Programming Calculate Square Root Continued Fraction

Introduction

A continued fraction is an expression of the form:

where a₀ is an integer and a₁, a₂, a₃, ... are positive integers. For square roots, we can use a special form of continued fraction called the simple continued fraction representation.

This method allows us to compute square roots with arbitrary precision by iteratively improving the approximation. It's particularly useful in programming because it provides a balance between speed and accuracy, and it can be implemented efficiently using simple arithmetic operations.

Continued Fraction Basics

Continued fractions provide a way to represent numbers as a sequence of integer coefficients. For square roots, the continued fraction representation is particularly elegant because it can represent irrational numbers exactly.

The general form for the square root of a non-square integer n is:

where a₀ is the integer part of √n, and the subsequent coefficients a₁, a₂, a₃, ... can be computed using a recursive algorithm.

This representation is particularly useful because it allows us to compute square roots with arbitrary precision by truncating the continued fraction at any point.

Square Root Algorithm

The algorithm for computing the continued fraction representation of a square root involves the following steps:

1. Compute the integer part a₀ of √n.
2. Initialize the sequence with m₀ = 0 and d₀ = 1.
3. For each subsequent term, compute:
• mᵢ₊₁ = dᵢ * aᵢ - mᵢ
• dᵢ₊₁ = (n - mᵢ₊₁²) / dᵢ
• aᵢ₊₁ = floor((a₀ + mᵢ₊₁) / dᵢ₊₁)
4. Repeat until the desired precision is achieved or until the sequence starts repeating.

This algorithm generates the sequence of coefficients a₀, a₁, a₂, ... that make up the continued fraction representation of √n.

Programming Implementation

Here's a Python implementation of the algorithm to compute the continued fraction representation of a square root:

import math

def continued_fraction_sqrt(n, max_terms=10):
    a0 = int(math.sqrt(n))
    if a0 * a0 == n:
        return [a0]  # Perfect square

    m = 0
    d = 1
    a = a0
    result = [a]

    for _ in range(max_terms - 1):
        m = d * a - m
        d = (n - m * m) // d
        a = (a0 + m) // d
        result.append(a)

    return result

This function takes a positive integer n and returns the continued fraction representation of √n as a list of coefficients. The max_terms parameter controls how many terms of the continued fraction to compute.

The algorithm works by iteratively computing the coefficients of the continued fraction using the recurrence relations shown in the previous section. The loop continues until either the maximum number of terms is reached or the sequence starts repeating.

Examples

Let's look at a couple of examples to see how this works in practice.

Example 1: √2

For n = 2, the continued fraction representation of √2 is:

This corresponds to the sequence [1, 2, 2, 2, ...].

Using our Python function:

>> continued_fraction_sqrt(2, 5)
[1, 2, 2, 2, 2]

Example 2: √5

For n = 5, the continued fraction representation of √5 is:

This corresponds to the sequence [2, 4, 4, 4, ...].

Using our Python function:

>> continued_fraction_sqrt(5, 5)
[2, 4, 4, 4, 4]

These examples demonstrate how the continued fraction representation can provide both an approximation and an exact representation of the square root.