The Ngazi Method of Exponential Function Calculation.

Abstract

The Ngazi Method of Exponential Function CalculationMsc. Student, Research MethodsJomo Kenyatta University of Agriculture and Technology. Tel: +254-713 172 555 e-mail: tkuria19@gmail.com Abstract This paper presents an entirely new way of calculating exponential functions. This method uses the 2n sequence at its foundation. An example of the 2n sequence is 2,4,8,16,32,64 and so on. This method is vastly more time saving and energy saving because one performs very few multiplication operations once an exponent has been broken down into its 2n components. This method can be coded into a computer software and this will improve the speed with which computer libraries calculate exponential functions. This paper also explains how this new method of exponential function calculation can partly help to solve the discrete exponential and discrete logarithm problem through easier calculation of exponential functions.1.Introduction.“Neural network simulations often spend a large proportion of their time computing exponential functions. Since the exponentiation routines of typical math libraries are rather slow, their replacement with a fast approximation can greatly reduce the overall computation time.”[1]It is true that computer math libraries that deal with the exponential functions are rather slow but this is a function of using very archaic and long routine calculations which are time-consuming and energy wasting. If a more time-efficient method for calculating exponential functions can be found and then coded into a computer math library then this will save a lot of time for many mathematicians, engineers and computer scientists who use these exponential function libraries a lot. This author presents a new method of calculating exponential functions which is both accurate and time-saving. We no longer need to rely on time-saving approximations of exponential functions. We can now use time-saving accurate calculations of exponential functions and this could indeed change the field computational science.There is a relationship between exponential functions and logarithms. The power of logarithms as a computational device lies in the fact that by them multiplication and division are reduced to the simpler operations of addition and subtraction.[2]“There are many applications of exponential functions and logarithmic functions in science and technology. The voltage in a given circuit can be expressed using exponents. The value of money in an investment can be determined through the use of exponents. The intensity of earthquakes is measured by a logarithmic scale. The intensity of light related to the thickness of the material through which it passes can be expressed using exponents. The distinction between acids and bases in chemistry is measured in terms of logarithms.” [3]When it comes to exponential functions, the word exponent is often used instead of index, and functions in which the variable is in the index (such as 2x, 10sinx) are called exponential functions. [4] If b is a real number greater than zero, then for each real exponent x we assume bx is a unique real number. Since for each real x there is one and only one bx, the equation y=bx,(b>0)defines a function. We call such an equation an exponential function.[3]2. Body .To solve the exponential function f(x)=mx, the most common method of calculating exponential functions has been to directly multiply m, x number of times. This method of calculation is extremely slow especially if the exponent has a large value. For, example calculatingf(x) = 35123000 involves multiplying 35, 123,000 times. This is an extremely difficult task and is frankly almost impossible if calculated manually. Computers are best suited to calculate this function because computers do not get tired of performing repetitive tasks. However, there is a need to come up with a more efficient way of calculating the exponential function. This is important because exponential functions are very important in mathematics and engineering fields.This paper introduces a new, novel method of calculating the exponential function. This method is extremely efficient and it uses the 2n sequence at its foundation. One can solve the exponential function quicker if one uses the 2n table. A sample of the table will be displayed towards the end of the paper.I have decided to call this method the Ngazi method. Ngazi is a swahili word for ladder or stairs. Ngazi is a two syllable word which is pronounced as (nga-zi) where /n/ and /g/ are pronounced as one syllable.[5] I named this method ngazi or stairs because the results of the first multiplication are used in the second multiplication and the results of the second multiplication are used in the third multiplication and so on. Therefore the first multiplication is linked to the second multiplication and the second multiplication is linked to the third multiplication and this reminds the author of a flight of stairs where one stair leads to the next stair up to the final stair.