I'm writing a MATLAB program which accepts 3 inputs x (a vector containing the x values for interpolation), y (a vector containing the y values for interpolation) and a string specifying the type of cubic spline required ('natural', 'parabolically_terminated', 'not_a_knot') and then interpolates these points accordingly.
I've written a sample program (this clearly isn't the finished function) which computes the "a" vector (a vector containing the a coefficients that define the spline) of a natural cubic spline. The end conditions for a natural cubic spline are a1 = 0 and an = 0. The code is here:
x = [-2 -1 0 1 2];
y = [4 -1 2 1 8];
n=length(x) - 1;
q = length(y) - 2;
h = x(2:n+1)-x(1:n);
v1=[h(1:n-1) 0];
v2=[1 2*(h(1:n-1)+h(2:n)) 1];
v3=[0 h(2:n)] ;
M=diag(v1,-1)+diag(v2)+diag(v3,1);
d = [0 ((y(3: q + 2) - y(2: q + 1))./h(2:q + 1) - (y(2:q + 1) - y(1:q ))./h(1:q)) 0];
a = M\d'
I'm just wondering how I would alter this code to deal with a parabolically terminated cubic spline (which has end conditions -a_1 + a_2 = 0 and -a_{n - 1} + a_n = 0)? I know that the only changes will be to the first and last rows of M and d. I'm just not sure HOW they will differ from the natural cubic spline form. Thanks.
