2. Consider problem
$x^4 - 4x^2 = -2$
on [1.5, 2.5]. Use the Bisection method, the Fixed-Point Iteration, Newton's method,
and Secant method to determine a solution accurate to $10^{-8}$. Use the initial condition
$p_0 = 1.5$ for the Fixed-Point Iteration, Newton's method and Secant method. As a
second initial condition for the Secant method use the first approximation obtained
using Newton's method. Before running the Fixed-Point Iteration method, make sure
(and prove theoretically) that the method converges based on your definition of the
fixed point problem.
Note: Make sure to transform the equation into a root-finding problem or a fixed point
problem, depending on the method you are using to solve it. The equation has a unique
fixed point on a given interval, so make sure to define your functions correctly.
showing your choice of $f$ for $f(x) = 0$ problems, and $g$ for $g(x) = x$
problems, as well as the calculations showing that your choice of $g$ satisfies conditions
required for the fixed point iteration to converge. Write the approximations of the
solution of the problem obtained using each method. Compare the number of iterations
needed by each algorithm.
