And Polynomials begat Polyterms

[Namir]

“And Polynomials Begat Polyterms”

After several months of intense calculations, I was able to complete an important phase of studying polyterms and their role in curve fitting. Polyterms are non-orthogonal polynomials with mostly non-integer powers. As such, curve fitting with polyterms requires optimization of least-square models, because both the coefficients and powers need to be computed. The hp museum web site has a few HP Prime programs that illustrate polyterms using random search optimization.

Polyterms are the next step of using evolved polynomials to perform curve fitting between two or more variables.

I have posted a big ZIP file (close to 380 Mbytes) on my web site. When you visit the main page at namirshammas.com, click on the link titled “The New World of Polyterms (beta 1)”. The ZIP file unpacks a rich set of folders. Each folder studies a set of functions to fit using specific polyterms and a specific optimization method. The root folder has the file “Introducing the New World of Polyterms.pdf” which contains the general overview of polyterms, and a summary of results obtained. The file also lists MATLAB files for the various optimization methods used. If you are interested in the subject but have limited time, then just read the pdf file. Each folder has a pdf document file, MATLAB files to test fitting a set of functions, output graph files, and output text dumps from MATLAB while running various test files. Most of the tests relate the data between two variables. A few folders show the results for testing functions for two and three independent variables with a dependent variable.

Polyterms answer the question often asked at HHC conferences, “What would math and numerical methods look like if legacy mathematicians had programmable calculators?”. Polyterms also answer the question of whether these legacy mathematicians had today’s personal computers. I have mentioned in some of my HHC talks that the legacy numerical methods were limited by the computing resources the legacy mathematicians had—educated people who did math calculations by hand! Even so, the legacy mathematicians pushed algorithms to the limit like with the Gaussian quadrature methods for numerical integration and least square curve fitting.

Enjoy!

Namir