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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

>> 



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