Question

matlab

For this problem you will test a few interpolation approaches for the application of generating interpolated data. We'll do this by interpolating data that is sampled from a known mathematical function. The same function can then be used to compute true values at the interpolated points to use in error Consider the following mathematical function (Runge's function): 1+25r2 Write a function mfile that uses this formula to generate a set of data use those points along approaches outlined below to data (x i, f i) on a specified increment. Your function should accept the following inputs (in points. Your function should with the interpolation generate interpolated 1. A vector ofx values at which fix) is to be evaluated to generate the sampled data. 2. A scalar input specifying the increment to use for the interpolated data. Your function should use the following three approaches to generate interpolated data: . A polynomial of order (N-1) where N is the number of points in the sampled data .A linear spline . A clamped cubic spline with derivative end conditions set to values equal to the exact analytical derivative of f(x) evaluated at the end points. Your function should have the following three outputs (in order): 1. The interpolated data (f i) generated with polynomial interpolation (column vector) 2. The interpolated data (f i) generated with a linear spline (column vector) 3. The interpolated data (f i) generated with the clamped cubic spline (column vector) Note this p

matlab

Answer 1

Matlab Copde

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

close all,
clear all,
clc,

x=0:1:10;

Fx=[];
for n=1:length(x)
Fx(n) = 1/(1+(25*x(n)^2));
end
subplot(3,2,1); plot(x,Fx); title('Original Data Plot');

xi = 0:0.5:10;
Fxi=[];
for n=1:length(xi)
Fxi(n) = 1/(1+(25*xi(n)^2));
end
subplot(3,2,2); plot(x,Fx); title('Test Data used for Interpolation Testing');

%Linear Interpolation
Fi_L = interp1(x,Fx,xi,'linear');
subplot(3,2,3); plot(xi,Fxi,'r',xi,Fi_L,'b'); title('Test Data (Blue) and Interpolated Data (Red) using Linear Interpolation');

%Polynomial Interpolation
p = polyfit(x,Fx,(length(x)-1));
Fi_P = polyval(p,xi);
subplot(3,2,4); plot(xi,Fxi,'r',xi,Fi_P,'b'); title('Test Data (Blue) and Interpolated Data (Red) using Polynomial Interpolation');

%Polynomial Interpolation
Fi_S = spline(x,Fx,xi);

subplot(3,2,5); plot(xi,Fxi,'r',xi,Fi_S,'b'); title('Test Data (Blue) and Interpolated Data (Red) using Cubic Spline Interpolation');

ProjectPath = pwd;
FilePath = strcat(ProjectPath,'\InterpolationData.txt');
fpt = fopen(FilePath,'wt');
fprintf(fpt,'Interpolation Data Points\n');
for r=1:length(x)
fprintf(fpt,'%5d%10.4f\n',x(r),Fxi(r));
end
fprintf(fpt,'\n\nData Used for Interpolation Testing\n');
for r=1:length(xi)
fprintf(fpt,'%5d%10.4f%10.4f%10.4f%10.4f\n',xi(r),Fxi(r),Fi_L(r),Fi_P(r),Fi_S(r));
end

InterpolatedData=[];
disp([' xi Fxi Fi_L Fi_P Fi_S']);
InterpolatedData=[InterpolatedData xi' Fxi' Fi_L' Fi_P' Fi_S']

fclose(fpt);

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

Original Data Plot Test Data used for Interpolation Testing 0.5 0.5 2 3 45 6 78 910 2 3 4 5 678910 Test Data (Blue) and Interpolated Data (Red) using Linear Interpolation Test Data (Blue) and Interpolated Data (Red) using Polynomial Interpolation 1.5 0.5 0.5 0 123 5 6 78 910 0.5 Test Data (Blue) and Interpolated Data (Red) using Cubic Spline Interpolation 0.5 0.5 9 10

Matlab Output

Warning: Polynomial is badly conditioned. Add points with distinct X
values, reduce the degree of the polynomial, or try centering
and scaling as described in HELP POLYFIT.
> In polyfit at 80
In MatlabCodes at 25
xi Fxi Fi_L Fi_P Fi_S

InterpolatedData =

0 1.0000 1.0000 1.0000 1.0000
0.5000 0.1379 0.5192 0.2285 0.3309
1.0000 0.0385 0.0385 0.0385 0.0385
1.5000 0.0175 0.0242 0.0107 -0.0207
2.0000 0.0099 0.0099 0.0099 0.0099
2.5000 0.0064 0.0072 0.0076 0.0166
3.0000 0.0044 0.0044 0.0044 0.0044
3.5000 0.0033 0.0035 0.0028 0.0005
4.0000 0.0025 0.0025 0.0025 0.0025
4.5000 0.0020 0.0020 0.0022 0.0027
5.0000 0.0016 0.0016 0.0016 0.0016
5.5000 0.0013 0.0014 0.0011 0.0011
6.0000 0.0011 0.0011 0.0011 0.0011
6.5000 0.0009 0.0010 0.0012 0.0010
7.0000 0.0008 0.0008 0.0008 0.0008
7.5000 0.0007 0.0007 0.0003 0.0007
8.0000 0.0006 0.0006 0.0006 0.0006
8.5000 0.0006 0.0006 0.0019 0.0006
9.0000 0.0005 0.0005 0.0005 0.0005
9.5000 0.0004 0.0004 -0.0070 0.0004
10.0000 0.0004 0.0004 0.0004 0.0004

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