Application of Interpolation Newton’s Forward Difference formulae
Introduction
f(x0 + ph) = Ep f (x0 ) = (1+ Δ)p f (x )
= [ 1+pΔ+p p-1 Δ2 / 2! + p (p-1)(p -2) Δ3/ 3! +.]+ f (x0)
f(x0 + ph) = f(x0)+ pΔ f(x0) + f (x0) p p-1 Δ2 / 2! + f (x0) p (p-1)(p -2) Δ3/ 3! + .. +Δ f(x0) p p-1p-n+1/ n!+ Error
The above equation is called Interpolation Newton’s forward difference formula it can also be referred to as Newton-Gregory forward difference interpolation formula. (Buchanan 2012)
Abstract
The data to be analyzed is to be searched from the web. The type of data to be colected is the number of illegal immigrants in the country. The data is in form of tables which show immigrants from all over the world immigrate into the country. The data shows how the figures of immigrants have been fairing over the years. The formula, Newton interpolation will be helpful in generating data from the current available data for further analysis.
Some data cannot be easily estimated especially when one wanted to determine some rough estimates of the number of refugees in the future. For instance the number of immigrants in the year 2015 could not be found since that is the future. It is the job of the mathematicians involved to come up with a formula to make an estimate of the crucial figures needed by the government. The Newton interpolation formula is the most applicable for this task since it can generate a set of data that can be further be analyzed.
The formula is to be generally applied using the years as the standard intervals between times. However, this is dependent on the type of the data that one is using. The number of immigrants can be expressed as a function of years that have passed. From that particular point a statistician can use to estimate various figures one wants to investigate.
Statement of the problems
What is the expected number of the immigrants to be recorded if nothing is to be done keeping all other factors constant? This number is crucial since it will help the government through the department of homeland security to implement steps that will help control the number of immigrants coming into the country.
Using the Newton’s interpolation formulae, investigate the expected total number of males and females that are expected immigrate into the country if all the factors were held constant. From the estimated results one will make a good conclusion whether there is gender balance in the immigration process.
Again using Newton Interpolation formula, give the estimated number of children, below 12 years of age, who are expected to be in the country if, the government fails to address the issue of child immigration within the stipulated time of two years.
References
Buchanan Robert (2012) Math 603 Numerical Analysis. Lecture 21
Baker Bryan, Hoefer Michael and Rytina Nancy (Jan. 2012). Estimates of Unauthor8ized Immigrant Population Residing in the United States.
