Question

Using the Excel, fit a second-order polynomial to the data in Table Q2. Using the MATLAB. findfX) at x= 5.4 using the Lagrange and second order method. + Q2 Table Q2 2.5-| 3.5-| 5+' | 6' | 7.5-1 1001 12.5-1+' Vie | 70+ | 5.5- ו3.9 '| 3.6e | 3.1-וכ2.8 -וכ2.6 -וכ+'

Answer 1

Matlab code for Lagrange interpolating Polynomial clear all close all xi-2.5 3.5 5 6 7.5 10 12.5]; yi-7.0 5.5 3.9 3.6 3.1 2.8 2.61 plot (xi, yǐ, ' r* ' , . linewidth' , 5) lagrange interpolating polynomial for yi n-length(xi)-1; syms x S-0; for i-1:n+1 L-li for j-1: n+1 if j--i し=し* (x-xi(j) )/ ( xi ( i)-xi(j)); end end s-Styǐ (i) *L; end Printing the polynomial S-expand (S); fprintf (n tThe interpolating function y(x) using Lagrange polynomia isln' disp(vpa(S,2)) Creating the function syms x 1 (x)-S; hold on x1-2.5:0.1:12.5; for i= 1:leng th ( x 1) for that polynomial y1 (і,-double ( ї ( x1 ( i))); end plot (x1,yl, 1inewidth',2) xlabel ( ·x' ) ylabel ('E(x)' title( Plotting of f(x) vs. x') %Finding 2nd order polynomial matrix for 2nd order polynomial for i= 1: length ( xi ) end coefficient c=A\ b ; matrix

fprintf(ntThe interpolating function y(x) using 2nd order polynomial is\n') disp vpa (P,2)) x2-2.5:0.1:12.5; for i-l:length (x2) y2(i)-double (P ( x2 ( i))); plot (x2,y2, 'linewidth',2) legend 'Actual data',Lagrange fit', '2nd order polynomial') fprint f ("Atvalue of f(x) at x-5.4 using Lagrange înterpolation is %f. n,double (1(5.4))) fprint f ( '\tvalue of f(x) at x-5.4 using 2nd order interpolation is %f. n',double (P(5.4))) 888888888888888888888888 End of Code 옹88888888888888888888응 The interpolating function y(x) using Lagrange polynomial is -7.0e-4*x"6 + 0.029*x"5-0.48*x"4 + 4.0*x^3-17.0*x"2 + 35.0*x- 19.0 The interpolating function y(x) using 2nd order polynomial is 0.073*x21.5*x 9.9 symbolic function inputs: x Value of f(x) at x-5.4 using Lagrange interpolation is 3.743216. value of f(x) at x=5.4 using 2nd order interpolation is 4.061513

Plotting of f(x) vs. x Actual data Lagrange fit 6.5 -2nd order 5.5 4.5 3.5 2.5 10 12 14 Published with MATLAB R2018a

%%Matlab code for Lagrange interpolating Polynomial
clear all
close all
xi=[2.5 3.5 5 6 7.5 10 12.5];
yi=[7.0 5.5 3.9 3.6 3.1 2.8 2.6];
plot(xi,yi,'r*','linewidth',5)
%lagrange interpolating polynomial for yi
n=length(xi)-1;
syms x;
S=0;
for i=1:n+1
    L=1;
    for j=1:n+1
        if j~=i
            L=L*(x-xi(j))/(xi(i)-xi(j));
        end
    end
    S=S+yi(i)*L;
end
%Printing the polynomial
S=expand(S);
fprintf('\n\tThe interpolating function y(x) using Lagrange polynomial is\n')
disp(vpa(S,2))
%Creating the function for that polynomial
syms x
l(x)=S;
hold on
x1=2.5:0.1:12.5;
for i=1:length(x1)
    y1(i)=double(l(x1(i)));
end

plot(x1,y1,'linewidth',2)
xlabel('x')
ylabel('f(x)')
title('Plotting of f(x) vs. x')

%Finding 2nd order polynomial
%matrix for 2nd order polynomial

for i=1:length(xi)
    A(i,1)=1;
    A(i,2)=xi(i);
    A(i,3)=(xi(i)).^2;
  
    b(i,1)=yi(i);
end

%coefficient matrix
c=A\b;

P(x)=c(1)+c(2)*x+c(3)*x^2;

fprintf('\n\tThe interpolating function y(x) using 2nd order polynomial is\n')
disp(vpa(P,2))

x2=2.5:0.1:12.5;
for i=1:length(x2)
    y2(i)=double(P(x2(i)));
end

plot(x2,y2,'linewidth',2)
legend('Actual data','Lagrange fit','2nd order polynomial')

fprintf('\tValue of f(x) at x=5.4 using Lagrange interpolation is %f.\n',double(l(5.4)))
fprintf('\tValue of f(x) at x=5.4 using 2nd order interpolation is %f.\n',double(P(5.4)))

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