How to implement Horner’s scheme for multivariate polynomials?

Question

Background

I need to solve polynomials in multiple variables using Horner's scheme in Fortran90/95. The main reason for doing this is the increased efficiency and accuracy that occurs when using Horner's scheme to evaluate polynomials.

I currently have an implementation of Horner's scheme for univariate/single variable polynomials. However, developing a function to evaluate multivariate polynomials using Horner's scheme is proving to be beyond me.

An example bivariate polynomial would be: $12x^2y^2+8x^2y+6xy^2+4xy+2x+2y$ which would factorised to $x(x(y(12y+8))+y(6y+4)+2)+2y$ and then evaluated for particular values of x & y.

Research

I've done my research and found a number of papers such as:
staff.ustc.edu.cn/~xinmao/ISSAC05/pages/bulletins/articles/147/hornercorrected.pdf
citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.40.8637&rep=rep1&type=pdf
www.is.titech.ac.jp/~kojima/articles/B-433.pdf

Problem

However, I'm not a mathematician or computer scientist, so I'm having trouble with the mathematics used to convey the algorithms and ideas.

As far as I can tell the basic strategy is to turn a multivariate polynomial into separate univariate polynomials and compute it that way.

Can anyone help me? If anyone could help me turn the algorithms into pseudo-code that I can implement into Fortran myself, I would be very grateful.

Answer 1

The paper you cite, "On the multivariate Horner scheme" (Pena, Sauer) has an explicit algorithm specified on p.3. The remaining challenge is to penetrate the notation and conventions in the paper laid out in the first three pages far enough to turn their algorithm presentation into code.

It also seems that this paper (just reading the abstract) specifies an explicit algorithm: "Evaluation of Multivariate Polynomials and Their Derivatives," J. Carnicer and M. Gasca, Mathematics of Computation, Vol. 54, No. 189 (Jan., 1990), pp. 231-243. ResearchGate link to full text.

Abstract. An extension of Horner's algorithm to the evaluation of m-variate polynomials and their derivatives is obtained. The schemes of computation are represented by trees because this type of graph describes exactly in which order the computations must be done. Some examples of algorithms for one and two variables are given.

Answer 2

I implemented this in Python: multivar_horner

You can look at the approach used there and port it to Fortran.

NOTE: In contrast to the one dimensional case there are multiple possible Horner factorisations of multivariate polynomials. One can allow a search over the possible factorisations to find a minimal representation as described HERE. In most cases however it suffices to use a heuristic to find a single "good" factorisation. multivar_horner implements the greedy heuristic described in "Greedy Algorithms for Optimizing Multivariate Horner Schemes".

The authors of some related publications (including the above mentioned) claim to have an implementation of their proposed algorithms, but I was not able to find any publicly available ones.