More Web Proxy on the site http://driver.im/

research-article

Regular systems of linear functional equations and applications

Authors:

Moulay A. Barkatou,

Gary Broughton,

Eckhard PflügelAuthors Info & Claims

ISSAC '08: Proceedings of the twenty-first international symposium on Symbolic and algebraic computation

Pages 15 - 22

https://doi.org/10.1145/1390768.1390774

Published: 20 July 2008 Publication History

Abstract

The algorithmic classification of singularities of linear differential systems via the computation of Moser- and super-irreducible forms as introduced in [21] and [16] respectively has been widely studied in Computer Algebra ([8, 12, 22, 6, 10]). Algorithms have subsequently been given for other forms of systems such as linear difference systems [4, 3] and the perturbed algebraic eigenvalue problem [18]. In this paper, we extend these concepts to the general class of systems of linear functional equations. We derive a definition of regularity for these type of equations, and an algorithm for recognizing regular systems. When specialised to q-difference systems, our results lead to new algorithms for computing polynomial solutions and regular formal solutions.

References

[1]

S. Abramov. Rational solutions of linear difference and q-difference equations with polynomial coefficients. In Proceedings of ISSAC'95, pages 303--308. ACM Press, 1995.

Digital Library

[2]

S. Abramov and M. A. Barkatou. Rational solutions of first order linear difference systems. In Proceedings of ISSAC '98, pages 124--131, Rostock, Germany, 1998. ACM Press.

Digital Library

[3]

M. A. Barkatou. Contribution `a l'etude des equations differentiel les et de differences dans le champ complexe. PhD thesis, INPG, 1989.

[4]

M. A. Barkatou. On the reduction of linear systems of difference equations. In Proceedings of ISSAC '89, pages 1--6, Portland, Oregon, 1989. ACM Press.

Digital Library

[5]

M. A. Barkatou. A rational version of Moser's Algorithm. In Proceedings of ISSAC '95, pages 297--302, Montreal, Canada, 1995. ACM Press.

Digital Library

[6]

M. A. Barkatou. An algorithm to compute the exponential part of a formal fundamental matrix solution of a linear differential system. Journal of App. Alg. in Eng. Comm. and Comp., 8(1):1--23, 1997.

Digital Library

[7]

M. A. Barkatou. On rational solutions of systems of linear differential equations. Journal of Symbolic Computation, 28:547--567, 1999.

Digital Library

[8]

M. A. Barkatou. On super-irreducible forms of linear differential systems with rational function coefficients. Journal of Computational and Applied Mathematics, 162(1):1--15, 2004.

Digital Library

[9]

M. A. Barkatou. Factoring systems of linear functional equations using eigenrings. In I. S. Kotsireas and E. V. Zima, editors, Latest Advances in Symbolic Algorithms, pages 22--42. World Scientific, 2007.

[10]

M. A. Barkatou and E. Pflugel. An algorithm computing the regular formal solutions of a system of linear differential equations. Journal of Symbolic Computation, 28:569--588, 1999.

Digital Library

[11]

M. A. Barkatou and E. Pflugel. The ISOLDE package. A SourceForge Open Source project, http://isolde.sourceforge.net, 2006.

[12]

M. A. Barkatou and E. Pflugel. Computing super-irreducible forms of systems of linear differential equations via moser-reduction: a new approach. In Proceedings of ISSAC '07, pages 1--8, Waterloo, Canada, 2007. ACM Press.

Digital Library

[13]

M. A. Barkatou and E. Pflugel. On the Moser- and super-reduction algorithms of systems of linear differential equations and their complexity. Submitted to JSC, 2007.

Digital Library

[14]

M. Bronstein and M. Petkovsek. An Introduction to Pseudo-Linear Algebra. Theoretical Computer Science, 157(1):3--33, 1996.

Digital Library

[15]

V. Dietrich. Zur Reduktion von linearen Differentialgleichungssystemen. Math. Ann., 237:79--95, 1978.

[16]

A. Hilali and A. Wazner. Formes super--irreductibles des systemes differentiels lineaires. Numer. Math., 50:429--449, 1987.

Digital Library

[17]

N. Jacobson. Pseudo-linear transformations. Annals of Mathematics, 33(2):484--507, 1937.

[18]

C.-P. Jeannerod and E. Pflugel. A reduction algorithm for matrices depending on a parameter. In Proceedings of ISSAC '99, pages 121--128, Vancouver, Canada, 1999. ACM Press.

Digital Library

[19]

S. Lang. Algebra. New York; London: Springer, c2002, 2002.

[20]

A. Levelt. Stabilizing differential operators: a method for computing invariants at irregular singularities. In M. Singer, editor, Differential Equations and Computer Algebra, pages 181--228. Academic Press, 1991.

[21]

J. Moser. The order of a singularity in Fuchs' theory. Math. Z., pages 379--398, 1960.

[22]

E. Pflugel. Effective formal reduction of linear differential systems. Appl. Alg. Eng. Comm. Comp., 10(2):153--187, 2000.

Cited By

Barkatou MCluzeau TEl Hajj ADavenport JWang DKauers MBradford R(2019)Simple Forms and Rational Solutions of Pseudo-Linear SystemsProceedings of the 2019 International Symposium on Symbolic and Algebraic Computation10.1145/3326229.3326246(26-33)Online publication date: 8-Jul-2019
https://dl.acm.org/doi/10.1145/3326229.3326246
Barkatou MCluzeau TJalouli ARobertz DLinton SYokoyama K(2015)Formal Solutions of Linear Differential Systems with Essential Singularities in their CoefficientsProceedings of the 2015 ACM International Symposium on Symbolic and Algebraic Computation10.1145/2755996.2756669(45-52)Online publication date: 24-Jun-2015
https://dl.acm.org/doi/10.1145/2755996.2756669
Barkatou MMaddah SRobertz DLinton SYokoyama K(2015)Removing Apparent Singularities of Systems of Linear Differential Equations with Rational Function CoefficientsProceedings of the 2015 ACM International Symposium on Symbolic and Algebraic Computation10.1145/2755996.2756668(53-60)Online publication date: 24-Jun-2015
https://dl.acm.org/doi/10.1145/2755996.2756668
Show More Cited By

Index Terms

Regular systems of linear functional equations and applications
1. Computing methodologies
  1. Symbolic and algebraic manipulation
    1. Symbolic and algebraic algorithms

Recommendations

Computing super-irreducible forms of systems of linear differential equations via moser-reduction: a new approach
ISSAC '07: Proceedings of the 2007 international symposium on Symbolic and algebraic computation

The notion of irreducible forms of systems of linear differential equations as defined by Moser [14 ] and its generalisation, the super-irreducible forms introduced by Hilali/Wazner in [9 ] are important concepts in the context of the symbolic ...
On the Moser- and super-reduction algorithms of systems of linear differential equations and their complexity

The notion of irreducible forms of systems of linear differential equations with formal power series coefficients as defined by Moser [Moser, J., 1960. The order of a singularity in Fuchs' theory. Math. Z. 379-398] and its generalisation, the super-...
Symbolic methods for solving systems of linear ordinary differential equations
ISSAC '10: Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation

The main purpose of this tutorial is to present and explain symbolic methods for studying systems of linear ordinary differential equations with emphasis on direct methods and their implementation in computer algebra systems.

Comments

Please enable JavaScript to view thecomments powered by Disqus.

Information & Contributors

Information

Published In

cover image ACM Conferences

ISSAC '08: Proceedings of the twenty-first international symposium on Symbolic and algebraic computation

July 2008

348 pages

ISBN:9781595939043

DOI:10.1145/1390768

General Chair:
J. Rafael Sendra
Universidad de Alcalá, Spain
,
Program Chair:
Laureano Gonzalez-Vega
Universidad de Cantabria, Spain

Copyright © 2008 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

Sponsors

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 20 July 2008

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Author Tags

Qualifiers

Research-article

Conference

ISSAC '08

Sponsor:

ISSAC '08: International Symposium on Symbolic and Algebraic Computation

July 20 - 23, 2008

Linz/Hagenberg, Austria

Acceptance Rates

Overall Acceptance Rate 395 of 838 submissions, 47%

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

6
Total Citations
View Citations
143
Total Downloads

Downloads (Last 12 months)6
Downloads (Last 6 weeks)1

Reflects downloads up to 08 Mar 2025

Other Metrics

View Author Metrics

Citations

Cited By

Barkatou MCluzeau TEl Hajj ADavenport JWang DKauers MBradford R(2019)Simple Forms and Rational Solutions of Pseudo-Linear SystemsProceedings of the 2019 International Symposium on Symbolic and Algebraic Computation10.1145/3326229.3326246(26-33)Online publication date: 8-Jul-2019
https://dl.acm.org/doi/10.1145/3326229.3326246
Barkatou MCluzeau TJalouli ARobertz DLinton SYokoyama K(2015)Formal Solutions of Linear Differential Systems with Essential Singularities in their CoefficientsProceedings of the 2015 ACM International Symposium on Symbolic and Algebraic Computation10.1145/2755996.2756669(45-52)Online publication date: 24-Jun-2015
https://dl.acm.org/doi/10.1145/2755996.2756669
Barkatou MMaddah SRobertz DLinton SYokoyama K(2015)Removing Apparent Singularities of Systems of Linear Differential Equations with Rational Function CoefficientsProceedings of the 2015 ACM International Symposium on Symbolic and Algebraic Computation10.1145/2755996.2756668(53-60)Online publication date: 24-Jun-2015
https://dl.acm.org/doi/10.1145/2755996.2756668
Maddah SBarkatou MAbbas HNagasaka KWinkler FSzanto A(2014)On the reduction of singularly-perturbed linear differential systemsProceedings of the 39th International Symposium on Symbolic and Algebraic Computation10.1145/2608628.2608655(320-327)Online publication date: 23-Jul-2014
https://dl.acm.org/doi/10.1145/2608628.2608655
Barkatou MPflügel EStan F(2013)ISOLDEACM Communications in Computer Algebra10.1145/2429135.242917046:3/4(157-159)Online publication date: 15-Jan-2013
https://dl.acm.org/doi/10.1145/2429135.2429170
Barkatou MBroughton GPflügel E(2011)A Monomial-by-Monomial Method for Computing Regular Solutions of Systems of Pseudo-Linear EquationsMathematics in Computer Science10.1007/s11786-010-0056-z4:2-3(267-288)Online publication date: 20-Jan-2011
https://doi.org/10.1007/s11786-010-0056-z

View Options

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Publication

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

Figures

Tables

Media

View Table of Conten