Write a MATLAB function to implement the composite Gauss quadrature. The input is(f, a, b, m ,n)wheref, a, b, mhave the same meaning as in problem 3, andnis the degree of the interpolating polyno-mial in each subinterval. Use the gauss.m file on Canvas to generate the nodes and weights for the Gaussquadrature. Usen= 2

to do part (a).

What order of accuracy do you observe?

Now do (b),

now what are the digits that you believe are correct?

Code for the gauss.m file mentioned in the problem:

function [x,w]=gauss(a,b,N)

% gauss.m

%

% This script is for computing definite integrals using Legendre-Gauss

% Quadrature. Computes the Legendre-Gauss nodes and weights on an

interval

% [a,b] with truncation order N

%

% Suppose you have a continuous function f(x) which is defined on [a,b]

% which you can evaluate at any x in [a,b]. Simply evaluate it at all of

% the values contained in the x vector to obtain a vector f. Then compute

% the definite integral using sum(f.*w);

%

% Written by Greg von Winckel - 02/25/2004

N=N-1;

N1=N+1; N2=N+2;

xu=linspace(-1,1,N1)';

% Initial guess

y=cos((2*(0:N)'+1)*pi/(2*N+2))+(0.27/N1)*sin(pi*xu*N/N2);

% Legendre-Gauss Vandermonde Matrix

L=zeros(N1,N2);

% Derivative of LGVM

Lp=zeros(N1,N2);

% Compute the zeros of the N+1 Legendre Polynomial

% using the recursion relation and the Newton-Raphson method

y0=2;

% Iterate until new points are uniformly within epsilon of old points

while max(abs(y-y0))>eps

L(:,1)=1;

Lp(:,1)=0;

L(:,2)=y;

Lp(:,2)=1;

for k=2:N1

L(:,k+1)=( (2*k-1)*y.*L(:,k)-(k-1)*L(:,k-1) )/k;

end

Lp=(N2)*( L(:,N1)-y.*L(:,N2) )./(1-y.^2);

y0=y;

y=y0-L(:,N2)./Lp;

end

% Linear map from[-1,1] to [a,b]

x=(a*(1-y)+b*(1+y))/2;

% Compute the weights

w=(b-a)./((1-y.^2).*Lp.^2)*(N2/N1)^2;

(a) Test your code on J 2 xe-rcos2rdz 3(e-2x 1) 10??2 Use a for loop to calculate the error for m = 2, 22, 23, . . . , 28 and empirically calculate the order of accuracy. Put m, the error and the order as columns in a matrix and display your results. (a) Test your code on J 2 xe-rcos2rdz 3(e-2x 1) 10??2 Use a for loop to calculate the error for m = 2, 22, 23, . . . , 28 and empirically calculate the order of accuracy. Put m, the error and the order as columns in a matrix and display your results