eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2006-10-25
6271
1
16
10.4230/DagSemProc.06271.1
article
06271 Abstracts Collection – Challenges in Symbolic Computation Software
Decker, Wolfram
Dewar, Mike
Kaltofen, Erich
Watt, Stephen M.
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.
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol06271/DagSemProc.06271.1/DagSemProc.06271.1.pdf
Symbolic computation
computer algebra
computational algebraic geometry
combinatorial methods in algebra
hybrid
symbolic-numerical methods
algorithm design
symbolic computation languages
systems and user interfaces
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2006-10-25
6271
1
2
10.4230/DagSemProc.06271.2
article
06271 Executive Summary - Challenges in Symbolic Computation Software
Decker, Wolfram
Dewar, Mike
Kaltofen, Erich
Watt, Stephen M.
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.
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol06271/DagSemProc.06271.2/DagSemProc.06271.2.pdf
Symbolic computation
computer algebra
computational algebraic geometry
combinatorial methods in algebra
hybrid symbolic-numerical methods
algori
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2006-10-25
6271
1
18
10.4230/DagSemProc.06271.3
article
Adaptive Triangular System Solving
Dumas, Jean-Guillaume
Pernet, Clément
Roch, Jean-Louis
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.
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol06271/DagSemProc.06271.3/DagSemProc.06271.3.pdf
Adaptive and hybrid algorithms; triangular system solving; parallel and sequential degenerations
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2006-10-25
6271
1
9
10.4230/DagSemProc.06271.4
article
Bounds and algebraic algorithms in differential algebra: the ordinary case
Moreno Maza, Marc
Golubitsky, Oleg
Kondratieva, Marina V.
Ovchinnikov, Alexey
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.
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol06271/DagSemProc.06271.4/DagSemProc.06271.4.pdf
Differential algebra
Rosenfeld Groebner Algorithm
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2006-10-25
6271
1
12
10.4230/DagSemProc.06271.5
article
Challenges in Computational Commutative Algebra
Abbott, John
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.
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol06271/DagSemProc.06271.5/DagSemProc.06271.5.pdf
Academic recognition implementation OpenMath CoCoA
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2006-10-25
6271
1
6
10.4230/DagSemProc.06271.6
article
Computation of the Minimal Associated Primes
Laplagne, Santiago
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.
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol06271/DagSemProc.06271.6/DagSemProc.06271.6.pdf
Minimal associated primes
groebner basis
polynomail equations
radical
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2006-10-25
6271
1
8
10.4230/DagSemProc.06271.7
article
Computational Aspects of the Resolution of Singularities
Frühbis-Krüger, Anne
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.
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol06271/DagSemProc.06271.7/DagSemProc.06271.7.pdf
Resolution of Singularities
algorithmic desingularization,
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2006-10-25
6271
1
14
10.4230/DagSemProc.06271.8
article
Coxeter Lattice Paths
Ashby, Thomas J.
Kennedy, Anthony D.
Watt, Stephen M.
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.
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol06271/DagSemProc.06271.8/DagSemProc.06271.8.pdf
Parallel computing
code generation
Coxeter groups
regular lattices
lattice paths
path optimisation
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2006-10-25
6271
1
10
10.4230/DagSemProc.06271.9
article
Decomposition of Differential Polynomials
Gao, Xiao-Shan
Zhang, Mingbo
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.
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol06271/DagSemProc.06271.9/DagSemProc.06271.9.pdf
Decomposition
differential polynomial
difference polynomial
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2006-10-25
6271
1
3
10.4230/DagSemProc.06271.10
article
GNU TeXmacs
van der Hoeven, Joris
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.
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol06271/DagSemProc.06271.10/DagSemProc.06271.10.pdf
Scientific text editor
Mathematics
Computer algebra system
Front-end
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2006-10-25
6271
1
8
10.4230/DagSemProc.06271.11
article
MathBrush: An Experimental Pen-Based Math System
Labahn, George
MacLean, Scott
Marzouk Mirette
Rutherford, Ian
Tausky, David
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.
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol06271/DagSemProc.06271.11/DagSemProc.06271.11.pdf
PC Tablets
pen-based devices
computer algebra systems
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2006-10-25
6271
1
6
10.4230/DagSemProc.06271.12
article
Notes on computing minimal approximant bases
Storjohann, Arne
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 $Theta(N/m)$.
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol06271/DagSemProc.06271.12/DagSemProc.06271.12.pdf
Hermite Pade approximation
minimal approximant bases
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2006-10-25
6271
1
0
10.4230/DagSemProc.06271.13
article
Pivot-Free Block Matrix Inversion
Watt, Stephen M.
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.
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol06271/DagSemProc.06271.13/DagSemProc.06271.13.pdf
Linear algebra
block matrices
matrix inverse
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2006-10-25
6271
1
11
10.4230/DagSemProc.06271.14
article
Probabilistically Stable Numerical Sparse Polynomial Interpolation
Giesbrecht, Mark
Labahn, George
Lee, Wen-Shin
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.
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol06271/DagSemProc.06271.14/DagSemProc.06271.14.pdf
Symbolic-numeric computing
multivariate interpolation
sparse polynomial
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2006-10-25
6271
1
20
10.4230/DagSemProc.06271.15
article
Two Families of Algorithms for Symbolic Polynomials
Watt, Stephen M.
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.
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol06271/DagSemProc.06271.15/DagSemProc.06271.15.pdf
Computer algebra
symbolic computation
factorization
gcd
symbolic exponents
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dagstuhl Seminar Proceedings
1862-4405
2006-10-25
6271
1
5
10.4230/DagSemProc.06271.16
article
Using fast matrix multiplication to solve structured linear systems
Schost, Eric
Bostan, Alin
Jeannerod, Claude-Pierre
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.
https://drops.dagstuhl.de/storage/16dagstuhl-seminar-proceedings/dsp-vol06271/DagSemProc.06271.16/DagSemProc.06271.16.pdf
Structured matrices
matrix multiplication
Hermite-Pade
bivariate interpolation