Question

Explain using Matlab code but also why you used the linear system please

1 Quadrature Rule A quadrature rule is a way to approximate integrals numerically i.e. using a computer). Many such quadrature rules can be derived by solving a simple linear system. Set up a linear system and then use Matlab to find the coefficients wo, W1, W2, W3, W4, W5 such that | f(x)dx = wof(0) + wif(0.2) + w2f(0.4) + w3f(0.6) +w4f(0.8) + w5f (1) for each of f(x) = 1, 2, 22, 23, 24, 25. Show work how you found your linear system, and what the resulting system was that you solved. Give the coefficients and the resulting rule. Additionally, use the rule you have developed to approximate the integral letax dx

Answer 1

5124t 1- o2/6 3 + =ON+ O oo16 t o 2 56 S. + 0.0777 6 w+ o.327 68 wp +/04 per To solve thi-s systen Matlab

% matlab programe

x=0:0.2:1;% vector of x values

A=[[1,1,1,1,1,1];x;x.^2;x.^3;x.^4;x.^5]% coefficient matrix of the linear %system
B=[1;1/2;1/3;1/4;1/5;1/6]; % right hand side vector of the system
w=inverse(A)*B; % where w = [w0, w1, w2, w3, w4, w5]

% output

w =

   0.065972
   0.260417
   0.173611
   0.173611
   0.260417
   0.065972

% w0 =0.065972, w1= 0.260417 , w2 = 0.173611, w3 = 0.173611 , w4 =0.260417 , w5 =0.065972