Fixed-point iteration

Fixed-point iteration is like playing a guessing game with numbers. You have a special number that's called a fixed point, and you try to find another number that when you do some math with it, it gets closer and closer to the fixed point.

It's like when you play a game with a dog and you throw a ball, and the dog runs to catch it, but it gets closer and closer to where you are standing. In the same way, we start with a number and do some math with it to get a new number that is closer to the fixed point.

Say we are trying to find the square root of a number. We pick a starting number, let's say 4, and we do some math with it to get a new number, which is closer to the true square root. We take the average of the original number and the starting number divided by the original number, and we get a new number, let's say 2.5. Then we do the same thing again, we take the average of 2.5 and the original number divided by 2.5, and we get 2.05. We keep doing this again and again until we get very, very close to the true square root of the number.

So fixed-point iteration is like trying to find the target with a bow and arrow; you start by shooting the arrow in a direction that gets it closer to the target and then repeat the process, each time getting closer to the target. Similarly, we start with a number and do some math with it to get closer to the fixed point with each new number, until we get as close as possible.