Question

1. (25 pts) Given the following start for a Matlab function: function [answer] = NewtonForm(m,x,y,z) that inputs • number of data points m; • vectors x and y, both with m components, holding x- and y-coordinates, respectively, of data points; • location z; and uses divided difference tables and Newton form to output the value of the Lagrange polynomial, interpolating the data points, at z.

1. (25 pts) Given the following start for a Matlab function: function [answer] NewtonForm(m.x.yz) that inputs » number of data points m; vectors x and y, both with m components, holding r- and y-coordinates, respec tively, of data points; location 2 and uses divided difference tables and Newton form to output the value of the Lagrange polynomial, interpolating the data points, at

Answer 1

clear all close all %x and y values for finding the %function for which interpolation have to do func-e (x) 1./(1+25.x.*2); x1-1 in space (-1 , 1 ,2 1 ) ; yl-func (x1); all interpolating points zl-linspace(-1,1,101) edisplaying the function fprintf The function is f (x) disp(func) %Function for Newton Divide difference [Value] NewtonForm (21,x1,y1,z1); figure(1) scatter(x1,yl, 'filled , 'Linewidth',2) hold orn plot (z1,Value, 'r' xlabel ( ' x' ) ylabel(x) title( 'x vs. f(x) plot') legend( Actual data', Interpolating data') %Function for Newton Divided difference interpolation function [Value]-NewtonForm(m,xl,yl, zl) %creat in the Newton Divided difference formu lae for variable x symsX y2-yl; 2z-zeros (m,m+1); for i-1:m-1 n1-length (y2); for j-l:nl-l y3 (j) (y2 (j+1)-y2(j))/(x1 (i+j)-x1(j) ); end Y2-y3; clear y3; end nn-length (zz);

loop for creating the function n-length (z); for i=1:n s1-1; for j= 1 i s1-(x-x1)) s1 end end f_newton (x)-sum(2z1) tyl(l) Value-double(f_newton(z1)); end 888888888888옹 End of Code 88888888888888 The function is f(x)- x vs. fx) plot 10 ● Actual data data 10 -30 -50 -60 08 0.6 -04 0.2 0.2 0.4 0.6 0.8 Published with MATLAB R2018a

clear all
close all
%x and y values for finding the
%function for which interpolation have to do
func=@(x) 1./(1+25.*x.^2);
x1=linspace(-1,1,21);
y1=func(x1);

%all interpolating points
z1=linspace(-1,1,101);

%displaying the function
fprintf('The function is f(x)=\n')
disp(func)

%Function for Newton Divide difference
[Value]=NewtonForm(21,x1,y1,z1);

figure(1)
scatter(x1,y1,'filled','Linewidth',2)
hold on
plot(z1,Value,'r')

xlabel('x')
ylabel('f(x)')
title('x vs. f(x) plot')
legend('Actual data','Interpolating data')

%Function for Newton Divided difference interpolation
function [Value]=NewtonForm(m,x1,y1,z1)
  
    %Creatin the Newton Divided difference formulae for variable x
    syms x
    y2=y1;
    zz=zeros(m,m+1);

    zz(:,1)=x1'; zz(:,2)=y1';

    for i=1:m-1

        n1=length(y2);
        for j=1:n1-1
            y3(j)=(y2(j+1)-y2(j))/(x1(i+j)-x1(j));
            zz(j,i+2)=y3(j);
        end
        z(i)=y3(1);
        y2=y3;
        clear y3;
    end

    nn=length(zz);

    %loop for creating the function

    n=length(z);
        for i=1:n
            s1=1;
            for j=1:i
                    s1=(x-x1(j))*s1;
             end
             zz1(i)=(s1)*z(i);
        end
    f_newton(x)=sum(zz1)+y1(1);
    Value=double(f_newton(z1));
  
end

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