KMU206 Numerical Analysis Instructor: Dr. Selis Önel HOMEWORK 5 Print and submit to the course instructor

Q1. Consider data points (0,0), (1,1), (2,4), (3,3) and apply piecewise polynomial interpoplation:

§ Set up the linear system of equations using piecewise quadratic interpolation

§ Place knots at the midpoints of the data intervals

§ Show all your equations

§ Set up the linear equations in matrix form [A][c]=[y]

§ Determine the coefficients [c] of the linear system ( [c]=[A]\[y] )

§ Write the interpolating piecewise polynomial for each data interval

§ Write a script in MATLAB® to plot a graph of the interpolating polynomials in each interval

§ Copy the script and the graph on a word document, print and submit with the homework

   For example:

%plotting the quadratic piecewise polynomial solution

format short

syms x

xi=[0;1;2;3]; yi=[0;1;4;3];

A=[1 -1 0 0 2 0; 0 1 -1 0 2 2; 0 0 1 -1 0 2; 1 1 0 0 -1 0; 0 1 1 0 1 -1; 0 0 1 1 0 1];

r=[4;12;-4;0;0;0];

c=A\r;  %since b1=0 and b4=0:

c=[c(1); c(2); c(3); c(4); 0; c(5); c(6); 0];

n=length(xi);

for k=1:n,

if k<n, h(k)=(xi(k)+xi(k+1))/2;end

   a(k)=c(k); b(k)=c(k+n); C(k)=yi(k);

   P(k)=c(k)*(x-xi(k))^2+b(k)*(x-xi(k))+C(k);

end

h1=[xi(1):.01:h(1)];  h2=[h(1):.01:h(2)];

h3=[h(2):.01:h(3)];  h4=[h(3):.01:xi(4)];

plot(h1,subs(P(1),h1),h2,subs(P(2),h2),h3,…

       subs(P(3),h3),h4,subs(P(4),h4),xi,yi,'*m')

legend('P1','P2','P3','P4','Data points',4);

xlabel('x'); ylabel('y');

grid on