Question

3. Consider the function f(x) = cos(x) in the interval [0,8]. You are given the following 3 points of this function: 10.5403 2 -0.4161 6 0.9602 (a) (2 points) Calculate the quadratic Lagrange interpolating polynomial as the sum of the Lo(x), L1(x), L2(x) polynomials we defined in class. The final answer should be in the form P)a2 bx c, but with a, b, c known. DELIVERABLES: All your work in constructing the polynomial. This is to be done by hand not MATLAB (b) (2 points) Plot f(x) cos() in the interval [0, 8] by creating vectors x 0:0, 1:8; y cos (x) ; Use the MATLAB hold on command to retain the plot and plot the three data points provided above as stars. There is a similar example in the Lecture Notes for lecture 15. To plot the first star for example use plot(1, 0.5403, *) DELIVERABLES: The commands and the resulting plot from MATLAB (c) (2 points) Plot each polynomial Lo(x)/(20),L,(z)/(x),L2(x)/(#2) for these specific values using the same axis as part (b) x = 0:0.1:8 Note: Use fplot for this part DELIVERABLES: The commands and the resulting plot from MATLAB. (d) (2 points) Make a new figure with f(x) and the points as stars as well as the sum of Lo(x)f (xo), Li()f(x), L2(x)f(r2) using the same x = 0:0.1:8 but without showing Lo(x), L1 (x), L2(x) DELIVERABLES: The commands and the resulting plot from MATLAB.

Answer 1

mkupolotio polyn Lio) 17 20-M Hesu 2 0.4 16 I L20n G)-2) 0- S403

Matlab code for Lagrange polynomial clear all close all %All given data point for x and y xx-[1 2 6]; yy-[0.5403 -0.4161 0.9602] tanswering question b. %function for which interpolation have to do f- (x) cos (x) tall given x data and y data x1-0:0.1:8; yl-f (x1); %plotting data hold on plot (xl,yl, 'Linewidth',2) plot (xx, yy, r*') xlabel(x ylabel(f(x) title 'f (x) vs. x plot') legend ( 'Actual data', 'Given data point') answering question c %function for all L(x) し0-e (x) ((x-2 ) ."(x-6 ) ) ./5; し2-e (x) ((x-1)."(x-2 ) ) ./20; figure (2) hold on plot (x1,LO (x1) *f (xx(1)), 'Linewidth',2) plot (xl, Ll (xl).*f (xx (2)), 'Linewidth',2) plot (x1, L2 (x1) .*f (xx (3)), Linewidth',2) legend('LO (x), Ll (x), L2 (x)') xlabel (·x' ) ylabel'L(x)) title('L(x) vs. x plot') answering question d syms x summ (x)-L0 (x)*f(xx (1))+L1 (x) *f (xx (2))+L2 (x)*f (xx (3)); for i-l:length(xl) y2 (i)-double (summ(xl(i))); end %displaying Lagrange function fprintf (displaying Lagrange function\n) disp (vpa (summ, 2)) figure (3) hold on

plot (x1,yl, Linewidth',2) plot (xx, yy,r* ') plot (x1,y2, Linewidth',2) legend'Actual function, 'given data point', 'Lagrange interpolation') titlex vs. f(x) plot' ,,8888888888,,88 End of Code 88888888888888움 displaying Lagrange function 0.048*(x - 1.0)*(x-2.0) + 0.1*(x - 1.0)*(x-6.0) + 0.11*(x - 2.0) (x- 6.0) symbolic function inputs: x f(x) vs. x pot Actual data Given data point 0.8 0.6 0.4 02 0.2 0.4 0.6 0.8 0 6

L(x) vs. x plot 2.5 -LI(x) 1.5 0.5 0.5 xvs。t(x) plot Actual function given data point nterpolation

%%Matlab code for Lagrange polynomial
clear all
close all
%All given data point for x and y
xx=[1 2 6];
yy=[0.5403 -0.4161 0.9602];

%answering question b.

%function for which interpolation have to do
f=@(x) cos(x);
%all given x data and y data
x1=0:0.1:8;
y1=f(x1);

%plotting data
hold on
plot(x1,y1,'Linewidth',2)
plot(xx,yy,'r*')
xlabel('x')
ylabel('f(x)')
title('f(x) vs. x plot')
legend('Actual data','Given data point')
%answering question c.
%function for all L(x)
L0=@(x) ((x-2).*(x-6))./5;
L1=@(x) -((x-1).*(x-6))./4;
L2=@(x) ((x-1).*(x-2))./20;

figure(2)
hold on
plot(x1,L0(x1).*f(xx(1)),'Linewidth',2)
plot(x1,L1(x1).*f(xx(2)),'Linewidth',2)
plot(x1,L2(x1).*f(xx(3)),'Linewidth',2)
legend('L0(x)','L1(x)','L2(x)')
xlabel('x')
ylabel('L(x)')
title('L(x) vs. x plot')

%answering question d.
syms x
summ(x)=L0(x)*f(xx(1))+L1(x)*f(xx(2))+L2(x)*f(xx(3));
for i=1:length(x1)
    y2(i)=double(summ(x1(i)));
end

%displaying Lagrange function
fprintf('displaying Lagrange function\n')
disp(vpa(summ,2))

figure(3)
hold on
plot(x1,y1,'Linewidth',2)
plot(xx,yy,'r*')
plot(x1,y2,'Linewidth',2)
legend('Actual function','given data point','Lagrange interpolation')
title('x vs. f(x) plot')

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