Natural cubic spline interpolation using MATLAB

MATLAB Program:

% Natural cubic spline interpolation

 % Find the approximate value of f(1.5) from

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

 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

 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,m+1);

 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) = 1;

 xu(1) = 0;

 xz(1) = 0;

 for i = 1:m

   xl(i+1) = 2*(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) = 1;

 xz(n+1) = 0;

 b = zeros(1,n+1);

 c = zeros(1,n+1);

 d = zeros(1,n+1);

 c(n+1) = xz(n+1);

 for i = 0:m

   j = m-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('\nThe 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:

>> Natural_spline

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

 7.29

Enter x(3) and f(x(3)) on separate lines:  

 3

 19.68

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   1.44933333   0.00000000   0.25066667 

 2.70000000   2.20133333   0.75200000   1.63666667 

 7.29000000   8.61533333   5.66200000  -1.88733333 

>>