Question

1. (25 pts) Write a Matlab function with the header function [value- polynomialpiece(m,x.y,k,z) that inputs number of data points m; vectors a and y, both with m components, holding r- and y-coordinates, respec- tively, of data points, and where the components of r are evenly spaced and in increasing order (rk/2,z,n+1-k/2) an even number k and a location distinct from the 0, > z . nodes; and outputs, using Lagrange form, the value at z of the deg S k-1 Lagrange polynomial that interpolates only the k data points with nodes closest to z.

Answer 1

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AS FOR GIVEN DATA.

MATLAB:

%%Matlab example for Lagrange Interpolation function
clear all
close all
%Example for using function [value]=polynomialpiece(m,x,y,k,z)
m=100;
x1=linspace(-5,5,m);
y1=x1.^2;
%plotting of actual data
plot(x1,y1);
%interpolated data points
k=10;
z=linspace(-3,0,k);
p=polynomialpiece(m,x1,y1,10,z);
hold on
%plotting of interpolated data
plot(z,p,'linewidth',2)
title('Data plotting using Lagrange Interpolation')
xlabel('X value')
ylabel('Y value')
legend('Actual Data','Interpolated data')

%%%%%%%%%% Matlab function for Lagrange interpolation %%%%%%%%%%
function [value]=polynomialpiece(m,x,y,k,z)
    %Data points colses to z
    for i=1:k
        %finding data points close to z
        c=abs(x-z(i));
        [bb,ii]=sort(c);
        x1(i)=x(ii(1));
        y1(i)=y(ii(1));
    end
    syms xx
    %loop for creating the Lagrange Interpolation function
        for i=1:k
            s1=1;
            s2=1;
            for j=1:k
                if i~=j
                    s1=(xx-x1(j))*s1;
                    s2=(x1(i)-x1(j))*s2;
                end
             end
             zz(i)=(s1./s2)*y1(i);
        end
        f(xx)=sum(zz);
    %interpolated data using Lagrange Method
    for i=1:k
        value(i)=double(f(x1(i)));
    end  
end

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