Polynomial Evaluation Algorithm: Real World Uses and Solutions

Polynomial evaluation algorithm
Introduction
Polynomials are always applied in high performance DSP algorithms to estimate the computation
of functions or parametrical modelling of the systems. Several basic digital circuits has the
polynomial evaluation algorithm such as high precision elementary functional evaluation circuits
and digital filters. In fact, at a system level, many applications such as cryptography, speech
recognition and communications, involve the computation of polynomials. Researchers have
always investigated the polynomials with the major challenge being speed of computation.
Polynomials are majorly beneficial as they only apply multiplication and addition. High degree
polynomials usually involve multiple word length multiplications which are time consuming.
This can always be reduced through a low latency application that can be through hard or
software approaches (Xu, 2013).
Problem statement of the polynomial evaluation algorithm
The problem of polynomial evaluation can be summarized as follow. Using a general format for
kth degree polynomial evaluation,
f ( x ) = ∫
i =1
k
ai xi ; x=input of the polynomial with a set of coefficient a1 of a constant value
The coefficient can be obtained by various algorithms. The coefficient will not be frequently
changed although they could be updated from one time to another. The range of the coefficient
and input x is defined to be (0, 1) enhancing accuracy in analysis to be performed later
(Higham, 2002).

Real world uses of polynomials
Polynomials are often applied in a supermarket for instance, when one wants to know the
different good prices, let’s say a price for one bundle of flour, five crates of whiskey and one box
of milk then there is need to design a simple polynomial before going to prices. With the algebra
ready, one will be able to fix prices and determine the amount of cash that he/she will be able to
spend (Gohberg, Lancaster, & Rodman, 2009).
Professionals who are most likely to apply the polynomials on their day to day activities are
those who make complex calculations. A roller coaster designing engineer would use
polynomials to model curves as well as civil engineers who apply polynomials in road, building
and other structure designing.
Polynomials can as well be used in predicting the traffic patterns enabling appropriate control
measures on the traffic for example traffic lights.
Economists apply polynomial in modelling economic growth patterns as well as medical
researchers using the polynomials to establish and designate the bacteria colony.
The drivers also apply polynomials in calculating distance covered versus the amount of money
to be paid.
Solution to the problem
In the field of economics and statistics, there is always need to determine trends and make of
relevant conclusions. Economists often try to use polynomials in predicting demand and price
relationships (Pan, 2016). Suppose a simplified polynomial predicting the demand for a new
gadget as a function of price p (in dollars) over the interval 0 < p < 150 given by,

D ( p )=0.006 p2−3.5 p+250
Then the average rate in change in demand as price goes from 75 dollars to 80 dollars then we
can apply the polynomial,
Average rate∈change=D ( p )=0.006 p2−3.5 p+250
D ( p1=75 ) =0.006(75)2−3.5 ( 75 ) +250=21.25
D ( p1=80 ) =0.006(80)2−3.5 ( 80 ) +250=8.4
Therefore, the average rate of change is given as follows;
∆= 8.4−21.25
(80−75) =−2.57
This shows that the demand changes by 2.57 whenever the price increases.
Possible algorithm (How computing can be useful in providing the solution discussed in
section 4)
The Horner rule is the algorithm that can be used in solving the above problem (Cahen &
Chabert, 2017).
Considering the following polynomial
p ( x )=a0 + a1 x1 +a2 x2+ a3 x3 +...+ an xn
The idea behind the above polynomial is to evaluate the number of x values.
The standard way to evaluate the algorithm is to write it in some form of algorithmic format as
follows;
poly=a0