MultivariatePowerSeries

The MultivariatePowerSeries package, which is among new features of Maple 2021, is an object-oriented implementation of multivariate formal power series and univariate polynomials over such series (UPoPS).

This package is written in Maple and provides the ability to create and manipulate multivariate power series with rational or algebraic number coefficients, as well as UPoPS objects whose coefficients are multivariate power series. Through lazy evaluation techniques and a careful implementation, this package achieves high performance in comparison with software counterparts. These power series and UPoPS are employed in optimized implementations of Weierstrass Preparation Theorem and factorization of UPoPS via Hensel’s lemma.

This figure shows all methods in Maple:

Installation Guide

This package is shipped with Maple 2021 and can be reached by with(MultivariatePowerSeries) command in Maple. However, for those using older versions of Maple, there is another way to install this manually as follows:

1. Clone or Download this repository
2. Go to the repository folder (i.e. cd ./MultivariatePowerSeries)
3. Run make in your terminal
4. Add libname := "/package/address", libname: to ~/.mapleinit

A successful installation creates a new file MultivariatePowerSeries.mla at the same directory. Note that /package/address is the complete path to where MultivariatePowerSeries.mla is located. To test or remove this package, run make test or make clean.

Package Structure

Both power series and UPoPS types are implemented as Maple classes and named, respectively,

• PowerSeriesObject (./PowerSeries/src)
• UnivariatePolynomialOverPowerSeriesObject (./UPoPS/src)

The MultivariatePowerSeries package contains these two classes along with procedures to construct and manipulate objects of those classes. In fact, these additional procedures are used to "hide" the object-oriented nature of the library behind simple procedure calls. The code below shows an overview of this package.

MultivariatePowerSeries := module()
option package;
  local PowerSeriesObject,
      UnivariatePolynomialOverPowerSeriesObject;

  # create a power series:
  export PowerSeries := proc(...)

  # create a UPoPS:
  export UnivariatePolynomialOverPowerSeries := proc(...)
  
  # Additional procedures to interface these two classes

  module PowerSeriesObject() 
  option object;
    local hpoly :: Array,
          precision :: nonnegint,
          generator :: procedure;
    # other members and methods
  end module;

  module UnivariatePolynomialOverPowerSeriesObject()
  option object;
    local upoly :: Array, 
          vname :: name;
    # other members and methods
  end module;
end module;

Documentation

A complete documentation is available on Maple Online Help and here.

Examples

The Maple worksheet MPS-demo.mw is a demo of this package presenting examples such as,

• creating PowerSeries objects from a polynomial or an anonymous function:

• decomposing a UPoPS object using Weierstrass Preparation Theorem:

• factoring a UPoPS object:

Credit

This package is designed and developed by M. Asadi under the supervision of M. Moreno Maza and E. Postma follows the lazy evaluation scheme of multivariate power series in the BPAS library written in C language and implemented by A. Brandt, M. Kazemi, and M. Moreno Maza.

To cite this work, please use

@inproceedings{asadi2020MC,
  author = {Asadi, Mohammadali and 
      Brandt, Alexander and 
      Kazemi, Mahsa and 
      Moreno Maza, Marc and 
      Postma, Erik},
  title = {{M}ultivariate {P}ower {S}eries in {M}aple},
  booktitle = {Maple in Mathematics Education and Research},
  series = {Communications in Computer and Information Science},
  publisher = {Springer (accepted)},
  year = {2021}
}