Clamped Cubic Spline Interpolation

% Clamped cubic spline interpolation

 % Find the approximate value of f(1.5) from

 % (x,y)= (0,1), (1,e), (2,e^2) & (3,e^3).

 % Also f'(0)=1, f'(3)=e^3

 close all;

 clc;

 n = input('Enter n for (n+1) nodes, n:  ');

 x = zeros(1,n+1);

 a = zeros(1,n+1);

 for i = 0:n

   fprintf('Enter x(%d) and f(x(%d)) on separate lines:  \n', i, i);

   x(i+1) = input(' ');

   a(i+1) = input(' ');

 end

 fprintf('Enter f''(x(0)) and f''(x(n)) on separate lines\n');

 fp0 = input(' ');

 fpn = input(' ');

 m = n - 1;

 h = zeros(1,m+1);

 for i = 0:m

   h(i+1) = x(i+2) - x(i+1);

 end

 xa = zeros(1,n+1);

 xa(1) = 3.0 * (a(2) - a(1)) / h(1) - 3.0 * fp0;

 xa(n+1) = 3.0 * fpn - 3.0 * (a(n+1) - a(n)) / h(n);

 for i = 1:m

   xa(i+1) = 3.0*(a(i+2)*h(i)-a(i+1)*(x(i+2)-x(i))+a(i)*h(i+1))/(h(i+1)*h(i));

 end

 xl = zeros(1,n+1);

 xu = zeros(1,n+1);

 xz = zeros(1,n+1);

 xl(1) = 2.0 * h(1);

 xu(1) = 0.5;

 xz(1) = xa(1) / xl(1);

 for i = 1:m

  xl(i+1) = 2.0 * (x(i+2) - x(i)) - h(i) * xu(i);

  xu(i+1) = h(i+1) / xl(i+1);

  xz(i+1) = (xa(i+1) - h(i) * xz(i)) / xl(i+1);

 end

 xl(n+1) = h(n) * (2.0 - xu(n));

 xz(n+1) = (xa(n+1) - h(n) * xz(n)) / xl(n+1);

 c = zeros(1,n+1);

 b = zeros(1,n+1);

 d = zeros(1,n+1);

 c(n+1) = xz(n+1);

 for i = 1:n

  j = n - i;

  c(j+1) = xz(j+1) - xu(j+1) * c(j+2);

  b(j+1) = (a(j+2)-a(j+1))/h(j+1)-h(j+1)*(c(j+2)+2.0*c(j+1))/3.0;

  d(j+1) = (c(j+2) - c(j+1)) / (3.0 * h(j+1));

 end 

 fprintf('The numbers x(0), ..., x(n) are:\n');

 for i = 0:n

   fprintf(' %5.4f', x(i+1));

 end

 fprintf('\n\nThe coefficients of the spline on the subintervals are: \n');

 fprintf('     a(i)          b(i)           c(i)         d(i)\n');

 for i = 0:m 

 fprintf('%11.8f %11.8f %11.8f %11.8f\n',a(i+1),b(i+1),c(i+1),d(i+1));

 end



OutPut: 

Enter n for (n+1) nodes, n:  3
Enter x(0) and f(x(0)) on separate lines:  
 0
 1
Enter x(1) and f(x(1)) on separate lines:  
 1
 2.7
Enter x(2) and f(x(2)) on separate lines:  
 2
 2.7^2
Enter x(3) and f(x(3)) on separate lines:  
 3
 2.7^3
Enter f'(x(0)) and f'(x(n)) on separate lines
 1
 2.7^3
The numbers x(0), ..., x(n) are:
 0.0000 1.0000 2.0000 3.0000

The coefficients of the spline on the subintervals are: 

     a(i)          b(i)           c(i)         d(i)
 1.00000000  1.00000000  0.41906667  0.28093333
 2.70000000  2.68093333  1.26186667  0.64720000
 7.29000000  7.14626667  3.20346667  2.04326667

>> 

Join us on Telegram:
https://t.me/matlabirawen 

Join us on Facebook Group:
https://www.facebook.com/groups/matlabcodes