Introduction to polexpr — polexpr 0.8.7a documentation

# Introduction to polexpr

polexpr is a TeX and LaTeX macro package.

To use it with Plain or formats other than LaTeX:

    \input polexpr.sty

To use it with LaTeX:

    \usepackage{polexpr}

polexpr loads xintexpr and extends its syntax with a polynomial type.

Polynomials are defined via the \poldef parser and can then be manipulated via TeX macros for various operations such as obtaining G.C.D.s, computing derivatives, or indefinite integrals… many of these functionalities are available directly via functions in the syntax recognized by the `\poldef` parser.

Apart from the final step of internally creating some TeX macros associated with the polynomial name, this is all done completely expandably, and a polynomial expression can be fetched directly to `\xinteval` for a completely expandable operation from start to finish.

For polynomials fully declared via `\poldef`, root localization implementing the Sturm algorithm is available via macros. Arbitrarily long decimal expansion of all real roots can be obtained, and all rational roots can be found exactly.

The legacy documentation section which showed examples of root localization has been moved to the polexpr-examples.pdf document, where both the TeX code and its output are displayed.

To access this document (if the previous link for some reason does not work):

    texdoc polexpr-examples

## Usage via xintsession

The simplest manner to get a feeling for the package abilities regarding in particular root localization is to use it on the command line via the xintsession interface. Here is an example:

    $ rlwrap etex xintsession
    [...xintsession welcome message...]
    Starting in exact mode (floating point evaluations use 16 digits)
    >>> &pol
    pol mode (i.e. function definitions use \poldef)
    >>> f(x) := x^7 - x^6 - 2x + 1;
    f = x^7-x^6-2*x+1
        --> &GenFloat(f) lets f become usable as function in fp mode
        --> &ROOTS(f) (resp. &ROOTS(f,N)) finds all rational roots exactly and
            all irrational roots with at least 10 (resp. N) fractional digits
    >>> &ROOTS(f)
    Solving for real roots of f and assigning them (please wait...)
    (mult. 1) Rootf_1 = -1.0719678841...
    (mult. 1) Rootf_2 = 0.4962386948...
    (mult. 1) Rootf_3 = 1.3151140860...
    Square-free irrational part: x^7-x^6-2*x+1
        --> &REFINEROOTS(f,N) to extend real irr. roots to N fractional digits
    >>> &REFINEROOTS(f,40)
    Refining real roots of f to 40 digits (please wait...)
    (mult. 1) Rootf_1 = -1.0719678841080266034109100331975342338141...
    (mult. 1) Rootf_2 = 0.4962386948771497344730618510143671567979...
    (mult. 1) Rootf_3 = 1.3151140860165192656960005018679846354234...

The xintsession interface allows to define polynomial variables via its `&pol` mode. It also exposes all polynomial functions added to the xintexpr syntax by `polexpr`. Further, as seen in the example above it also covers some of the `polexpr` capabilities currently implemented via user macros. This is to be considered a work in progress, the above `&ROOTS` and `&REFINEROOTS` may be renamed into something else, and may have been so already since this documentation was written.

Any input on the command line at the xintsession `>>>` prompt which starts with a backslash is executed as TeX macros, so all macros of `polexpr` are in fact already available, including those typesetting material in background of the interactive session on command line. The `&ROOTS` and `&REFINEROOTS` direct their outputs to the terminal rather than to the TeX page as would the typesetting macros defined by `polexpr` itself.

## License is LPPL 1.3c

This work is distributed under the conditions of the LaTeX Project Public License version 1.3c.

See README.md for details.

## In memoriam: Jürgen Gilg (1966-2022)

My first contact with Jürgen was in January 2018, and his little question about usage of xintexpr for differentiating polynomials proved the direct cause for the creation of polexpr. His regular comments and questions (often, jointly with Thomas Söll) largely contributed to motivate the author into investing time and effort into the somewhat strange occupation of developing such unusual TeX/LaTeX packages. The topics of our discussions evolved over the years and were not limited to TeX, LaTeX, PSTricks or PostScript matters.

I learned to discover and appreciate his unassuming and kind character. He was very conscientious in all his projects with others and was often overly generous in his appreciation of people’s contributions. On receiving the sad news of his unexpected and untimely death, I initially thought I had been knowing him for at least 10 years, but it is actually only a bit more than 4 years… I will sorely miss this friendship with a gentle soul from over the Rhine.

I dedicate the work which has gone into polexpr to Jürgen’s memory.