Show that Newton's method applied to x^2-c=0 (where c>0 is...

Show that Newton's method applied to $x^{2}-c=0$ (where $c>0$ is some constant) produces the iterative scheme $x_{n+1}=\frac{1}{2}\left(x_{n}+c / x_{n}\right)$ for approximating $\sqrt{c}$. This scheme has been known for over 2000 years. To understand why it works, suppose that your initial guess $\left(x_{0}\right)$ for $\sqrt{c}$ is a little too small. How would $c / x_{0}$ compare to $\sqrt{c}$ ? Explain why the average of $x_{0}$ and $c / x_{0}$ would give a better approximation to $\sqrt{c}$.

Key Concepts

Newton's Method

Newton’s method is an iterative algorithm used for finding successively better approximations to the roots of a real-valued function. It works by linearizing the function at a current guess, using the tangent line, and then finding where this line crosses the x-axis. This method leverages the derivative to predict where the function might be zero, and under appropriate conditions, it converges rapidly to the true root.

Iterative Schemes

Iterative schemes are processes that generate a sequence of approximations to a solution, refining an initial guess through a repeated application of a specific update rule. In problems where an exact solution is difficult or impossible to compute directly, these schemes are particularly valuable because they progressively improve the approximation until a sufficiently accurate answer is reached.

Error Correction via Averaging

When approximating a square root, if the initial guess is too low, its reciprocal (scaled by the constant) becomes too high, resulting in one underestimation and one overestimation. Averaging these two values capitalizes on this error correction by balancing the under and over estimates, thereby producing a new approximation that lies closer to the actual value. This concept of averaging to correct estimation errors is a common strategy in numerical methods.

Convergence and Quadratic Convergence

The use of averaging in this context not only corrects a biased initial guess, but it also contributes to the method’s rapid convergence to the true value. In many cases, including this one, the algorithm exhibits quadratic convergence, meaning that the number of correct digits roughly doubles with each step. This property is a significant advantage in achieving high accuracy quickly.

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