Discipline: Mathematics and Statistics
Subcategory: Mathematics and Statistics
Darryle Cyrille - University of the Virgin Islands
Polynomial interpolation is the construction of a polynomial that passes through a given set of points, a procedure that is useful in applications where continuous models of phenomena are desired but only finite amounts of data are available. Such models can then be used to predict results at points not in the original data set. In the real world, measurements are never exact. Rounding errors are bound to occur. What is the impact of such errors on the polynomial interpolant the data is used to produce? Can this impact be quantified in a simple way? In this study, measurement error was represented by a small shift in the value of the dependent variable at individual points. The change in the interpolant produced was measured using the L^1-norm, the definite integral of the absolute value of the difference between the interpolants, a quantity which takes into account the global behavior of the functions in question. Using MATLAB, various nodes were perturbed slightly to simulate measurement error. The result was surprising: the L^1-difference due to such a change depends only on the size of the perturbation and the distribution of x-values, not on the y-values of the nodes. This fact, first noted during numerical simulations, is also demonstrated algebraically. The effects of rounding errors are more pronounced at nodes nearer to the center of the distribution of x-values. In particular, for uniform distributions of nodes, the effects appear to adhere to a binomial distribution. Future research will investigate this hypothesis algebraically and then will seek to generalize the result to non-uniform distributions of nodes. References Atkinson, Kendall E. An Introduction to Numerical Analysis. John Wiley & Sons, 2008. Binegar, Birne. “Errors in Polynomial Interpolation.” Oklahoma State University, http://math.okstate.edu/people/binegar/4513-F98/4513-l16.pdf. Epperson, James F. An Introduction to Numerical Methods and Analysis. John Wiley & Sons, 2013 Funded by NSF HBCU-UP Scholars – 1137472 Faculty mentor: Andrew Gard, Andrew.email@example.com
Funder Acknowledgement(s): NSF HBCU-UP Scholars 1137472
Faculty Advisor: Dr. Andrew Gard, firstname.lastname@example.org
Role: I learned the theory of polynomial interpolation and rounding errors that occur while interpolating. Then I applied what I learned into my research and created a set of codes that corresponded to what I needed the program to do. However, my mentor did give me little hints when I needed the guidance.