Newton iteration formula for square root approximation

Key concept: This is the classic Newton or Babylonian iteration for square roots. The goal is to write the recursive update rule clearly and interpret its purpose in numerical approximation.

Step by step

Recognize the iteration rule

The method shown is an iterative formula for approximating $\sqrt{y_n}$. The update rule is

$x_{n+1}=\frac{1}{2}\left(x_n+\frac{y_n}{x_n}\right).$

This formula takes the current estimate $x_n$ and improves it by averaging two values:

1. the estimate itself, $x_n$
2. the quotient $\frac{y_n}{x_n}$

That average is what drives the next estimate closer to the true square root.

State the formula in standard form

If the question asks for the iteration formula, the answer is simply

$\boxed{x_{n+1}=\frac{x_n+\frac{y_n}{x_n}}{2}}.$

This is often called the Babylonian method or Newton's method for square roots. It is one of the most important numerical algorithms because it converges quickly when the starting guess $x_0$ is reasonable and positive.

Why the formula works

If $x_n$ is too large, then $\frac{y_n}{x_n}$ is too small; if $x_n$ is too small, then $\frac{y_n}{x_n}$ is too large. Averaging those two quantities balances the error. For positive $y_n$, the method repeatedly corrects the estimate until it stabilizes near $\sqrt{y_n}$.

Key properties to remember

The method assumes $x_n \neq 0$ because division by zero is undefined. In practical computation, the starting value should also be positive if the target is a real square root. The iteration is especially efficient because each step roughly doubles the number of correct digits once it is close enough to the answer.

How to present it on homework or a test

If the prompt says "State the iteration formula," do not add extra derivation unless requested. Write the recurrence exactly and, if needed, name the method. A clear response is:

$x_{n+1}=\frac{1}{2}\left(x_n+\frac{y_n}{x_n}\right).$

That is the full iteration statement.

Pitfall alert

A common mistake is to swap the roles of $x_n$ and $y_n$ and write $\frac{x_n}{y_n}$, which is not the square-root iteration. Another error is forgetting the factor of $\frac{1}{2}$, which is essential because the formula is an average. Some students also try to substitute a numerical value for $y_n$ when the question only asks for the iteration rule. If the prompt asks for the formula, keep it symbolic and exact.

Try different conditions

If the same method were written for a fixed number $a$ instead of $y_n$, the iteration would become $x_{n+1}=\frac{1}{2}\left(x_n+\frac{a}{x_n}\right).$ For example, to approximate $\sqrt{10}$, you would use $x_{n+1}=\frac{1}{2}\left(x_n+\frac{10}{x_n}\right).$ If the task asked for an iteration to solve cube roots instead, the update rule would change completely and would not use this averaging formula. The square-root version is specific to quadratic convergence for $\sqrt{\,\cdot\,}$ problems.

Further reading

Babylonian method, Newton's method, iterative approximation