{"@context":"https:\/\/schema.org\/","@type":"PublicationVolume","@id":"#volume615","volumeNumber":6271,"name":"Dagstuhl Seminar Proceedings, Volume 6271","dateCreated":"2006-10-25","datePublished":"2006-10-25","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":{"@type":"Periodical","@id":"#series119","name":"Dagstuhl Seminar Proceedings","issn":"1862-4405","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume615"},"hasPart":[{"@type":"ScholarlyArticle","@id":"#article1504","name":"06271 Abstracts Collection \u2013 Challenges in Symbolic Computation Software","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.\r\nDuring the seminar, several participants presented their current\r\nresearch, and ongoing work and open problems were discussed. Abstracts of\r\nthe presentations given during the seminar as well as abstracts of\r\nseminar results and ideas are put together in this paper. The first section\r\ndescribes the seminar topics and goals in general.\r\nLinks to extended abstracts or full papers are provided, if available.","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"],"author":[{"@type":"Person","name":"Decker, Wolfram","givenName":"Wolfram","familyName":"Decker"},{"@type":"Person","name":"Dewar, Mike","givenName":"Mike","familyName":"Dewar"},{"@type":"Person","name":"Kaltofen, Erich","givenName":"Erich","familyName":"Kaltofen"},{"@type":"Person","name":"Watt, Stephen M.","givenName":"Stephen M.","familyName":"Watt"}],"position":1,"pageStart":1,"pageEnd":16,"dateCreated":"2006-11-03","datePublished":"2006-11-03","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Decker, Wolfram","givenName":"Wolfram","familyName":"Decker"},{"@type":"Person","name":"Dewar, Mike","givenName":"Mike","familyName":"Dewar"},{"@type":"Person","name":"Kaltofen, Erich","givenName":"Erich","familyName":"Kaltofen"},{"@type":"Person","name":"Watt, Stephen M.","givenName":"Stephen M.","familyName":"Watt"}],"copyrightYear":"2006","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.06271.1","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume615"},{"@type":"ScholarlyArticle","@id":"#article1505","name":"06271 Executive Summary - Challenges in Symbolic Computation Software","abstract":"Symbolic computation software allows mathematicians,\r\nscientists, engineers, or educators to deal with\r\nelaborate calculations using a computer. The applications\r\nrange from introducing the experimental method in fields of\r\npure mathematics to practical applications, for instance,\r\nin cryptology, robotics, or signal theory. The\r\nsoftware includes mainstream commercial products\r\nsuch as Maple or Mathematica and highly specialized,\r\npublic domain systems such as CoCoa, Macaulay2, or Singular.\r\n\r\nSymbolic computation software implements a variety\r\nof sophisticated algorithms on polynomials, matrices,\r\ncombinatorial structures, and other mathematical\r\nobjects in a multitude of different dense, sparse,\r\nor implicit (black box) representations.\r\n\r\nThe subject of the seminar was innovation in algorithms\r\nand software, bringing algorithm designers,\r\nsoftware builders, and software users together.","keywords":["Symbolic computation","computer algebra","computational algebraic geometry","combinatorial methods in algebra","hybrid symbolic-numerical methods","algori"],"author":[{"@type":"Person","name":"Decker, Wolfram","givenName":"Wolfram","familyName":"Decker"},{"@type":"Person","name":"Dewar, Mike","givenName":"Mike","familyName":"Dewar"},{"@type":"Person","name":"Kaltofen, Erich","givenName":"Erich","familyName":"Kaltofen"},{"@type":"Person","name":"Watt, Stephen M.","givenName":"Stephen M.","familyName":"Watt"}],"position":2,"pageStart":1,"pageEnd":2,"dateCreated":"2006-10-25","datePublished":"2006-10-25","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Decker, Wolfram","givenName":"Wolfram","familyName":"Decker"},{"@type":"Person","name":"Dewar, Mike","givenName":"Mike","familyName":"Dewar"},{"@type":"Person","name":"Kaltofen, Erich","givenName":"Erich","familyName":"Kaltofen"},{"@type":"Person","name":"Watt, Stephen M.","givenName":"Stephen M.","familyName":"Watt"}],"copyrightYear":"2006","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.06271.2","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume615"},{"@type":"ScholarlyArticle","@id":"#article1506","name":"Adaptive Triangular System Solving","abstract":"Large-scale applications and software systems are\r\ngetting increasingly complex. To deal with this complexity, those\r\nsystems must manage themselves in accordance with high-level guidance\r\nfrom humans. Adaptive and hybrid algorithms enable this\r\nself-management of resources and structured inputs.\r\n\r\nIn this talk, we first propose a classification of the different\r\nnotions of adaptivity. For us, an algorithm is adaptive (or a\r\npoly-algorithm) when there is a choice at a high level between at\r\nleast two distinct algorithms, each of which could solve the same\r\nproblem. The choice is strategic, not tactical. It is motivated by\r\nan increase of the performance of the execution, depending on both\r\ninput\/output data and computing resources. \r\n\r\nThen we propose a new adaptive algorithm for the exact simultaneous\r\nresolution of several triangular systems over finite fields. The\r\nresolution of such systems is e.g. one of the two main operations in block\r\nGaussian elimination. For solving triangular systems over finite\r\nfields, the block algorithm reduces to matrix multiplication and\r\nachieves the best known algebraic complexity. Exact matrix\r\nmultiplication, together with matrix factorizations, over finite\r\nfields can now be performed at the speed of the highly optimized\r\nnumerical BLAS routines. This has been established by the FFLAS and\r\nFFPACK libraries. In this talk we propose several practicable variants\r\nsolving these systems: a pure recursive version, a reduction to the\r\nnumerical dtrsm routine and a delaying of the modulus operation. Then\r\na cascading scheme is proposed to merge these variants into an\r\nadaptive sequential algorithm.\r\n\r\nWe then propose a parallelization of this resolution. The adaptive\r\nsequential algorithm is not the best parallel algorithm since its\r\nrecursion induces a dependancy. A better parallel algorithm would be\r\nto first invert the matrix and then to multiply this inverse by the\r\nright hand side. Unfortunately the latter requires more total\r\noperations than the adaptive algorithm. We thus propose a coupling of\r\nthe sequential algorithm and of the parallel one in order to get the\r\nbest performances on any number of processors. The resulting cascading\r\nis then an adaptation to resources. \r\n\r\nThis shows that the same process has been used both for adaptation to\r\ndata and to resources. We thus propose a generic framework for the\r\nautomatic adaptation of algorithms using recursive cascading.","keywords":"Adaptive and hybrid algorithms; triangular system solving; parallel and sequential degenerations","author":[{"@type":"Person","name":"Dumas, Jean-Guillaume","givenName":"Jean-Guillaume","familyName":"Dumas"},{"@type":"Person","name":"Pernet, Cl\u00e9ment","givenName":"Cl\u00e9ment","familyName":"Pernet"},{"@type":"Person","name":"Roch, Jean-Louis","givenName":"Jean-Louis","familyName":"Roch"}],"position":3,"pageStart":1,"pageEnd":18,"dateCreated":"2006-10-25","datePublished":"2006-10-25","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Dumas, Jean-Guillaume","givenName":"Jean-Guillaume","familyName":"Dumas"},{"@type":"Person","name":"Pernet, Cl\u00e9ment","givenName":"Cl\u00e9ment","familyName":"Pernet"},{"@type":"Person","name":"Roch, Jean-Louis","givenName":"Jean-Louis","familyName":"Roch"}],"copyrightYear":"2006","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.06271.3","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume615"},{"@type":"ScholarlyArticle","@id":"#article1507","name":"Bounds and algebraic algorithms in differential algebra: the ordinary case","abstract":"Consider the Rosenfeld-Groebner algorithm for computing a regular \r\n decomposition of a radical differential ideal generated by a set\r\n of ordinary differential polynomials. This algorithm inputs a\r\n system of differential polynomials and a\r\n ranking on derivatives and constructs finitely many regular systems\r\n equivalent to the original one. The property of\r\n regularity allows to check consistency of the systems and\r\n membership to the corresponding differential ideals. \r\n\r\n We propose a bound on the orders of derivatives \r\n occurring in all intermediate and final systems computed by the \r\n Rosenfeld-Groebner algorithm and outline its proof. \r\n\r\n We also reduce the problem of conversion of \r\n a regular decomposition of a radical\r\n differential ideal from one ranking to another to a purely\r\n algebraic problem.","keywords":["Differential algebra","Rosenfeld Groebner Algorithm"],"author":[{"@type":"Person","name":"Moreno Maza, Marc","givenName":"Marc","familyName":"Moreno Maza"},{"@type":"Person","name":"Golubitsky, Oleg","givenName":"Oleg","familyName":"Golubitsky"},{"@type":"Person","name":"Kondratieva, Marina V.","givenName":"Marina V.","familyName":"Kondratieva"},{"@type":"Person","name":"Ovchinnikov, Alexey","givenName":"Alexey","familyName":"Ovchinnikov"}],"position":4,"pageStart":1,"pageEnd":9,"dateCreated":"2007-05-21","datePublished":"2007-05-21","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Moreno Maza, Marc","givenName":"Marc","familyName":"Moreno Maza"},{"@type":"Person","name":"Golubitsky, Oleg","givenName":"Oleg","familyName":"Golubitsky"},{"@type":"Person","name":"Kondratieva, Marina V.","givenName":"Marina V.","familyName":"Kondratieva"},{"@type":"Person","name":"Ovchinnikov, Alexey","givenName":"Alexey","familyName":"Ovchinnikov"}],"copyrightYear":"2007","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.06271.4","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume615"},{"@type":"ScholarlyArticle","@id":"#article1508","name":"Challenges in Computational Commutative Algebra","abstract":"In this paper we consider a number of challenges from the point of view of\r\nthe CoCoA project one of whose tasks is to develop software specialized for\r\ncomputations in commutative algebra. Some of the challenges extend\r\nconsiderably beyond the boundary of commutative algebra, and are addressed\r\nto the computer algebra community as a whole.","keywords":"Academic recognition implementation OpenMath CoCoA","author":{"@type":"Person","name":"Abbott, John","givenName":"John","familyName":"Abbott"},"position":5,"pageStart":1,"pageEnd":12,"dateCreated":"2006-10-25","datePublished":"2006-10-25","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":{"@type":"Person","name":"Abbott, John","givenName":"John","familyName":"Abbott"},"copyrightYear":"2006","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.06271.5","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume615"},{"@type":"ScholarlyArticle","@id":"#article1509","name":"Computation of the Minimal Associated Primes","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. \r\nIn 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.\r\nWe study the existing algorithms, and propose some modifications. \r\n\r\nA 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 $I = (I : h^infty) cap langle I, h^m \r\nangle$ for some specific polynomial $h$ and integer $m$. This splitting introduces a number of redundant components that are not part of the original ideal.\r\n\r\nIn the algorithm we present here, we use the reduction to the zero dimensional case, but we avoid working with the ideal $langle I, h^m \r\nangle$. As a result, when the ideal has components of different dimensions, our algorithm is usually more efficient.","keywords":["Minimal associated primes","groebner basis","polynomail equations","radical"],"author":{"@type":"Person","name":"Laplagne, Santiago","givenName":"Santiago","familyName":"Laplagne"},"position":6,"pageStart":1,"pageEnd":6,"dateCreated":"2006-10-25","datePublished":"2006-10-25","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":{"@type":"Person","name":"Laplagne, Santiago","givenName":"Santiago","familyName":"Laplagne"},"copyrightYear":"2006","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.06271.6","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume615"},{"@type":"ScholarlyArticle","@id":"#article1510","name":"Computational Aspects of the Resolution of Singularities","abstract":"The task of resolution of singularities has been one of the central topics in Algebraic \r\nGeometry for many decades. After results in low dimension in the first half of the 20th \r\ncentury, it was Hironaka's monumental article in 1964 which solved the porblem in the\r\ngeneral case in chareacteristic zero. The case of characteristic $p > 0$ is still unsolved \r\nexcept in partial results in low dimension.\r\n\r\nBut Hironaka's proof did not put an end to the interest in characteric zero, instead it shifted\r\nthe focus toward the task of finding a more constructive approach. Such algorithmic\r\napproaches appeared at the end of the 1980's independently by Villamayor and by\r\nBierstone and Milman. In this talk we consider the computational tasks arising from \r\nVillamayor's algorithm and present an implementation.","keywords":["Resolution of Singularities","algorithmic desingularization,"],"author":{"@type":"Person","name":"Fr\u00fchbis-Kr\u00fcger, Anne","givenName":"Anne","familyName":"Fr\u00fchbis-Kr\u00fcger"},"position":7,"pageStart":1,"pageEnd":8,"dateCreated":"2006-10-25","datePublished":"2006-10-25","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":{"@type":"Person","name":"Fr\u00fchbis-Kr\u00fcger, Anne","givenName":"Anne","familyName":"Fr\u00fchbis-Kr\u00fcger"},"copyrightYear":"2006","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.06271.7","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume615"},{"@type":"ScholarlyArticle","@id":"#article1511","name":"Coxeter Lattice Paths","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. \r\n\r\nIn 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.","keywords":["Parallel computing","code generation","Coxeter groups","regular lattices","lattice paths","path optimisation"],"author":[{"@type":"Person","name":"Ashby, Thomas J.","givenName":"Thomas J.","familyName":"Ashby"},{"@type":"Person","name":"Kennedy, Anthony D.","givenName":"Anthony D.","familyName":"Kennedy"},{"@type":"Person","name":"Watt, Stephen M.","givenName":"Stephen M.","familyName":"Watt"}],"position":8,"pageStart":1,"pageEnd":14,"dateCreated":"2006-10-25","datePublished":"2006-10-25","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Ashby, Thomas J.","givenName":"Thomas J.","familyName":"Ashby"},{"@type":"Person","name":"Kennedy, Anthony D.","givenName":"Anthony D.","familyName":"Kennedy"},{"@type":"Person","name":"Watt, Stephen M.","givenName":"Stephen M.","familyName":"Watt"}],"copyrightYear":"2006","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.06271.8","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume615"},{"@type":"ScholarlyArticle","@id":"#article1512","name":"Decomposition of Differential Polynomials","abstract":"We present an algorithm to decompose nonlinear differential\r\npolynomials in one variable and with rational functions as\r\ncoefficients. The algorithm is implemented in Maple for the {em\r\nconstant field} case. The program can be used to decompose\r\ndifferential polynomials with more than one thousand terms\r\neffectively.","keywords":["Decomposition","differential polynomial","difference polynomial"],"author":[{"@type":"Person","name":"Gao, Xiao-Shan","givenName":"Xiao-Shan","familyName":"Gao"},{"@type":"Person","name":"Zhang, Mingbo","givenName":"Mingbo","familyName":"Zhang"}],"position":9,"pageStart":1,"pageEnd":10,"dateCreated":"2006-10-25","datePublished":"2006-10-25","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Gao, Xiao-Shan","givenName":"Xiao-Shan","familyName":"Gao"},{"@type":"Person","name":"Zhang, Mingbo","givenName":"Mingbo","familyName":"Zhang"}],"copyrightYear":"2006","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.06271.9","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume615"},{"@type":"ScholarlyArticle","@id":"#article1513","name":"GNU TeXmacs","abstract":"GNU TeXmacs is a free scientific editing platform with special features for mathematicians.\r\nThe editor can be used to produce documents with a professional typesetting quality\r\n(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,\r\nlike a presentation mode, a technical picture editor, a typed linking tool, etc.\r\nThe editor can be extended by users in several ways: using style files, plug-ins or\r\nvia the Scheme extension language. Converters exist for LaTeX, Xhtml and MathML.","keywords":["Scientific text editor","Mathematics","Computer algebra system","Front-end"],"author":{"@type":"Person","name":"van der Hoeven, Joris","givenName":"Joris","familyName":"van der Hoeven"},"position":10,"pageStart":1,"pageEnd":3,"dateCreated":"2006-10-25","datePublished":"2006-10-25","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":{"@type":"Person","name":"van der Hoeven, Joris","givenName":"Joris","familyName":"van der Hoeven"},"copyrightYear":"2006","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.06271.10","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume615"},{"@type":"ScholarlyArticle","@id":"#article1514","name":"MathBrush: An Experimental Pen-Based Math System","abstract":"It is widely believed that mathematics will be one of the major applications\r\nfor Tablet PCs and other pen-based devices. In this paper we discuss many of\r\nthe issues that make doing mathematics on such pen-based devices a hard task.\r\nWe give a preliminary description of an experimental system, currently named\r\nMathBrush, for working with mathematics\r\nusing pen-based devices. The system allows a user to enter mathematical\r\nexpressions with a pen and to then do mathematical computation using a\r\ncomputer algebra system. The system provides a simple and easy way for\r\nusers to verify the correctness of their handwritten expressions and, if\r\nneeded, to correct any errors in recognition. Choosing mathematical operations\r\nis done making use of context menus, both with input and output expressions.","keywords":["PC Tablets","pen-based devices","computer algebra systems"],"author":[{"@type":"Person","name":"Labahn, George","givenName":"George","familyName":"Labahn"},{"@type":"Person","name":"MacLean, Scott","givenName":"Scott","familyName":"MacLean"},{"@type":"Person","name":"Marzouk Mirette","familyName":"Marzouk Mirette"},{"@type":"Person","name":"Rutherford, Ian","givenName":"Ian","familyName":"Rutherford"},{"@type":"Person","name":"Tausky, David","givenName":"David","familyName":"Tausky"}],"position":11,"pageStart":1,"pageEnd":8,"dateCreated":"2006-10-25","datePublished":"2006-10-25","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Labahn, George","givenName":"George","familyName":"Labahn"},{"@type":"Person","name":"MacLean, Scott","givenName":"Scott","familyName":"MacLean"},{"@type":"Person","name":"Marzouk Mirette","familyName":"Marzouk Mirette"},{"@type":"Person","name":"Rutherford, Ian","givenName":"Ian","familyName":"Rutherford"},{"@type":"Person","name":"Tausky, David","givenName":"David","familyName":"Tausky"}],"copyrightYear":"2006","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.06271.11","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume615"},{"@type":"ScholarlyArticle","@id":"#article1515","name":"Notes on computing minimal approximant bases","abstract":"We show how to transform the problem of computing solutions \r\nto a classical Hermite Pade approximation problem for an input\r\nvector of dimension $m \times 1$, arbitrary degree constraints\r\n$(n_1,n_2,ldots,n_m)$, and order $N := (n_1 + 1) + cdots +\r\n(n_m + 1) - 1$, to that of computing a minimal approximant\r\nbasis for a matrix of dimension $O(m) \times O(m)$, uniform\r\ndegree constraint $Theta(N\/m)$, and order $Theta(N\/m)$.","keywords":["Hermite Pade approximation","minimal approximant bases"],"author":{"@type":"Person","name":"Storjohann, Arne","givenName":"Arne","familyName":"Storjohann"},"position":12,"pageStart":1,"pageEnd":6,"dateCreated":"2006-10-25","datePublished":"2006-10-25","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":{"@type":"Person","name":"Storjohann, Arne","givenName":"Arne","familyName":"Storjohann"},"copyrightYear":"2006","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.06271.12","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume615"},{"@type":"ScholarlyArticle","@id":"#article1516","name":"Pivot-Free Block Matrix Inversion","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.","keywords":["Linear algebra","block matrices","matrix inverse"],"author":{"@type":"Person","name":"Watt, Stephen M.","givenName":"Stephen M.","familyName":"Watt"},"position":13,"pageStart":1,"pageEnd":0,"dateCreated":"2006-11-02","datePublished":"2006-11-02","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":{"@type":"Person","name":"Watt, Stephen M.","givenName":"Stephen M.","familyName":"Watt"},"copyrightYear":"2006","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.06271.13","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume615"},{"@type":"ScholarlyArticle","@id":"#article1517","name":"Probabilistically Stable Numerical Sparse Polynomial Interpolation","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.","keywords":["Symbolic-numeric computing","multivariate interpolation","sparse polynomial"],"author":[{"@type":"Person","name":"Giesbrecht, Mark","givenName":"Mark","familyName":"Giesbrecht"},{"@type":"Person","name":"Labahn, George","givenName":"George","familyName":"Labahn"},{"@type":"Person","name":"Lee, Wen-Shin","givenName":"Wen-Shin","familyName":"Lee"}],"position":14,"pageStart":1,"pageEnd":11,"dateCreated":"2006-10-25","datePublished":"2006-10-25","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Giesbrecht, Mark","givenName":"Mark","familyName":"Giesbrecht"},{"@type":"Person","name":"Labahn, George","givenName":"George","familyName":"Labahn"},{"@type":"Person","name":"Lee, Wen-Shin","givenName":"Wen-Shin","familyName":"Lee"}],"copyrightYear":"2006","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.06271.14","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume615"},{"@type":"ScholarlyArticle","@id":"#article1518","name":"Two Families of Algorithms for Symbolic Polynomials","abstract":"We wish to work with polynomials where the exponents are not known \r\nin advance, such as $x^{2n} - 1$. There are various operations we will\r\nwant to be able to do, such as squaring the value to get $x^{4n}-2x^{2n}+1$,\r\nor differentiating it to get $2nx^{2n-1}$. Expressions of this sort\r\narise frequently in practice, for example in the analysis of algorithms,\r\nand it is very difficult to work with them effectively in current computer\r\nalgebra systems.\r\n\r\nWe consider the case where multivariate polynomials can have exponents\r\nwhich are themselves integer-valued multivariate polynomials, and we present\r\nalgorithms to compute their GCD and factorization. The algorithms fall into\r\ntwo families: algebraic extension methods and interpolation methods.\r\nThe first family of algorithms uses the algebraic independence of $x$, $x^n$,\r\n$x^{n^2}$, $x^{nm}, etc, to solve related problems with more indeterminates. \r\nSome subtlety is needed to avoid problems with fixed divisors of the exponent\r\npolynomials. The second family of algorithms uses evaluation and interpolation\r\nof the exponent polynomials. While these methods can run into unlucky\r\nevaluation points, in many cases they can be more appealing. Additionally,\r\nwe also treat the case of symbolic exponents on rational coefficients\r\n(e.g. $4^{n^2+n}-81$) and show how to avoid integer factorization.","keywords":["Computer algebra","symbolic computation","factorization","gcd","symbolic exponents"],"author":{"@type":"Person","name":"Watt, Stephen M.","givenName":"Stephen M.","familyName":"Watt"},"position":15,"pageStart":1,"pageEnd":20,"dateCreated":"2006-11-07","datePublished":"2006-11-07","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":{"@type":"Person","name":"Watt, Stephen M.","givenName":"Stephen M.","familyName":"Watt"},"copyrightYear":"2006","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.06271.15","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume615"},{"@type":"ScholarlyArticle","@id":"#article1519","name":"Using fast matrix multiplication to solve structured linear systems","abstract":"Structured linear algebra techniques are a versatile set of tools;\r\nthey enable one to deal at once with various types of matrices, with\r\nfeatures such as Toeplitz-, Hankel-, Vandermonde- or Cauchy-likeness.\r\n\r\nFollowing Kailath, Kung and Morf (1979), the usual way of measuring to\r\nwhat extent a matrix possesses one such structure is through its\r\ndisplacement rank, that is, the rank of its image through a suitable\r\ndisplacement operator. Then, for the families of matrices given above,\r\nthe results of Bitmead-Anderson, Morf, Kaltofen, Gohberg-Olshevsky,\r\nPan (among others) provide algorithm of complexity $O(alpha^2 n)$, up\r\nto logarithmic factors, where $n$ is the matrix size and $alpha$ its\r\ndisplacement rank.\r\n\r\nWe show that for Toeplitz- Vandermonde-like matrices, this cost can be\r\nreduced to $O(alpha^(omega-1) n)$, where $omega$ is an exponent for\r\nlinear algebra. We present consequences for Hermite-Pad'e approximation\r\nand bivariate interpolation.","keywords":["Structured matrices","matrix multiplication","Hermite-Pade","bivariate interpolation"],"author":[{"@type":"Person","name":"Schost, Eric","givenName":"Eric","familyName":"Schost"},{"@type":"Person","name":"Bostan, Alin","givenName":"Alin","familyName":"Bostan"},{"@type":"Person","name":"Jeannerod, Claude-Pierre","givenName":"Claude-Pierre","familyName":"Jeannerod"}],"position":16,"pageStart":1,"pageEnd":5,"dateCreated":"2006-10-25","datePublished":"2006-10-25","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Schost, Eric","givenName":"Eric","familyName":"Schost"},{"@type":"Person","name":"Bostan, Alin","givenName":"Alin","familyName":"Bostan"},{"@type":"Person","name":"Jeannerod, Claude-Pierre","givenName":"Claude-Pierre","familyName":"Jeannerod"}],"copyrightYear":"2006","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.06271.16","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume615"}]}