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23
Hybrid GaussTrapezoidal Quadrature Rules
 SIAM Journal on Scientific Computing
, 1999
"... . A new class of quadrature rules for the integration of both regular and singular functions is constructed and analyzed. For each rule the quadrature weights are positive and the class includes rules of arbitrarily highorder convergence. The quadratures result from alterations to the trapezoidal r ..."
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. A new class of quadrature rules for the integration of both regular and singular functions is constructed and analyzed. For each rule the quadrature weights are positive and the class includes rules of arbitrarily highorder convergence. The quadratures result from alterations to the trapezoidal rule, in which a small number of nodes and weights at the ends of the integration interval are replaced. The new nodes and weights are determined so that the asymptotic expansion of the resulting rule, provided by a generalization of the EulerMaclaurin summation formula, has a prescribed number of vanishing terms. The superior performance of the rules is demonstrated with numerical examples and application to several problems is discussed. Key words. EulerMaclaurin formula, Gaussian quadrature, highorder convergence, numerical integration, positive weights, singularity AMS subject classifications. 41A55, 41A60, 65B15, 65D32 PII. S1064827597325141 1. Introduction. Recent advances in algor...
The computation of Galois groups over function fields
, 1992
"... Abstract. Symmetric function theory provides a basis for computing Galois groups which is largely independent of the coefficient ring. An exact algorithm has been implemented over Q(t1,t2,...,tm) in Maple for degree up to 8. A table of polynomials realizing each transitive permutation group of degre ..."
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Abstract. Symmetric function theory provides a basis for computing Galois groups which is largely independent of the coefficient ring. An exact algorithm has been implemented over Q(t1,t2,...,tm) in Maple for degree up to 8. A table of polynomials realizing each transitive permutation group of degree 8 as a Galois group over the rationals is included.
Polynomial RootFinding : Analysis and Computational Investigation of a Parallel Algorithm
 Proceedings of the 4th Annual Symposium on Parallel Algorithms and Architectures
, 1992
"... A practical version of a parallel algorithm that approximates the roots of a polynomial whose roots are all real is developed using the ideas of an existing NC algorithm. An new elementary proof of correctness is provided and the complexity of the algorithm is analyzed. A particular implementation o ..."
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A practical version of a parallel algorithm that approximates the roots of a polynomial whose roots are all real is developed using the ideas of an existing NC algorithm. An new elementary proof of correctness is provided and the complexity of the algorithm is analyzed. A particular implementation of the algorithm that performs well in practice is described and its runtime behaviour is compared with the analytical predictions. 1 Introduction In this paper we describe and analyze the behaviour of an implementation of a parallel algorithm that approximates the roots of a polynomial which has only real roots. The polynomial root approximation problem we consider can be defined as follows. Given a positive integer ¯, and a polynomial p 0 (x) of degree n, whose coefficients are mbit integers and whose roots x 1 ; x 2 ; . . . ; x n are all real, we wish to compute ¯approximations ~ x 1 ; ~ x 2 ; . . . ; ~ x n respectively to these roots, where the ¯approximation ~ x i to the root x i i...
Gaïa: A Package for the Random Generation of Combinatorial Structures
 MapleTech
, 1994
"... This article explains how to define a class of decomposable combinatorial structures with Gaia, how to count the number of structures of a given size, how to generate a random structure and how to use it. Details about the algorithms used will be found in [5] and [6]. ..."
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This article explains how to define a class of decomposable combinatorial structures with Gaia, how to count the number of structures of a given size, how to generate a random structure and how to use it. Details about the algorithms used will be found in [5] and [6].
Process Scheduling in DSC and the Large Sparse Linear Systems Challenge
 J. Symbolic Comput
, 1998
"... this paper appeared in "Design and Implementation of Symbolic Computation Systems," A. Miola (ed.), Springer Lect. Notes Comput. Science, 722, 6680 (1993). ..."
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this paper appeared in "Design and Implementation of Symbolic Computation Systems," A. Miola (ed.), Springer Lect. Notes Comput. Science, 722, 6680 (1993).
Worst Cases and Lattice Reduction
"... We propose a new algorithm to find worst cases for correct rounding of an analytic function. We first reduce this problem to the real small value problem — i.e. for polynomials with real coefficients. Then we show that this second problem can be solved efficiently, by extending Coppersmith’s work on ..."
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We propose a new algorithm to find worst cases for correct rounding of an analytic function. We first reduce this problem to the real small value problem — i.e. for polynomials with real coefficients. Then we show that this second problem can be solved efficiently, by extending Coppersmith’s work on the integer small value problem — for polynomials with integer coefficients — using lattice reduction [4, 5, 6]. For floatingpoint numbers with a mantissa less than, and a polynomial approximation of ¡ degree, our algorithm finds all worst cases ¢ at distance a machine number �� � § ¥�©������� � in time ¡��¤ �. For, this improves �� � �� � � on the complexity from Lefèvre’s algorithm
Efficient computation of full Lucas sequences
, 1996
"... odd, then the computation of Uk does not require the computation of U l j (j 1). Proof : Since k is odd (i.e. k0 = 1), Uk(= U l 0 ) = Uh 1 V l 1 l 1 . Thus, only the value of Uh 1 is needed. We only need to show that the value of Uh j1 can be derived from Uh j . By Eq. (5) and depending on ..."
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odd, then the computation of Uk does not require the computation of U l j (j 1). Proof : Since k is odd (i.e. k0 = 1), Uk(= U l 0 ) = Uh 1 V l 1 l 1 . Thus, only the value of Uh 1 is needed. We only need to show that the value of Uh j1 can be derived from Uh j . By Eq. (5) and depending on the value of k j1 , we have the following cases: . if k j1 = 0, then (l j1 , h j1 ) = (2l j , l j + h j ); . if k j1 = 1, then (l j1 , h j1 ) = (l j + h j , 2h j ). Hence, if k j1 = 0, then h j1(= h j + l j = 2l j + 1) is odd and Uh j1 = Uh j V l j l j ; otherwise, h j1(= 2h j ) is even and Uh j1 = Uh j Vh j . We now are ready to give the algorithm that we shall extend to the case where k is even. Inputs: k = 2 s i=s k i 2 is , (ks = 1) P, Q Outputs: (Uk , Vk ) Uh = 1; V l = 2; Vh = P ; Q l = 1; Qh = 1; for j from n 1 to s + 1 by 1 if k[j] == 1 then Qh = Q l Vh ; Vh Qh else Qh = Q l ; Q l fi Qh ; Qh =
Small limit points of Mahler’s measure
 Experiment. Math
"... Abstract. Let M(P (z1,..., zn)) denote Mahler’s measure of the polynomial P (z1,..., zn). Measures of polynomials in n variables arise naturally as limiting values of measures of polynomials in fewer variables. We describe several methods for searching for polynomials in two variables with integer c ..."
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Abstract. Let M(P (z1,..., zn)) denote Mahler’s measure of the polynomial P (z1,..., zn). Measures of polynomials in n variables arise naturally as limiting values of measures of polynomials in fewer variables. We describe several methods for searching for polynomials in two variables with integer coefficients having small measure, demonstrate effective methods for computing these measures, and identify 48 polynomials P (x, y) with integer coefficients, irreducible over Q, for which 1 < M(P (x, y)) < 1.37. 1.
Some Primality Testing Algorithms
 Notices of the AMS
, 1993
"... We describe the primality testing algorithms in use in some popular computer algebra systems, and give some examples where they break down in practice. 1 Introduction In recent years, fast primality testing algorithms have been a popular subject of research and some of the modern methods are now i ..."
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We describe the primality testing algorithms in use in some popular computer algebra systems, and give some examples where they break down in practice. 1 Introduction In recent years, fast primality testing algorithms have been a popular subject of research and some of the modern methods are now incorporated in computer algebra systems (CAS) as standard. In this review I give some details of the implementations of these algorithms and a number of examples where the algorithms prove inadequate. The algebra systems reviewed are Mathematica, Maple V, Axiom and Pari/GP. The versions we were able to use were Mathematica 2.1 for Sparc, copyright dates 19881992; Maple V Release 2, copyright dates 19811993; Axiom Release 1.2 (version of February 18, 1993); Pari/GP 1.37.3 (Sparc version, dated November 23, 1992). The tests were performed on Sparc workstations. Primality testing is a large and growing area of research. For further reading and comprehensive bibliographies, the interested re...
Cs: a MuPAD package for counting and randomly generating combinatorial structures
 In Proceedings of FPSAC'98
, 1998
"... We present a new computer algebra package which permits to count and to generate combinatorial structures of various types, provided that these structures can be described by a speci cation, as de ned in [7]. Resume Nous presentons un nouveau module de calcul formel dedie audenombrement etala genera ..."
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We present a new computer algebra package which permits to count and to generate combinatorial structures of various types, provided that these structures can be described by a speci cation, as de ned in [7]. Resume Nous presentons un nouveau module de calcul formel dedie audenombrement etala generation aleatoire uniforme de structures combinatoires decomposables. 1 What is CS? CS is a computer algebra package devoted to the handling of combinatorial structures. Its main features are the following: given a combinatorial speci cation of a class of decomposable structures (in the sense of [7]), CS is able to count and uniformly draw at random the structures of any given size n. It can also give some properties of the associated generating series, like recurrences and di erential equations. A speci cation of a class of combinatorial structures, as de ned in [7], is a set of productions made from basic objects (atoms) (Epsilon and Z of size 0 and 1 respectively) and from