Question

matlab

The error function is a mathematical function that frequently arises in probability and statistics. It also can show up in the solution to some partial differential equations, particularly those arising in heat and mass transfer applications. The error function is defined as 2 e-t dt picture attached This function is actually built-in to MATLAB as the command erf, and here we'll use that function to compute a "true value" with which we can compare results of two interpolation approaches. Note that this value in MATLAB is actually the result of a numerical integration since the integral defining the error function cannot be solved analytically. More on that in week 7… For this problem consider the following tabulated valucs of the error function attached table 0.5 1.0 1.5 2.0 0.0 0.9953 erf(x)0.0 0.5205 0.8427 0.9661 Write a function mfile tha t accepts a single, scalar input for x and uses these tabulated values to calculate and output the following fou results (in order and all scalars) 1. An interpolated value for erfx) using a 2nd order Lagrange interpolating polynomial. 2. The true error associated with the first output 3. An interpolated value for erfix) generated using a spline with "not-a-knot" end conditions 4. The true error associated with the third output. Notes: For the Lagrange interpolating polynomial, you will need to formulate the polynomial by hand and enter it as a mathematical expression in your function. Make sure your function always uses the three data points closest to the input value for the Lagrange interpolation. Use MATLAB's built-in erf(x) function to compute the true value for the error calculations.

Answer 1

`Hey,

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% a. Define the necessary symbolic variables.
clear, clc
[L,LE,S,SE]=interpolated(1)

function [L,LE,S,SE]=interpolated(x)
XX=[0:0.5:2];
Y=[0,0.5205,0.8427,0.9661,0.9953];
min=inf;
ind=-1;
for i=1:length(XX)-2
if(min>norm(XX(i:i+2)-x))
ind=i;
end
end
L=lagrangeInterp(XX(ind:ind+2),Y(ind:ind+2),x);
LE=abs(L-erf(x));
S=interp1(XX,Y,x,'spline');
SE=abs(S-erf(x));
function [yhat]=lagrangeInterp(x,y,interpx)

X=x;
f=y;
x=interpx;

l=0;
for i=1:length(X)
li=1;
for j=1:length(X)
if i~=j
li=(li).*((x-X(j))/(X(i)-X(j)));
end
end
l=(l)+((li)*f(i));
end
yhat=l;
  
end
end

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