Question

8. (Matlab) Suppose we have a function "hw5f.m" that takes as input r and outputs the value for a function f(x). Write a Matlab program that inputs: interval la, b ·m, the number of data points with evenly spaced nodes from xi = a to xm-b, and values from f(x); . location z satisfying x2 < z < x'n-l, where h = (b-a)/(m-1); and outputs the value of the interpolaton polynomial using only the four data points (a) Write out or print out your program. You may find Matlab's "Aloor" or "ceil" commands to be helpful. (b) Apply your program to the case m = 101 and r, evenly spaced, from-π to π, and yi-sina, and write out or print out your results when s =-3,-1,05,2

(Matlab) Suppose we have a function “hw5f.m” that takes as input x and outputs the value for a function f(x). Write a Matlab program that inputs: • interval [a, b]; • m, the number of data points with evenly spaced nodes from x1 = a to xm = b, and values from f(x); • location z satisfying x2 < z < xm−1, where h = (b − a)/(m − 1); and outputs the value of the interpolaton polynomial using only the four data points with nodes closest to z. 2 (a) Write out or print out your program. You may find Matlab’s “floor” or “ceil” commands to be helpful. (b) Apply your program to the case m = 101 and xi evenly spaced, from −π to π, and yi = sin xi , and write out or print out your results when z = −3, −1, 0.5, 2.

Answer 1

z-2; [PP]-ppoly4 (f,interval,m,z); fprintfFor 2-82.2f the function value becomes P (x)-,z) disp (vpa(PP, 3)) for i-1length(x1) y2(1)-double (PP ( xl (i))); end plot(x1,y2) ylim(-2 2]) xlabel ( ·x' ) ylabel('Y (X) sin(X)) title ('x vs. y(x) plot') legend( Actual data,'interp for z-3",. . . interp for z--1, interp for z-0.5,interp for z-2) %function for interpolating polynomial function [PP]-ppoly4 (f,interval,m,z) f is function for which polynomial have to find interval is data interval [a,b] m is number of data points %z is data point around which polynomial have to find a-interval (1); b-interval(2); h= (b-a ) / (m-1); %all equidistant nodes x-a: h:b; %function values corresponding y=f(x); x %finding 4 data point close to z c-abs (x-z); Ibb, i]-sort(c) four data points arou nd which interpolation have to do for j-l:4 %finding polynomial intepolation p-polyfit (xx,yy, 3) syms x PP (x)- p (1)x 3 + p(2) *x*2 + p (3)*x + p(4) end

,,,,움움움응움움움응움움움응움움움움움움움움움움움움움움各움움움움움움움움움움움움움움움움움움움움움움움움움움움움움움各各움움움 For z-3.00 the function value becomes P(x)-0.165*x*3 + 1.55 *x2 + 3.88*x + 1.97 symbolic function inputs: x For z-1.00 the function value becomes P(x)0.0936 *x*3 1.1 *x 0.0257 0.14*x*2+ symbolic function inputs: x z=0.50 the function value becomes 0.148*x^3-0.017*x"2 For P(x)-- + 1.01*x- 7.21e-4 symbolic function inputs: x function value P(x)=0.0662*x^3-0.851*x^2 For z=2.00 the becomes + 2.2*x0.605 symbolic function inputs: x x vs. y(x) plot Actual data -interp tr 2.-3 interp for - interp for 2-0.5 1.5 0.5 0.5 1.5 -3 -2 Published with MATLAB® R2018a

%%Matlab code for interpolating polynomial
clear all
close all

%function for which interpolation have to do
f=@(x) sin(x);
interval=[-pi,pi];
m=101;
z=-3;

[PP]=ppoly4(f,interval,m,z);
fprintf('For z=%2.2f the function value becomes P(x)=',z)
disp(vpa(PP,3))

%plotting the function
x1=linspace(-pi,pi,101);
y1=f(x1);
plot(x1,y1,'linewidth',2)

for i=1:length(x1)
    y2(i)=double(PP(x1(i)));
end

hold on
plot(x1,y2)

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
z=-1;

[PP]=ppoly4(f,interval,m,z);
fprintf('For z=%2.2f the function value becomes P(x)=',z)
disp(vpa(PP,3))

for i=1:length(x1)
    y2(i)=double(PP(x1(i)));
end

plot(x1,y2)

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
z=0.5;

[PP]=ppoly4(f,interval,m,z);
fprintf('For z=%2.2f the function value becomes P(x)=',z)
disp(vpa(PP,3))

for i=1:length(x1)
    y2(i)=double(PP(x1(i)));
end
plot(x1,y2)

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
z=2;

[PP]=ppoly4(f,interval,m,z);
fprintf('For z=%2.2f the function value becomes P(x)=',z)
disp(vpa(PP,3))

for i=1:length(x1)
    y2(i)=double(PP(x1(i)));
end

plot(x1,y2)
ylim([-2 2])
xlabel('X')
ylabel('Y(X)=sin(X)')
title('x vs. y(x) plot')
legend('Actual data','interp for z=-3',...
    'interp for z=-1','interp for z=0.5','interp for z=2')

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%function for interpolating polynomial
function [PP]=ppoly4(f,interval,m,z)
      
        %f is function for which polynomial have to find
        %interval is data interval [a,b]
        %m is number of data points
        %z is data point around which polynomial have to find
      
        a=interval(1); b=interval(2);
        h=(b-a)/(m-1);
        %all equidistant nodes
        x=a:h:b;
        %function values corresponding x
        y=f(x);
      
        %finding 4 data point close to z
        c=abs(x-z);
        [bb,i]=sort(c);
      
        %four data points around which interpolation have to do
        for j=1:4
            xx(j)=x(i(j));
            yy(j)=y(i(j));
        end
      
        %finding polynomial intepolation
        p=polyfit(xx,yy,3);
        syms x
      
        PP(x)= p(1)*x^3 + p(2)*x^2 + p(3)*x + p(4);
      
end
      
      
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%