fixed point & iterative algorithm

fixed point

In mathematics, a fixed point of a function is an element of the function’s domain that is mapped to itself by the function. That means if you have a function f, a number x is a fixed point if f(x) = x.

To give a simple example, consider the function f(x) = x². The fixed points of this function are 0 and 1, because f(0) = 0 and f(1) = 1. If you input either of these numbers into the function, you’ll get the same number out.

Fixed points have important applications in many areas of mathematics and computer science. For instance, in the field of algorithms and computation, fixed point theory can be used to analyze and design iterative algorithms, which start with an initial guess and then repeatedly apply a function to get closer to the desired result. These algorithms typically converge to a fixed point, which is the solution to the problem.

iterative algorithm

An iterative algorithm is a method of solving a problem that works by repeatedly applying a function or a set of steps until a desired condition is met. This is usually done by starting with an initial value and then progressively improving it in each iteration, based on some rule or function.

Here’s a simple example of an iterative algorithm: suppose you want to find the square root of a number (N) and you decide to use the Newton-Raphson method, which is an iterative method.

1. You start with an initial guess, x₀ (it could be N/2, for instance).
2. Then, in each iteration, you apply the function x = x - (x² - N) / (2x). This function gives you a new guess, which is closer to the actual square root of N.
3. You repeat this process, using the output of the previous iteration as the input for the next, until your guess is close enough to the square root (or until it doesn’t change significantly between iterations).

Iterative algorithms are used extensively in various fields of computer science and mathematics, such as in numerical methods for solving equations, optimization problems, and in machine learning algorithms. They can be contrasted with recursive algorithms, which solve problems by breaking them down into smaller subproblems.