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
calculating
f(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.