Multivariate Horner scheme implementation in MATLAB

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Mohammad Shojaei Arani 2022-3-6

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评论： Mohammad Shojaei Arani 2022-3-7

Hello friends!

I have very long symbolic expressions (mainly polynomials or rational types) and I need to evaluate them many many times. Therefore,

in order to reduce the computational time I would like to re-write my expressions by their equivalent horner form. For instance, the expression

needs f=2*x^3-3*x^2+6*x-5 requires 8 multiplications and 3 additions while its following horner equivalent requires only 3 multiplications and 3 additions:

>> horner(f,x)

ans =

x*(x*(2*x - 3) + 6) - 5

Unfortunately, matlab only supports a univariate horner scheme. But, my expressions are multivariate and I need a multivariate Horner scheme

to simplify my long multivariate expressions. For instance, consider the following bi-variate expression:

syms x y

f = x^2+2*x*y+y^2;

If I go for a uni-variate Horner scheme then I get

>> horner(f,x)

ans =

x*(x + 2*y) + y^2

which requires less multiplications. This is, of course, an improvement. But, this is not the best improvement. A bi-variate Horner scheme

should give us (x+y)^2 as the best representation which needs only 2 arithmatics!

So, in sum, I need a matlab code which can implement a multivariate Horner scheme. I could find a python code but nothing in matlab. Unfortunately, I do not know python very well and really need a matlab code.

Any help is GREATLY appreciated!

Thanks in advance,

Babak

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Torsten 2022-3-6

A multivariate Horner scheme under MATLAB does not exist.

And the package under Python also will not be useful for your purpose because it can only handle very basic cases. But you have unstructured and very long expressions as you wrote.

So I think you will have to be satisfied with what MATLAB can offer, namely "matlabFunction". Or you can completely restructure your code to numerical, not symbolic computation (which I would recommend).

Mohammad Shojaei Arani 2022-3-7

Hi Torsten,

You are, most probably, right that perhaps there does not exit a perfect package to handle the problem of

finding a multivariate Horner scheme. It seems to me that a nuclear physics group in the Netherlands

called 'Nikhef' (see the link https://www.nikhef.nl/~form/) have programmed the best methods to tackle this problem.

Unfortunately, it is not written in matlab. In my opinion, it is essential to make a nice matlab multivariate Horner package as matlab is not very fast. Think of the expansion of an expression like (a+b+c)^30 which has many terms (of course matlab can handle this). If matlab wants to survive then such a package is really needed. Unfortunately, at this moment I am extremely busy with my research but whenever I find time I will try to make such a package. And, I hope that others will contribute to this.

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