EMPLOY FIXED-POINT ITERATION TO LOCATE THE ROOT OF F(X)sin?

Answer 1

To locate the root of the function ( F(x) ) using fixed-point iteration, we first need to rearrange the equation ( F(x) = 0 ) into the form ( x = g(x) ). We then choose an initial guess ( x_0 ) and iteratively apply the function ( g(x) ) to get the next approximation: ( x_{n+1} = g(x_n) ). This process continues until the difference between successive approximations is sufficiently small, indicating convergence to the root. It's essential to ensure that the chosen function ( g(x) ) leads to convergence, typically requiring that ( |g'(x)| < 1 ) in the neighborhood of the root.