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% 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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