Question

Find the Newton formula for approximating \(1 / \sqrt{a}\). What is the remarkable difference to the Newton formula for \(\sqrt{a}\) ?

Short answer

The main difference is the convergence approach: \( \sqrt{a} \) uses averaging, while \( \frac{1}{\sqrt{a}} \) adjusts using a linear factor.

Step-by-step solution

Understand the Problem

We aim to derive the Newton-Raphson formula for approximating the reciprocal of the square root of a number, specifically for finding \( \frac{1}{\sqrt{a}} \), and compare it with the Newton-Raphson method for approximating \( \sqrt{a} \).

Recall Newton's Method Formula

Newton's Method formula is given by \( x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \). We need to find appropriate functions \( f(x) \) for both \( \frac{1}{\sqrt{a}} \) and \( \sqrt{a} \).

Choose Function for \( \frac{1}{\sqrt{a}} \)

Define the function \( f(x) = x^{-2} - \frac{1}{a} \). The root of this function corresponds to the value of \( x \) which makes \( f(x) = 0 \).

Compute Derivative for \( \frac{1}{\sqrt{a}} \)

The derivative of \( f(x) = x^{-2} - \frac{1}{a} \) is \( f'(x) = -2x^{-3} \).

Apply Newton's Method for \( \frac{1}{\sqrt{a}} \)

Substitute \( f(x_n) \) and \( f'(x_n) \) into Newton's method to get the formula: \[ x_{n+1} = x_n - \frac{x_n^{-2} - \frac{1}{a}}{-2x_n^{-3}} = x_n + \frac{1}{2}x_n[1 - ax_n^2] \].

Choose Function for \( \sqrt{a} \)

For \( \sqrt{a} \), use the function \( g(x) = x^2 - a \). The root of \( g(x) \) corresponds to the value of \( x \) such that \( x^2 = a \).

Compute Derivative for \( \sqrt{a} \)

The derivative of \( g(x) = x^2 - a \) is \( g'(x) = 2x \).

Apply Newton's Method for \( \sqrt{a} \)

Substitute \( g(x) \) and \( g'(x) \) into Newton's method to get the formula:\[ x_{n+1} = x_n - \frac{x_n^2 - a}{2x_n} = \frac{1}{2}(x_n + \frac{a}{x_n}) \].

Compare the Two Methods

The Newton formula for \( \sqrt{a} \) is \( x_{n+1} = \frac{1}{2}(x_n + \frac{a}{x_n}) \) whereas for \( \frac{1}{\sqrt{a}} \) it is \( x_{n+1} = x_n + \frac{1}{2}x_n[1 - ax_n^2] \). The method for \( \frac{1}{\sqrt{a}} \) adjusts \( x_n \) based on a deviation from \( 1 \), rather than combining \( x_n \) and \( \frac{a}{x_n} \) as in finding \( \sqrt{a} \).

Identify Remarkable Difference

The remarkable difference is that the formula for \( \sqrt{a} \) converges by averaging, whereas the formula for \( \frac{1}{\sqrt{a}} \) corrects based on a linear factor, gauging against the square of the current approximation.

Key concepts

Reciprocal Square Root Approximation

Understanding the concept of Reciprocal Square Root Approximation is essential in numerical methods and computational mathematics. The aim here is to approximate \( \frac{1}{\sqrt{a}} \) using the Newton-Raphson Method. This technique is widely used because it helps in efficiently computing operations that are computationally expensive, like division and taking square roots, especially within the context of graphics programming and scientific computing.

The idea is to consider a function whose root corresponds to the reciprocal square root you want to find. We define the function \( f(x) = x^{-2} - \frac{1}{a} \). The solution to \( f(x) = 0 \) gives us the approximation of \( \frac{1}{\sqrt{a}} \).

Upon applying Newton's method to this function, we calculate the derivative \( f'(x) = -2x^{-3} \). Substituting these into the Newton-Raphson formula, we derive:

• \( x_{n+1} = x_n + \frac{1}{2}x_n[1 - ax_n^2] \)

This approximation is particularly useful because it leverages the current estimate and adjusts it by a factor dependent on how far off the square of the current estimate is from the goal \( 1/a \). This results in more efficient iteration steps toward the true value.

Newton's Formula Derivation

Newton's Formula Derivation is a cornerstone of the Newton-Raphson method, which is an iterative numerical technique. It's used to find successively closer approximations to the roots (or zeroes) of a real-valued function. To apply this for \( \frac{1}{\sqrt{a}} \), we begin by defining an appropriate function \( f(x) \) whose root will give us the desired reciprocal square root.

The Newton-Raphson update formula is given by:

• \( x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \)

For the reciprocal square root, \( f(x) = x^{-2} - \frac{1}{a} \) and its derivative \( f'(x) = -2x^{-3} \). Substituting into the Newton-Raphson formula provides:

• \( x_{n+1} = x_n + \frac{1}{2}x_n[1 - ax_n^2] \)

This results in an update step that modifies the estimate \( x_n \) based on a correction term. The correction involves the discrepancy from \( 1 \) scaled by \( x_n^2 \), thus aligning the estimate closer to the true value of the reciprocal square root with each iteration.

Algorithm Convergence Analysis

Algorithm Convergence Analysis in the context of the Newton-Raphson method is crucial for understanding how quickly and accurately the method reaches the desired approximation. When considering the reciprocal square root approximation, convergence refers to how the iterative process approaches the actual reciprocal square root of \( a \).

The Newton-Raphson method generally exhibits quadratic convergence, meaning the number of correct digits roughly doubles with each step, provided a good initial estimate. For \( \frac{1}{\sqrt{a}} \), this rapid convergence is maintained because the update formula \( x_{n+1} = x_n + \frac{1}{2}x_n[1 - ax_n^2] \) effectively adjusts the current approximation based on a direct measurement of the error \( ax_n^2 - 1 \).

Each iteration attempts to correct this error robustly. Here is why this is beneficial:

• It results in relatively fewer iterations needed to achieve a desired precision.
• Handles initial estimates with significant deviations from the actual value.
• Takes advantage of computational efficiency due to simple operations involved in update steps.

The key is setting a suitable initial guess, allowing the algorithm to proceed swiftly to a highly accurate approximation of \( \frac{1}{\sqrt{a}} \).