~= 1
   error('inputs x and y must be column vectors of equal length');
end

n = length(x)

h = x(2:n) - x(1:n-1);
d = (y(2:n) - y(1:n-1))./h;

lower = h(1:end-1);
main  = 2*(h(1:end-1) + h(2:end));
upper = h(2:end);

T = spdiags([lower main upper], [-1 0 1], n-2, n-2);
rhs = 6*(d(2:end)-d(1:end-1));

m = T\rhs;

% Use natural boundary conditions where second derivative
% is zero at the endpoints

m = [ 0; m; 0];

s0 = y;
s1 = d - h.*(2*m(1:end-1) + m(2:end))/6;
s2 = m/2;
s3 =(m(2:end)-m(1:end-1))./(6*h);

Here is some code to plot a cubic spline:

function plot_cubic_spline(x,s0,s1,s2,s3)

n = length(x);

inner_points = 20;

for i=1:n-1
   xx = linspace(x(i),x(i+1),inner_points);
   xi = repmat(x(i),1,inner_points);
   yy = s0(i) + s1(i)*(xx-xi) + ... 
     s2(i)*(xx-xi).^2 + s3(i)*(xx - xi).^3;
   plot(xx,yy,'b')
   plot(x(i),0,'r');
end

Here is a function that constructs a cubic spline and plots in on the famous Runge function:

function cubic_driver(num_points)

runge = @(x) 1./(1+ 25*x.^2);

x = linspace(-1,1,num_points);
y = runge(x);

[s0,s1,s2,s3] = cubic_spline(x',y');

plot_points = 1000;
xx = linspace(-1,1,plot_points);
yy = runge(xx);

plot(xx,yy,'g');
hold on;
plot_cubic_spline(x,s0,s1,s2,s3);

You can see it in action by running the following at the Matlab prompt

 >> cubic_driver(5)
 >> clf 
 >> cubic_driver(10)
 >> clf
 >> cubic_driver(20)

By the time you have twenty nodes your interpolant is visually indistinguishable from the Runge function.

Some comments on the Matlab code: I don't use any for or while loops. I am able to vectorize all operations. I quickly form the sparse tridiagonal matrix with spdiags. I solve it using the backslash operator. I counting on Tim Davis's UMFPACK to handle the decomposition and forward and backward solves.

Hope that helps. The code is available as a gist on github https://gist.github.com/1269709