Question

1. This question concerns finding the roots of the scalar non-linear function f(x) = r2-1-sinx (1 mark) (b) Apply two iterations of the bisection method to f(x) 0 to find the positive root. (3 marks) (c) Apply two iterations of the Newton-Raphson method to find the positive root. Choose (3 marks) (d) Use the Newton-Raphson method and Matlab to find the positive root to 15 significant (3 marks) (a) Use Matlab to obtain a graph of the function that shows its roots. Choose suitable starting values. Do your calculations by hand a suitable initial value. Do your calculations by hand. figures

Answer 1

a)

f=@(x) x^2-x-sin(x);
fplot(f,[-1 2]);
grid on
xlabel('x');
ylabel('f(x)');

2.5 2 1.5 3 0.5 0 -0.5 -1 0.5 0.5 1.5 0 2

(b) and (c)

bisecton medhocl. 2. 2. 2. 2 2.4チ11 .6 2 3.41

(d)

tol=10^(-15);
syms x;
s(1)=2;
f= x^2-x-sin(x);
for n=1:1000
l1(n)=subs(f,s(n));
l2(n)=subs(diff(f),s(n));
s(n+1)=s(n)-l1(n)/l2(n);
if ( abs(s(n+1)-s(n)) < tol )
break;
end
end
fprintf('roots using NR method = %1.15f \n',s(end)');

OUTPUT:

roots using NR method = 1.617545286062255