An automated tool designed to approximate solutions to equations by repeatedly applying a function. This process begins with an initial guess and iteratively refines it, each time inputting the previous result back into the function. The goal is to converge on a value that remains unchanged when the function is applied, representing a fixed point and, therefore, a solution to the equation. As an illustration, consider an equation rearranged into the form x = g(x). Starting with an initial estimate, the tool calculates g(x), then uses that result as the new input for g, repeating until the output stabilizes within a defined tolerance.
Such tools provide a valuable method for solving equations that may be difficult or impossible to solve analytically. They enable approximation of solutions in diverse fields such as engineering, economics, and physics, where complex mathematical models often arise. Historically, these iterative methods predate modern computing, but their implementation became significantly more efficient and accessible with the advent of electronic calculation. The benefit lies in its ability to provide practical solutions to otherwise intractable problems, facilitating progress in many scientific and technological areas.
