Exer. 65-66: Some calculators use an algorithm similar to...

Exer. $65-66:$ Some calculators use an algorithm similar to the following to approximate $\sqrt{N}$ for a positive real number $N:$ Let $x_{1}=N / 2$ and find successive approximations $x_{2}, x_{3}, \ldots$ by using $$ x_{2}=\frac{1}{2}\left(x_{1}+\frac{N}{x_{1}}\right), \quad x_{3}=\frac{1}{2}\left(x_{2}+\frac{N}{x_{2}}\right), \quad \dots $$ until the desired accuracy is obtained. Use this method to approximate the radical to six-decimal-place accuracy. $$\sqrt{5}$$

Exer. $65-66:$ Some calculators use an algorithm similar to the following to approximate $\sqrt{N}$ for a positive real number $N:$ Let $x_{1}=N / 2$ and find successive approximations $x_{2}, x_{3}, \ldots$ by using
$$
x_{2}=\frac{1}{2}\left(x_{1}+\frac{N}{x_{1}}\right), \quad x_{3}=\frac{1}{2}\left(x_{2}+\frac{N}{x_{2}}\right), \quad \dots
$$
until the desired accuracy is obtained. Use this method to approximate the radical to six-decimal-place accuracy.
$$\sqrt{5}$$

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Newton's Method

A more general technique for finding roots of functions, Newton's Method applies to a wide range of equations. The Babylonian method for square roots is a specific case of Newton's Method applied to the function f(x) = x² - N, where the iterative update formula quickly converges to the correct square root.

Convergence and Error Tolerance

In numerical methods, convergence refers to the process of approaching a final value as iterations proceed. Error tolerance is the predetermined threshold used to decide when the iterative process should stop, ensuring that the solution is accurate to within a specified margin, such as six-decimal-place precision.

Babylonian Method

Also known as the method of averaging, this algorithm is one of the oldest techniques for finding square roots. It works by iteratively averaging an initial guess with the quotient of the number and the guess, rapidly converging to the true square root.

Iterative Approximation

This concept involves generating a sequence of values that progressively get closer to the desired solution. Each successive value is computed based on the previous one using a defined recurrence relation, ensuring that the process refines the approximation until an acceptable level of accuracy is reached.

Transcript

Please give Ace some feedback