Which of the following fixed point iterations will converge

Which of the following fixed point iterations will converge, and why? Also if possible please give the rate of convergence.

(a) x _n+1 =cos(x _n )

(b) x _n+1 =sin(x _n )

(c) x _n+1 =tan(x _n )

Answer only please. no long explanations.

Solution

1.

g(x)=cosx

g\'(x)=-sinx

|-sinx|<=1

which may be equal to 1.hence it doesn\'t converge.

2.

g(x)=sinx

g\'(x)=cosx

|cosx|<=1

which may be equal to 1.hence doesn\'t converge.

3.

g(x) = tanx

g\'(x)=sec^2(x)

same here it may be equal to 1.so doesn\'t converge.