LAST (Linear Algebra Symbolic Toolbox)

This is a module for the LAST (Linear Algebra Symbolic Toolbox) module. It contains the functions to solve linear systems of equations with large symbolic expressions. The module uses symbolic full pivoting LU and QR decompositions to solve linear systems. The LEM (Large Expressions Management) module is used to avoid expression swell.

The code in this repository is hopefully an improved version of the code provided in Wenqin Zhou's PhD thesis Symbolic Computation Techniques for Solving Large Expressions.

We would like to thank Jacques Carette for providing the original code that we used to develop this module.

Dependencies

The LAST Maple module depends on the LEM (Large Expressions Management) module. This module contains functions to avoid expression swell and it is freely available at this link.

Installation

Firstly, install the dependency. The LEM module is freely available at this link. For the installation of this dependency module, refer to the instructions present on the repository homepage.

To install the module you must have first installed Maple. Then open the PackAndGo.mw file and use the !!! button to execute the entire worksheet.

Then test the library in a Maple worksheet or document by typing:

> LAST:-Info(LAST); # Fot Maple versions up to 2020
> LAST:-Info();     # Fot Maple versions starting from 2021

Alternatively, you can use one of the tests file provided in the tests folder. If the module is loaded without errors, it is done!

Module description

If you want a full description of the LAST module type:

> Describe(LAST);

This command will generate a brief description of the module and all the procedures and other objects present in the LAST.mpl file.

Usage

In case you have no time to read the description and realize how it should or should not work, refer to the files Test_00_LU.mw and Test_01_QR.mw in the tests folder.

🚧 Attention! 🚧

Maple object-oriented programming features have slightly changed in 2021, which online documentation states:

As of Maple 2021, if the method has a formal parameter named _self, references to its object's local or exported variables may be written without prefixing them. That is, _self:-variable may be written as just variable. Maple will add the self:- prefix internally when the method is simplified.

As of Maple 2021, a message-passing form of method call can be used, which will automatically pass the object as an argument if the method has a formal parameter named _self.

Another way to invoke a method, similar to that used in other object-oriented languages, was introduced in Maple 2021. In this form, the object name is qualified by the method name, object_name:-method_name(argument) just as it can be in the function mechanism described above. However, the object can be omitted from the argument sequence.

For further information please refer to this link.

Authors

Current version authors:

• Davide Stocco, Department of Industrial Engineering, University of Trento
  email: davide.stocco@unitn.it

• Matteo Larcher, Department of Industrial Engineering, University of Trento
  email: matteo.larcher@unitn.it

• Enrico Bertolazzi, Department of Industrial Engineering, University of Trento
  email: enrico.bertolazzi@unitn.it

Inspired by the work of:

• Wenqin Zhou (former affiliation), Department of Applied Mathematics, University of Western Ontario

• David J. Jeffrey, Department of Applied Mathematics, University of Western Ontario

• Jacques Carette, Department of Computing and Software, McMaster University

• Robert M. Corless, Department of Applied Mathematics, University of Western Ontario

References

@phdthesis{zhou2007symbolic,
  author = {Zhou, Wenqin},
  title = {Symbolic Computation Techniques for Solving Large Expression Problems from Mathematics
    and Engineering},
  year = {2007},
  school = {University of Western Ontario}
}

@inproceedings{zhou2006hierarchical,
  title = {Hierarchical representations with signatures for large expression management},
  author = {Zhou, Wenqin and Carette, Jacques and Jeffrey, David J. and Monagan, Michael B.},
  booktitle = {Proceedings AISC 2006, LNCS 4120}
  editor = {Calmet, J. and Ida, T. and Wang, D.},
  pages = {254--268},
  year = {2006},
  organization = {Springer}
}

@inproceedings{carette2006linear,
  title = {Linear algebra using Maple’s LargeExpressions package},
  author = {Carette, Jacques and Zhou, Wenqin and Jeffrey, David J. and Monagan, Michael B.},
  booktitle = {Proceedings of Maple Conference},
  pages = {14--25},
  year = {2006}
}

@article{zhou2008fraction,
  title = {Fraction-free matrix factors: new forms for LU and QR factors},
  author = {Zhou, Wenqin and Jeffrey, David J.},
  journal = {Frontiers of Computer Science in China},
  volume = {2},
  pages = {67--80},
  year = {2008},
  publisher = {Springer}
}

Documented by Davide Stocco