Final answer:
The Fixed-Point method is a numerical approach used to find an approximate solution to nonlinear equations, implemented by rewriting the equation as x=g(x) and applying iterative transformation starting from an initial guess. The iterative process continues until the solution converges within a given tolerance or reaches the maximum iterations set.
Step-by-step explanation:
Implementing the Fixed-Point Method
To implement the Fixed-Point method for finding an approximate solution to a nonlinear equation f(x)=0, we can define a function fixedPointMethod with the following inputs: a function that calculates both f(x) and a transformation of x into g(x) such that x=g(x), an initial guess, a convergence tolerance, and the maximum number of iterations allowed for the process. The function would then output the approximate root after applying the method and the number of iterations that were needed to meet the convergence criterion or the maximum iterations set.
The Fixed-Point method operates on the principle that we can rewrite f(x)=0 in the form x=g(x), and then iteratively apply g(x) starting from an initial guess to get closer to the actual root. The pseudocode for this implementation could look like this:
-
- Define the function fixedPointMethod which accepts the parameters: function f_g, initial guess x0, tolerance tol, and max iterations maxIter.
-
- Create variables to hold the current approximation x, and a counter for iterations iter.
-
- The function enters a loop that will continue until the convergence criterion is met or maximum iterations are reached.
-
- In each iteration, calculate x_new = g(x) and check if the absolute difference between x_new and x is within the specified tolerance.
-
- If convergence is achieved or maximum iterations are reached, return the approximation x and the iteration count iter.