On Computing General Root Algorithms Based on Binomial Expansion, Bernoulli’s Method of Continuous Compound Interest, Series Expansion and Other Modification Methods.

Most of the P th root algorithms exhibit high latency or small convergence rates when iteratively computing the roots. Here we present a slew of algorithms based on series expansions of binomial form, and exponential terms that show low latency or small computational cost for finding the P th root. The latency decreases if the family of series taken are truncated at higher terms. We show that Babylonian method converges quadratically while the binomial series show an increase in convergence rate from quadratic, cubic and higher order 𝑂(𝑡 𝑁 ) convergence rate as the series is sequentially truncated at higher order terms.


Introduction
Square rooting technology is ubiquitously implemented in IEEE, science, and mathematics platforms such as in computing for root mean square, solving linear equations, in vision apparatus and computer graphics.Square rooting algorithms are also used in engineering platforms like field programmable gates arrays and spectrum analysers [1,2,3,4,5,6,7,8,9,10] The Babylonians were able to formulate a remarkable iterative loop [11,12], for computation of positive square roots in circa 1500 BC this primitive mathematical program emanating from the ancient Babylonian epoch is still implemented today in many computing models.Kosheleva [13], explains how the Babylonian method emerges naturally from somehow loose and archaic methods, he shows that by just taking the first term of the binomial series and making a slight mathematical tweak will generate a method that links well to the primitive Babylonian algorithm.Osler [14], somehow did extend the Babylonian method to high order roots not just square root using a brute force method where he introduces wellconditioned coefficients to fit the wanted algorithm using a technology the is quasi graph fitting or brute force through trial and error and observation.In addition, Johnson [15], provided an iterative procedure for computing square root.Knill [16], on the other hand developed a modified Babylonian method for calculating square root and Dubeau [17], used a double iteration method to calculate general roots.Bagala et.al [18], showed using binomial expansion that square root can be computed from the first term of the series which is analogous to the Babylonian method.The Babylonian method which was later refined by Heron of Alexandria is still in use today in calculators and other computing units [16].
The Heron method furnishes the average of a two guess roots.In the past, a lot of mathematicians have shown a proficient approximation technique for square roots, and the general p th roots using binomial expansion, and brute force [16,18] approach while generating generic computational algorithms which are analogous to Newton-Rhapson and the Babylonian method.These methods show case the Babylonian method when the binomial series is truncated at the second term, or when authors furnish a brute force method to approximate the Babylonian iterative loop.Several methods for computing square roots exist in the literature, including a paleo technique that was formulated by an ancient Indian mathematician Aryabhata using a long division algorithm known as digitby-digit calculation to compute square roots, and cube roots documented in his manuscript aryabhatiya [19].Another paleo document with origins from India was written by Bhakshali that gives an algorithm for computing square roots which was shown to have quartic convergence because it's an envelope of two iterations of Newton-Rhapson method [20,21].This method is also equivalent to two iterations of Heron's algorithm which converges quadratically when two iterations are taken the convergence rate is quartic.Taylor expansion is yet another technique that can be furnished to compute square roots which is like binomial expansion.In digital signal processing and field programmable gate array robust square rooting techniques that surpass the need to use division in calculations are employed as division tends to serve as a bottle neck in code execution time leading to increased computational cost, or high network latency.Most practitioners want to use cheaper calculation units that have efficient software rather than spend money on hard wares that centralize expensive divisors [22].It's true that most computer hardware has effectively powerful multipliers that's why algorithms like, single clock and reciprocal of square root methods are employed in digital signal processing and field programmable gate array.In our work we focus on algorithms that have small latency thus they converge rapidly rather than rely on the hardware specifications.Johnson et al. [23] discussed convergence rate of the Babylonian algorithm quantitatively using relative error of the t th iteration showing that the error will diminish quickly following subsequent iterative terms.
Given   = (1 +   ) where  = √  , is the exact solution, A relative error expression   = |  −|  given the Babylonian algorithm as viz: Set A = 1 and   = 1 +   .It can be shown that.
On expanding the term as a geometric series, one obtains, where,   3 is negligibly small.After some arithmetic manipulations we get the following: ) so, it has been shown that the Babylonian algorithm converges quadratically since higher terms of order 3 and beyond are negligibly small.Rubin et al [24] provided a proof of convergence of monotonic sequence when it is bounded both above and below converging to a limit √  ,.He detailed in his book principles of mathematical analysis 3 axioms to confirm the convergence of a sequence.
2) A monotonically decreasing series that is bounded below converges if  1 < √  , and above if  2 > √  , for  > 2. This is a monotonically decreasing sequence.
3) The limit  = lim →∞   is a Babylonian algorithm given by, Whiteside et al. [7] gave an outline of binomial series for the P th power crediting Briggs, partial discovery, the 17 th century English mathematician, in anticipation of Newton of the general binomial expansion.
Johnson et al [13] proposed a technique that shows that by extracting the first two terms of the binomial series yields a Babylonian algorithm as viz, [25]: ) When employing the first two terms of the binomial series, they gave a crude approximation of the Babylonian algorithm as vide infra trying to justify the discovery or mechanics of the algorithm from conventional mathematical technique.This is a method that was discovered by the Babylonians before Heron of Alexandria modified it to what is currently known formula for the Babylonian algorithm.
Square rooting algorithm is often employed in field programmable gate array applications of image processing [20], spectrum analyser [26] and many others.Bagala et al [17,27] proposed a single clock square root algorithm applicable in field programmable gate array.

Proposed Methods
In this section, we derive Binomial Expansion of P th root and different algorithms as subsequently extracted by truncating the series at different levels of performance or the N th term.
We want to solve for  0 , we start by adding and subtracting a dummy term  1 − 1, then. 0 = ( + 1) 1   1 (12) We then expand the binomial series in terms of x.
If we truncate the binomial expansion at the first term, we get the general Babylonian algorithm (or the first-generation algorithm) of the P th root.
If we set P = 2, we consequently derive the Babylonian loop for square roots Second generation algorithm is extracted from truncating the binomial series at the second term.
The series when truncated at consecutively higher terms yields even more robust iterative algorithms that perform better than their predecessors and it is noteworthy that the series converges to the P th root when all the terms are taken.Rate of convergence of the second-generation algorithm as it was derived from binomial expansion and compared to quadratically converging Babylonian method for square roots is shown as viz: The second-generation algorithm is depicted below: Substitute  0 = 1 and   =   + 1 into the above equation, where tn is a relative error term.
By taking geometric series of reciprocal terms and simplifying the equation we get We have shown that our algorithm converges cubically and thus subsequent algorithms in the binomial expansion have high order convergence rate better than the Babylonian algorithm.

A Rooting algorithm derived from the mathematics of compound interest (Bernoulli's method)
Here we are going to derive the modified Babylonian iterative method for the p th root using a formula for continuous compound interest.In addition, we also derive an extended second-generation iterative method for the P th root.We want to approximate  0 by adding a fudge term to a guess function A1.A mathematical technique analogous to the method of gradient descent where the step size is 1 and a 1-dimensional gradient is denoted by N. we first introduce a thought process or a heuristic argument of how the method works.This is a simple mental algorithm or mathematical thought process to show how we arrive at the square root of  0 2 .
• Choose Trial root function,  1 = 5 • Compute  `1 2 = 25 We take the limiting case of compound interest that applies when the power is infinite not just 2 for square roots but for general positive root of real numbers.
Then we excerpt Bernoulli formula for compound interest or exponentials power series expansion of N is derived by expressing ln (

Second generation Modified Babylonian method.
The second-generation Babylonian method is derived by truncating the ln (  0   1  ) at the second term.We obtain this using equation [24] to get: Substitute N into  0 =  1 − , from gradient descent algorithm (20), we obtain: We prove that   is a monotonically converging function and is bounded.
Prove that  +1 ≤An or  0 ≤  +1 The following is a second-generation Babylonian algorithm for square roots extracted from the Bernoulli method of compound interest power series.
(34) Hence shown that  0 is bounded above.The only wrinkle is the insight that this method (Bernoulli series of continuous compound interest) does not converge to  0 but to some numerical value close to  0 .Each truncation at subsequent term is an iterative loop but it's not necessarily true that successive terms perform better than their predecessors like with the method of binomial expansion.There is a sweet spot somewhere in the series progression when it comes to a more robust iterative algorithm.

General roots here are obtained by extracting only the odd terms of the Bernoulli series.
A 0 p = (A 1 − N) p (35) Again, we employ the Bernoulli formula of compound interest.
(37) The root is expressed as vide infra.
Note that N represents the number of iterations and P is the power of A Let us make N the subject of the argument.
Power series expansion of We take all the odd terms of the series and drop the even powers:

Power series expansion of equation [43]
We show that equation [45] of odd terms is numerically expressed as below.
Let's extract the odd terms only.(50) We then solve for y numerically as an approximate value of the sum by taking an integration instead of the sum.
This is the upper bound of y which shows that the odd terms of the series converge to.
When we excerpt the first term of the series, we succinctly derive the general Babylonian iteration loop for the roots.
Substituting an expression of x into equation [54] The general Babylonian iteration loop is typified by the formula above for  0 .Now let us truncate the series at the second odd term so that we get yet another iterative algorithm.

Discussion
The computational cost of selected algorithms increases in the following fashion: surrogate method algorithm, second generation binomial series derived algorithm,, odd terms only Bernoulli algorithm, general Bernoulli algorithm, Babylonian algorithm, even terms only Bernoulli algorithm.But the study of their convergence rates show that higher generation binomial terms have higher Order rate of convergence rates.Babylonian series converges quadratically, while second, third and Nth -generation binomial algorithms have progressive higher convergence rates as O (t 3 ), (t 4 ) …(t N ) respectively.The Babylonian method is simpler and has satisfactory convergence rate and computational cost.Other methods are sophisticated and more robust in computing roots.But the even only terms algorithm is poor and has high latency, so it is less useful in computing roots, it is a tortoise algorithm while higher order binomial algorithm and surrogate method have low latency.The methods furnished here can be used to compute root mean square and solve linear equations or serve as go to methods in IEEE platforms that require small computational costs.

Conclusion
Several methods have been applied to calculate the roots of positive real numbers.Classical methods include the Babylonian and Newton Raphson techniques.The current paper has employed series expansion as a method of generating algorithms for roots truncated at different levels of accuracy.The work has deduced several formulations that calculates the roots.The binomial expansion is somehow more superior to other series expansion methods.This paper wishes to bridge the gap in which classical methods did not exhaust.

2 5 • 16 •Figure 1 .
Figure 1.A map of square root A against square

1 𝑃− 1 ,
We only show the truncated power series of ln ( this is the general Babylonian iterative algorithm.Both the binomial expansion and power series expansion of ln ( from the method of compound interest give the general Babylonian algorithm when truncated at the first term.
Note that equation [49] is a geometric series of the form:

General methodology when considering all the terms, odd and even of the series for Bernoulli method of continuous compound interest.
Now we show the expression of Babylonian for the 3 rd generation from the Bernoulli series of continuous compound interest.We employ geometric series to represent the reciprocal terms.Convergence rate is   2 + (  3 ), higher order terms disappear because: 0 <   < 1