June 2nd, 2011
I wrote a Mathematica code that produces a polynomial of best fit for a given set of points, computes the roots of the polynomial (both real and complex), plots them in the complex plane and creates the corresponding Voronoi Diagram. I am continuing general research on polynomiography and preparing my first presentation.
June 16th, 2011
I gave my intial presentation on polynomiography, focusing on the relevance of iteration functions and root-finding. I modified my Mathematica code to compute the polynomials of best fit for degress 1 through 20 for a given data set, then compute the roots of each of these polynomials and plot the respective Voronoi diagram. I am continuing to investigate trends in the roots of these functions and their diagrams for various data sets. To download my first presentation, click the image below:
June 24th, 2011
I continued developing my Mathematica code to include graphs of the polynomials of best fit alongside the points they approximate. I continued to investigate trends in the Voronoi diagrams and roots of the functions in question.
July 1st, 2011
I evolved my code to produce plots and diagrams in consistent scales to facilitate comparison between degrees. I also added a plot of the convex hull of the roots of each function as a point of further comparison. Lastly, I began considering a new coding problem: creating a code to perform Hermite Interpolation.
July 8th, 2011
I continued working on the Hermite Interpolation code and began considering a new optimization problem. Additionally, I modified my original program to be more interactive. Now, one can click on a graph to indicate the points they wish to compose the data set to be analyzed. The program, equipped with detailed instructions, can be downloaded for use by right-clicking the image below and saving the attached file:
Also, I have included 2 examples of my own. Comparisons between the traditional polynomial and the polynomiograph for my data sets can be found below as word documents:
July 12th, 2011
I worked on my final presentation and organized my work. I continued to work on my optimization problem and Hermite Interpolation code.
June 15th, 2011
I gave my final presentation on polynomiography, focusing on the program I developed and my new optimization problem. I continued to work on that problem as well as my work with Hermite Interpolation. To download my first presentation, click the image below:
July 22nd, 2011
I wrapped up most of my work with polynomiography and worked on my future plans with Dr. Kalantari. We plan on continuing our work this summer and into the next academic semester. Additionally, I continued my research with the optimization problem.