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Dagstuhl Seminar Proceedings, Volume 6271

Publication Details

  • published at: 2006-10-25
  • Publisher: Schloss Dagstuhl – Leibniz-Zentrum für Informatik

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DOI: 10.4230/DagSemProc.06271.1
06271 Abstracts Collection – Challenges in Symbolic Computation Software

Authors: Wolfram Decker, Mike Dewar, Erich Kaltofen, and Stephen M. Watt

Abstract
From 02.07.06 to 07.07.06, the Dagstuhl Seminar 06271 ``Challenges in Symbolic Computation Software'' was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available.
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Wolfram Decker, Mike Dewar, Erich Kaltofen, and Stephen M. Watt. 06271 Abstracts Collection – Challenges in Symbolic Computation Software. In Challenges in Symbolic Computation Software. Dagstuhl Seminar Proceedings, Volume 6271, pp. 1-16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)

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@InProceedings{decker_et_al:DagSemProc.06271.1,
  author =	{Decker, Wolfram and Dewar, Mike and Kaltofen, Erich and Watt, Stephen M.},
  title =	{{06271 Abstracts Collection – Challenges in Symbolic Computation Software}},
  booktitle =	{Challenges in Symbolic Computation Software},
  pages =	{1--16},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2006},
  volume =	{6271},
  editor =	{Wolfram Decker and Mike Dewar and Erich Kaltofen and Stephen Watt},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.06271.1},
  URN =		{urn:nbn:de:0030-drops-7814},
  doi =		{10.4230/DagSemProc.06271.1},
  annote =	{Keywords: Symbolic computation, computer algebra, computational algebraic geometry, combinatorial methods in algebra, hybrid, symbolic-numerical methods, algorithm design, symbolic computation languages, systems and user interfaces}
}
Document
DOI: 10.4230/DagSemProc.06271.2
06271 Executive Summary - Challenges in Symbolic Computation Software

Authors: Wolfram Decker, Mike Dewar, Erich Kaltofen, and Stephen M. Watt

Abstract
Symbolic computation software allows mathematicians, scientists, engineers, or educators to deal with elaborate calculations using a computer. The applications range from introducing the experimental method in fields of pure mathematics to practical applications, for instance, in cryptology, robotics, or signal theory. The software includes mainstream commercial products such as Maple or Mathematica and highly specialized, public domain systems such as CoCoa, Macaulay2, or Singular. Symbolic computation software implements a variety of sophisticated algorithms on polynomials, matrices, combinatorial structures, and other mathematical objects in a multitude of different dense, sparse, or implicit (black box) representations. The subject of the seminar was innovation in algorithms and software, bringing algorithm designers, software builders, and software users together.
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Wolfram Decker, Mike Dewar, Erich Kaltofen, and Stephen M. Watt. 06271 Executive Summary - Challenges in Symbolic Computation Software. In Challenges in Symbolic Computation Software. Dagstuhl Seminar Proceedings, Volume 6271, pp. 1-2, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)

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@InProceedings{decker_et_al:DagSemProc.06271.2,
  author =	{Decker, Wolfram and Dewar, Mike and Kaltofen, Erich and Watt, Stephen M.},
  title =	{{06271 Executive Summary - Challenges in Symbolic Computation Software}},
  booktitle =	{Challenges in Symbolic Computation Software},
  pages =	{1--2},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2006},
  volume =	{6271},
  editor =	{Wolfram Decker and Mike Dewar and Erich Kaltofen and Stephen Watt},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.06271.2},
  URN =		{urn:nbn:de:0030-drops-7778},
  doi =		{10.4230/DagSemProc.06271.2},
  annote =	{Keywords: Symbolic computation, computer algebra, computational algebraic geometry, combinatorial methods in algebra, hybrid symbolic-numerical methods, algori}
}
Document
DOI: 10.4230/DagSemProc.06271.3
Adaptive Triangular System Solving

Authors: Jean-Guillaume Dumas, Clément Pernet, and Jean-Louis Roch

Abstract
Large-scale applications and software systems are getting increasingly complex. To deal with this complexity, those systems must manage themselves in accordance with high-level guidance from humans. Adaptive and hybrid algorithms enable this self-management of resources and structured inputs. In this talk, we first propose a classification of the different notions of adaptivity. For us, an algorithm is adaptive (or a poly-algorithm) when there is a choice at a high level between at least two distinct algorithms, each of which could solve the same problem. The choice is strategic, not tactical. It is motivated by an increase of the performance of the execution, depending on both input/output data and computing resources. Then we propose a new adaptive algorithm for the exact simultaneous resolution of several triangular systems over finite fields. The resolution of such systems is e.g. one of the two main operations in block Gaussian elimination. For solving triangular systems over finite fields, the block algorithm reduces to matrix multiplication and achieves the best known algebraic complexity. Exact matrix multiplication, together with matrix factorizations, over finite fields can now be performed at the speed of the highly optimized numerical BLAS routines. This has been established by the FFLAS and FFPACK libraries. In this talk we propose several practicable variants solving these systems: a pure recursive version, a reduction to the numerical dtrsm routine and a delaying of the modulus operation. Then a cascading scheme is proposed to merge these variants into an adaptive sequential algorithm. We then propose a parallelization of this resolution. The adaptive sequential algorithm is not the best parallel algorithm since its recursion induces a dependancy. A better parallel algorithm would be to first invert the matrix and then to multiply this inverse by the right hand side. Unfortunately the latter requires more total operations than the adaptive algorithm. We thus propose a coupling of the sequential algorithm and of the parallel one in order to get the best performances on any number of processors. The resulting cascading is then an adaptation to resources. This shows that the same process has been used both for adaptation to data and to resources. We thus propose a generic framework for the automatic adaptation of algorithms using recursive cascading.
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Jean-Guillaume Dumas, Clément Pernet, and Jean-Louis Roch. Adaptive Triangular System Solving. In Challenges in Symbolic Computation Software. Dagstuhl Seminar Proceedings, Volume 6271, pp. 1-18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)

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@InProceedings{dumas_et_al:DagSemProc.06271.3,
  author =	{Dumas, Jean-Guillaume and Pernet, Cl\'{e}ment and Roch, Jean-Louis},
  title =	{{Adaptive Triangular System Solving}},
  booktitle =	{Challenges in Symbolic Computation Software},
  pages =	{1--18},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2006},
  volume =	{6271},
  editor =	{Wolfram Decker and Mike Dewar and Erich Kaltofen and Stephen Watt},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.06271.3},
  URN =		{urn:nbn:de:0030-drops-7704},
  doi =		{10.4230/DagSemProc.06271.3},
  annote =	{Keywords: Adaptive and hybrid algorithms; triangular system solving; parallel and sequential degenerations}
}
Document
DOI: 10.4230/DagSemProc.06271.4
Bounds and algebraic algorithms in differential algebra: the ordinary case

Authors: Marc Moreno Maza, Oleg Golubitsky, Marina V. Kondratieva, and Alexey Ovchinnikov

Abstract
Consider the Rosenfeld-Groebner algorithm for computing a regular decomposition of a radical differential ideal generated by a set of ordinary differential polynomials. This algorithm inputs a system of differential polynomials and a ranking on derivatives and constructs finitely many regular systems equivalent to the original one. The property of regularity allows to check consistency of the systems and membership to the corresponding differential ideals. We propose a bound on the orders of derivatives occurring in all intermediate and final systems computed by the Rosenfeld-Groebner algorithm and outline its proof. We also reduce the problem of conversion of a regular decomposition of a radical differential ideal from one ranking to another to a purely algebraic problem.
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Marc Moreno Maza, Oleg Golubitsky, Marina V. Kondratieva, and Alexey Ovchinnikov. Bounds and algebraic algorithms in differential algebra: the ordinary case. In Challenges in Symbolic Computation Software. Dagstuhl Seminar Proceedings, Volume 6271, pp. 1-9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2007)

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@InProceedings{morenomaza_et_al:DagSemProc.06271.4,
  author =	{Moreno Maza, Marc and Golubitsky, Oleg and Kondratieva, Marina V. and Ovchinnikov, Alexey},
  title =	{{Bounds and algebraic algorithms in differential algebra: the ordinary case}},
  booktitle =	{Challenges in Symbolic Computation Software},
  pages =	{1--9},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2007},
  volume =	{6271},
  editor =	{Wolfram Decker and Mike Dewar and Erich Kaltofen and Stephen Watt},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.06271.4},
  URN =		{urn:nbn:de:0030-drops-10219},
  doi =		{10.4230/DagSemProc.06271.4},
  annote =	{Keywords: Differential algebra, Rosenfeld Groebner Algorithm}
}
Document
DOI: 10.4230/DagSemProc.06271.5
Challenges in Computational Commutative Algebra

Authors: John Abbott

Abstract
In this paper we consider a number of challenges from the point of view of the CoCoA project one of whose tasks is to develop software specialized for computations in commutative algebra. Some of the challenges extend considerably beyond the boundary of commutative algebra, and are addressed to the computer algebra community as a whole.
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John Abbott. Challenges in Computational Commutative Algebra. In Challenges in Symbolic Computation Software. Dagstuhl Seminar Proceedings, Volume 6271, pp. 1-12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)

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@InProceedings{abbott:DagSemProc.06271.5,
  author =	{Abbott, John},
  title =	{{Challenges in Computational Commutative Algebra}},
  booktitle =	{Challenges in Symbolic Computation Software},
  pages =	{1--12},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2006},
  volume =	{6271},
  editor =	{Wolfram Decker and Mike Dewar and Erich Kaltofen and Stephen Watt},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.06271.5},
  URN =		{urn:nbn:de:0030-drops-7682},
  doi =		{10.4230/DagSemProc.06271.5},
  annote =	{Keywords: Academic recognition implementation OpenMath CoCoA}
}
Document
DOI: 10.4230/DagSemProc.06271.6
Computation of the Minimal Associated Primes

Authors: Santiago Laplagne

Abstract
Solving systems of polynomial equations is a main task in Computer Algebra, although the precise meaning of what is an acceptable solution depends on the context. In this talk, we interpret it as finding the minimal associated primes of the ideal generated by the polynomials. Geometrically, this is equivalent to decompose the set of solutions into its irreducible components. We study the existing algorithms, and propose some modifications. A common technique used is to reduce the problem to the zero dimensional case. In a paper by Gianni, Trager and Zacharias they use this technique, combined with the splitting tool $I = (I : h^infty) cap langle I, h^m angle$ for some specific polynomial $h$ and integer $m$. This splitting introduces a number of redundant components that are not part of the original ideal. In the algorithm we present here, we use the reduction to the zero dimensional case, but we avoid working with the ideal $langle I, h^m angle$. As a result, when the ideal has components of different dimensions, our algorithm is usually more efficient.
Cite as

Santiago Laplagne. Computation of the Minimal Associated Primes. In Challenges in Symbolic Computation Software. Dagstuhl Seminar Proceedings, Volume 6271, pp. 1-6, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)

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@InProceedings{laplagne:DagSemProc.06271.6,
  author =	{Laplagne, Santiago},
  title =	{{Computation of the Minimal Associated Primes}},
  booktitle =	{Challenges in Symbolic Computation Software},
  pages =	{1--6},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2006},
  volume =	{6271},
  editor =	{Wolfram Decker and Mike Dewar and Erich Kaltofen and Stephen Watt},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.06271.6},
  URN =		{urn:nbn:de:0030-drops-7741},
  doi =		{10.4230/DagSemProc.06271.6},
  annote =	{Keywords: Minimal associated primes, groebner basis, polynomail equations, radical}
}
Document
DOI: 10.4230/DagSemProc.06271.7
Computational Aspects of the Resolution of Singularities

Authors: Anne Frühbis-Krüger

Abstract
The task of resolution of singularities has been one of the central topics in Algebraic Geometry for many decades. After results in low dimension in the first half of the 20th century, it was Hironaka's monumental article in 1964 which solved the porblem in the general case in chareacteristic zero. The case of characteristic $p > 0$ is still unsolved except in partial results in low dimension. But Hironaka's proof did not put an end to the interest in characteric zero, instead it shifted the focus toward the task of finding a more constructive approach. Such algorithmic approaches appeared at the end of the 1980's independently by Villamayor and by Bierstone and Milman. In this talk we consider the computational tasks arising from Villamayor's algorithm and present an implementation.
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Anne Frühbis-Krüger. Computational Aspects of the Resolution of Singularities. In Challenges in Symbolic Computation Software. Dagstuhl Seminar Proceedings, Volume 6271, pp. 1-8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)

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@InProceedings{fruhbiskruger:DagSemProc.06271.7,
  author =	{Fr\"{u}hbis-Kr\"{u}ger, Anne},
  title =	{{Computational Aspects of the Resolution of Singularities}},
  booktitle =	{Challenges in Symbolic Computation Software},
  pages =	{1--8},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2006},
  volume =	{6271},
  editor =	{Wolfram Decker and Mike Dewar and Erich Kaltofen and Stephen Watt},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.06271.7},
  URN =		{urn:nbn:de:0030-drops-7713},
  doi =		{10.4230/DagSemProc.06271.7},
  annote =	{Keywords: Resolution of Singularities, algorithmic desingularization,}
}
Document
DOI: 10.4230/DagSemProc.06271.8
Coxeter Lattice Paths

Authors: Thomas J. Ashby, Anthony D. Kennedy, and Stephen M. Watt

Abstract
This talk concerns generating code for running computationally intensive numerical lattice QCD simulations on large parallel computers, using an approach based on the theory of Coxeter groups. Many physical systems have inherent symmetry, and this is usually implicit in the calculations needed to simulate them using discrete approximations, and thus in the associated code. By reversing this and basing the generation of code on the symmetry group of the lattice in question, we arrive at a very natural way of generating and reasoning about programs. The principal aim is a formal way of representing lattices and the paths on these lattices that correspond to the required calculations. This foundation allows the creation and manipulation of lattices and paths to be automated, obviating what can be a labour-intensive and errorprone task. In more detail, a method will be given for representing the points of a regular lattice as elements of the translation subgroup of an affine Coxeter group, by finding the subgroup generators starting from the Coxeter graph of the affine group. Similarly, step sequences are derived as words in the free group generated by the translation subgroup generators themselves. We introduce code generation techniques and the automation of two code optimisations; the grouping of paths into equivalence classes, and the factoring out of common path segments. The latter technique reduces the amount of communication necessary between nodes, and is thus likely to be important in practice.
Cite as

Thomas J. Ashby, Anthony D. Kennedy, and Stephen M. Watt. Coxeter Lattice Paths. In Challenges in Symbolic Computation Software. Dagstuhl Seminar Proceedings, Volume 6271, pp. 1-14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)

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@InProceedings{ashby_et_al:DagSemProc.06271.8,
  author =	{Ashby, Thomas J. and Kennedy, Anthony D. and Watt, Stephen M.},
  title =	{{Coxeter Lattice Paths}},
  booktitle =	{Challenges in Symbolic Computation Software},
  pages =	{1--14},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2006},
  volume =	{6271},
  editor =	{Wolfram Decker and Mike Dewar and Erich Kaltofen and Stephen Watt},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.06271.8},
  URN =		{urn:nbn:de:0030-drops-7695},
  doi =		{10.4230/DagSemProc.06271.8},
  annote =	{Keywords: Parallel computing, code generation, Coxeter groups, regular lattices, lattice paths, path optimisation}
}
Document
DOI: 10.4230/DagSemProc.06271.9
Decomposition of Differential Polynomials

Authors: Xiao-Shan Gao and Mingbo Zhang

Abstract
We present an algorithm to decompose nonlinear differential polynomials in one variable and with rational functions as coefficients. The algorithm is implemented in Maple for the {em constant field} case. The program can be used to decompose differential polynomials with more than one thousand terms effectively.
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Xiao-Shan Gao and Mingbo Zhang. Decomposition of Differential Polynomials. In Challenges in Symbolic Computation Software. Dagstuhl Seminar Proceedings, Volume 6271, pp. 1-10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)

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@InProceedings{gao_et_al:DagSemProc.06271.9,
  author =	{Gao, Xiao-Shan and Zhang, Mingbo},
  title =	{{Decomposition of Differential Polynomials}},
  booktitle =	{Challenges in Symbolic Computation Software},
  pages =	{1--10},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2006},
  volume =	{6271},
  editor =	{Wolfram Decker and Mike Dewar and Erich Kaltofen and Stephen Watt},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.06271.9},
  URN =		{urn:nbn:de:0030-drops-7726},
  doi =		{10.4230/DagSemProc.06271.9},
  annote =	{Keywords: Decomposition, differential polynomial, difference polynomial}
}
Document
DOI: 10.4230/DagSemProc.06271.10
GNU TeXmacs

Authors: Joris van der Hoeven

Abstract
GNU TeXmacs is a free scientific editing platform with special features for mathematicians. The editor can be used to produce documents with a professional typesetting quality (better than TeX/LaTeX) via a user-friendly front-end. The editor can be used as a front-end to several computer algebra systems and includes a lot of additional facilities, like a presentation mode, a technical picture editor, a typed linking tool, etc. The editor can be extended by users in several ways: using style files, plug-ins or via the Scheme extension language. Converters exist for LaTeX, Xhtml and MathML.
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Joris van der Hoeven. GNU TeXmacs. In Challenges in Symbolic Computation Software. Dagstuhl Seminar Proceedings, Volume 6271, pp. 1-3, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)

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@InProceedings{vanderhoeven:DagSemProc.06271.10,
  author =	{van der Hoeven, Joris},
  title =	{{GNU TeXmacs}},
  booktitle =	{Challenges in Symbolic Computation Software},
  pages =	{1--3},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2006},
  volume =	{6271},
  editor =	{Wolfram Decker and Mike Dewar and Erich Kaltofen and Stephen Watt},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.06271.10},
  URN =		{urn:nbn:de:0030-drops-7672},
  doi =		{10.4230/DagSemProc.06271.10},
  annote =	{Keywords: Scientific text editor, Mathematics, Computer algebra system, Front-end}
}
Document
DOI: 10.4230/DagSemProc.06271.11
MathBrush: An Experimental Pen-Based Math System

Authors: George Labahn, Scott MacLean, Marzouk Mirette, Ian Rutherford, and David Tausky

Abstract
It is widely believed that mathematics will be one of the major applications for Tablet PCs and other pen-based devices. In this paper we discuss many of the issues that make doing mathematics on such pen-based devices a hard task. We give a preliminary description of an experimental system, currently named MathBrush, for working with mathematics using pen-based devices. The system allows a user to enter mathematical expressions with a pen and to then do mathematical computation using a computer algebra system. The system provides a simple and easy way for users to verify the correctness of their handwritten expressions and, if needed, to correct any errors in recognition. Choosing mathematical operations is done making use of context menus, both with input and output expressions.
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George Labahn, Scott MacLean, Marzouk Mirette, Ian Rutherford, and David Tausky. MathBrush: An Experimental Pen-Based Math System. In Challenges in Symbolic Computation Software. Dagstuhl Seminar Proceedings, Volume 6271, pp. 1-8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)

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@InProceedings{labahn_et_al:DagSemProc.06271.11,
  author =	{Labahn, George and MacLean, Scott and Marzouk Mirette and Rutherford, Ian and Tausky, David},
  title =	{{MathBrush: An Experimental Pen-Based Math System}},
  booktitle =	{Challenges in Symbolic Computation Software},
  pages =	{1--8},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2006},
  volume =	{6271},
  editor =	{Wolfram Decker and Mike Dewar and Erich Kaltofen and Stephen Watt},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.06271.11},
  URN =		{urn:nbn:de:0030-drops-7733},
  doi =		{10.4230/DagSemProc.06271.11},
  annote =	{Keywords: PC Tablets, pen-based devices, computer algebra systems}
}
Document
DOI: 10.4230/DagSemProc.06271.12
Notes on computing minimal approximant bases

Authors: Arne Storjohann

Abstract
We show how to transform the problem of computing solutions to a classical Hermite Pade approximation problem for an input vector of dimension $m imes 1$, arbitrary degree constraints $(n_1,n_2,ldots,n_m)$, and order $N := (n_1 + 1) + cdots + (n_m + 1) - 1$, to that of computing a minimal approximant basis for a matrix of dimension $O(m) imes O(m)$, uniform degree constraint $Theta(N/m)$, and order $Theta(N/m)$.
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Arne Storjohann. Notes on computing minimal approximant bases. In Challenges in Symbolic Computation Software. Dagstuhl Seminar Proceedings, Volume 6271, pp. 1-6, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)

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@InProceedings{storjohann:DagSemProc.06271.12,
  author =	{Storjohann, Arne},
  title =	{{Notes on computing minimal approximant bases}},
  booktitle =	{Challenges in Symbolic Computation Software},
  pages =	{1--6},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2006},
  volume =	{6271},
  editor =	{Wolfram Decker and Mike Dewar and Erich Kaltofen and Stephen Watt},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.06271.12},
  URN =		{urn:nbn:de:0030-drops-7763},
  doi =		{10.4230/DagSemProc.06271.12},
  annote =	{Keywords: Hermite Pade approximation, minimal approximant bases}
}
Document
DOI: 10.4230/DagSemProc.06271.13
Pivot-Free Block Matrix Inversion

Authors: Stephen M. Watt

Abstract
We present a pivot-free deterministic algorithm for the inversion of block matrices. The method is based on the Moore-Penrose inverse and is applicable over certain general classes of rings. This improves on previous methods that required at least one invertible on-diagonal block, and that otherwise required row- or column-based pivoting, disrupting the block structure. Our method is applicable to any invertible matrix and does not require any particular blocks to invertible. This is achieved at the cost of two additional specialized matrix multiplications and, in some cases, requires the inversion to be performed in an extended ring.
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Stephen M. Watt. Pivot-Free Block Matrix Inversion. In Challenges in Symbolic Computation Software. Dagstuhl Seminar Proceedings, Volume 6271, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)

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@InProceedings{watt:DagSemProc.06271.13,
  author =	{Watt, Stephen M.},
  title =	{{Pivot-Free Block Matrix Inversion}},
  booktitle =	{Challenges in Symbolic Computation Software},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2006},
  volume =	{6271},
  editor =	{Wolfram Decker and Mike Dewar and Erich Kaltofen and Stephen Watt},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.06271.13},
  URN =		{urn:nbn:de:0030-drops-7806},
  doi =		{10.4230/DagSemProc.06271.13},
  annote =	{Keywords: Linear algebra, block matrices, matrix inverse}
}
Document
DOI: 10.4230/DagSemProc.06271.14
Probabilistically Stable Numerical Sparse Polynomial Interpolation

Authors: Mark Giesbrecht, George Labahn, and Wen-Shin Lee

Abstract
We consider the problem of sparse interpolation of a multivariate black-box polynomial in floating-point arithmetic. That is, both the inputs and outputs of the black-box polynomial have some error, and all values are represented in standard, fixed-precision, floating-point arithmetic. By interpolating the black box evaluated at random primitive roots of unity, we give an efficient and numerically robust solution with high probability. We outline the numerical stability of our algorithm, as well as the expected conditioning achieved through randomization. Finally, we demonstrate the effectiveness of our techniques through numerical experiments.
Cite as

Mark Giesbrecht, George Labahn, and Wen-Shin Lee. Probabilistically Stable Numerical Sparse Polynomial Interpolation. In Challenges in Symbolic Computation Software. Dagstuhl Seminar Proceedings, Volume 6271, pp. 1-11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)

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@InProceedings{giesbrecht_et_al:DagSemProc.06271.14,
  author =	{Giesbrecht, Mark and Labahn, George and Lee, Wen-Shin},
  title =	{{Probabilistically Stable Numerical Sparse Polynomial Interpolation}},
  booktitle =	{Challenges in Symbolic Computation Software},
  pages =	{1--11},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2006},
  volume =	{6271},
  editor =	{Wolfram Decker and Mike Dewar and Erich Kaltofen and Stephen Watt},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.06271.14},
  URN =		{urn:nbn:de:0030-drops-7759},
  doi =		{10.4230/DagSemProc.06271.14},
  annote =	{Keywords: Symbolic-numeric computing, multivariate interpolation, sparse polynomial}
}
Document
DOI: 10.4230/DagSemProc.06271.15
Two Families of Algorithms for Symbolic Polynomials

Authors: Stephen M. Watt

Abstract
We wish to work with polynomials where the exponents are not known in advance, such as $x^{2n} - 1$. There are various operations we will want to be able to do, such as squaring the value to get $x^{4n}-2x^{2n}+1$, or differentiating it to get $2nx^{2n-1}$. Expressions of this sort arise frequently in practice, for example in the analysis of algorithms, and it is very difficult to work with them effectively in current computer algebra systems. We consider the case where multivariate polynomials can have exponents which are themselves integer-valued multivariate polynomials, and we present algorithms to compute their GCD and factorization. The algorithms fall into two families: algebraic extension methods and interpolation methods. The first family of algorithms uses the algebraic independence of $x$, $x^n$, $x^{n^2}$, $x^{nm}, etc, to solve related problems with more indeterminates. Some subtlety is needed to avoid problems with fixed divisors of the exponent polynomials. The second family of algorithms uses evaluation and interpolation of the exponent polynomials. While these methods can run into unlucky evaluation points, in many cases they can be more appealing. Additionally, we also treat the case of symbolic exponents on rational coefficients (e.g. $4^{n^2+n}-81$) and show how to avoid integer factorization.
Cite as

Stephen M. Watt. Two Families of Algorithms for Symbolic Polynomials. In Challenges in Symbolic Computation Software. Dagstuhl Seminar Proceedings, Volume 6271, pp. 1-20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)

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@InProceedings{watt:DagSemProc.06271.15,
  author =	{Watt, Stephen M.},
  title =	{{Two Families of Algorithms for Symbolic Polynomials}},
  booktitle =	{Challenges in Symbolic Computation Software},
  pages =	{1--20},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2006},
  volume =	{6271},
  editor =	{Wolfram Decker and Mike Dewar and Erich Kaltofen and Stephen Watt},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.06271.15},
  URN =		{urn:nbn:de:0030-drops-7933},
  doi =		{10.4230/DagSemProc.06271.15},
  annote =	{Keywords: Computer algebra, symbolic computation, factorization, gcd, symbolic exponents}
}
Document
DOI: 10.4230/DagSemProc.06271.16
Using fast matrix multiplication to solve structured linear systems

Authors: Eric Schost, Alin Bostan, and Claude-Pierre Jeannerod

Abstract
Structured linear algebra techniques are a versatile set of tools; they enable one to deal at once with various types of matrices, with features such as Toeplitz-, Hankel-, Vandermonde- or Cauchy-likeness. Following Kailath, Kung and Morf (1979), the usual way of measuring to what extent a matrix possesses one such structure is through its displacement rank, that is, the rank of its image through a suitable displacement operator. Then, for the families of matrices given above, the results of Bitmead-Anderson, Morf, Kaltofen, Gohberg-Olshevsky, Pan (among others) provide algorithm of complexity $O(alpha^2 n)$, up to logarithmic factors, where $n$ is the matrix size and $alpha$ its displacement rank. We show that for Toeplitz- Vandermonde-like matrices, this cost can be reduced to $O(alpha^(omega-1) n)$, where $omega$ is an exponent for linear algebra. We present consequences for Hermite-Pad'e approximation and bivariate interpolation.
Cite as

Eric Schost, Alin Bostan, and Claude-Pierre Jeannerod. Using fast matrix multiplication to solve structured linear systems. In Challenges in Symbolic Computation Software. Dagstuhl Seminar Proceedings, Volume 6271, pp. 1-5, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)

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@InProceedings{schost_et_al:DagSemProc.06271.16,
  author =	{Schost, Eric and Bostan, Alin and Jeannerod, Claude-Pierre},
  title =	{{Using fast matrix multiplication to solve structured linear systems}},
  booktitle =	{Challenges in Symbolic Computation Software},
  pages =	{1--5},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2006},
  volume =	{6271},
  editor =	{Wolfram Decker and Mike Dewar and Erich Kaltofen and Stephen Watt},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.06271.16},
  URN =		{urn:nbn:de:0030-drops-7787},
  doi =		{10.4230/DagSemProc.06271.16},
  annote =	{Keywords: Structured matrices, matrix multiplication, Hermite-Pade, bivariate interpolation}
}

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