**Undergraduate #292**

**Discipline:**Mathematics and Statistics

**Subcategory:**Mathematics and Statistics

**Session:**2

**Nikkoiya Cromwell**

**- University of the Virgin Islands**

Polynomial interpolation is a method used to approximate unknown functions based on small numbers of data points. Such a procedure is desirable in any scientific endeavor seeking to relate two variables (for example time and temperature) based on a finite set of observations. One advantage of polynomial interpolation over other methods is that an error function can be computed explicitly. In many cases, the most significant part of this error function is a polynomial w_n (x) defined entirely by the values of the independent variable at the n sample points. Just how strongly does w_n (x) depend on the choice of sample points x_i? Given that scientists cannot always control exactly where their sample points are taken, this is an important question.

In this study, points were randomly sampled in a representative interval and used to compute the interpolating polynomial and its corresponding w_n (x) in each of 100,000 trials. Each time, the maximum value of |w_n (x)| was computed. The mean of these values gives a picture of the overall performance of the method of polynomial interpolation for the chosen interval and the particular number of nodes.

The results indicate that the mean maximum of |w_n (x)| decreases rapidly (apparently exponentially) as the number of sample points increases, despite the well-known fact that this decrease is not guaranteed for every distribution of nodes. In other words, in cases where nodes are randomly sampled, greater model accuracy can generally be expected for increasing n.

In this project, the mean of max {|w_n (x)|} was modeled using random sampling. Can this mean be explicitly computed? What about standard deviation? Future work will focus on these questions.

**Funder Acknowledgement(s):** This research was funded and made possible by the University of the Virgin Islands Emerging Caribbean Scientist Program, NSF HBCU-UP Grant award #1137472 and NSF ACE Grant award #1623126.

**Faculty Advisor: **Andrew Gard,
andrew.gard@uvi.edu

**Role:** With the help of my mentor, I created various Matlab functions that would compute the maximum value of the polynomial wn(x) by inputting random random sample points into the polynomial, ran the maximum of wn(x) 100,000 times which would compute the maximum of wn(x) each time, and computed the mean maximum of wn(x) on a specific interval with a specific amount of nodes.