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18
Efficient Rational Number Reconstruction
 Journal of Symbolic Computation
, 1994
"... this paper we describe how a variant of the algorithm in Jebelean [6] can be so adapted. In Section 2 we review the problem of rational reconstruction and the solution proposed by Wang, while fixing some notation and terminology along the way. We also discuss certain errors that have appeared in the ..."
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Cited by 19 (0 self)
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this paper we describe how a variant of the algorithm in Jebelean [6] can be so adapted. In Section 2 we review the problem of rational reconstruction and the solution proposed by Wang, while fixing some notation and terminology along the way. We also discuss certain errors that have appeared in the literature. Section 3 describes a multiprecision Euclidean algorithm for computing gcds that will be the basis of our algorithm. In Section 4 we discuss our algorithm and various details that are essential for an efficient implementation. 2 Reconstructing Rational Numbers
Modular Rational Sparse Multivariate Polynomial Interpolation
 In ISSAC ’90: Proceedings of the international symposium on Symbolic and algebraic computation
, 1990
"... The problem of interpolating multivariate polynomials whose coefficient domain is the rational numbers is considered. The effect of intermediate number growth on a speeded BenOr and Tiwari algorithm is studied. Then the newly developed modular algorithm is presented. The computing times for the spe ..."
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Cited by 19 (4 self)
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The problem of interpolating multivariate polynomials whose coefficient domain is the rational numbers is considered. The effect of intermediate number growth on a speeded BenOr and Tiwari algorithm is studied. Then the newly developed modular algorithm is presented. The computing times for the speeded BenOr and Tiwari and the modular algorithm are compared, and it is shown that the modular algorithm is markedly superior. 1 Introduction Symbolic expressions, that is multivariate polynomials with rational coefficients, are often difficult to manipulate explicitly due to exponential growth in their size. An example is the computation of the determinant of a matrix with polynomial entries. When using straightforward Gaussian elimination over the polynomial entry domain, it can happen that intermediate subdeterminants are very large polynomials while the final answer is an expression of modest size. In this case, however, we can obtain the value of the determinant for a specialization ...
Maximal quotient rational reconstruction: an almost optimal algorithm for rational reconstruction
 Proceedings of ISSAC ’04, ACM Press
, 2004
"... Let n/d ∈ Q, mbe a positive integer and let u = n/d mod m. Thus u is the image of a rational number modulo m. The rational reconstruction problem is; given u and m find n/d. A solution was first given by Wang in 1981. Wang’s algorithm outputs n/d when m>2M 2 where M =max(n,d). Because of the wide ..."
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Cited by 13 (3 self)
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Let n/d ∈ Q, mbe a positive integer and let u = n/d mod m. Thus u is the image of a rational number modulo m. The rational reconstruction problem is; given u and m find n/d. A solution was first given by Wang in 1981. Wang’s algorithm outputs n/d when m>2M 2 where M =max(n,d). Because of the wide application of this algorithm in computer algebra, several authors have investigated its practical efficiency and asymptotic time complexity. In this paper we present a new solution which is almost optimal in the following sense; with controllable high probability, our algorithm will output n/d when m is a modest number of bits longer than 2nd. This means that in a modular algorithm where m is a product of primes, the modular algorithm will need one or two primes more than the minimum necessary to reconstruct n/d; thusifn  ≪d or d ≪n the new algorithm saves up to half the number of primes. Further, our algorithm will fail with high probability when m<2nd.
A modular algorithm for computing greatest common right divisors of ore polynomials
 In Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation
, 1997
"... Abstract. This paper presents a modular algorithm for computing the greatest common right divisor (gcrd) of two univariate Ore polynomials over Z[t]. The subresultants of Ore polynomials are used to compute the evaluation homomorphic images of the gcrd. Rational number and rational function reconstr ..."
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Cited by 12 (2 self)
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Abstract. This paper presents a modular algorithm for computing the greatest common right divisor (gcrd) of two univariate Ore polynomials over Z[t]. The subresultants of Ore polynomials are used to compute the evaluation homomorphic images of the gcrd. Rational number and rational function reconstructions are used to recover coefficients. The experimental results illustrate that the present algorithm is markedly superior to the Euclidean algorithm and the subresultant algorithm for Ore polynomials. 1.
Efficient Parallel Solution of Sparse Systems of Linear Diophantine Equations
, 1997
"... We present a new iterative algorithm for solving large sparse systems of linear Diophantine equations which is fast, provably exploits sparsity, and allows an efficient parallel implementation. This is accomplished by reducing the problem of finding an integer solution to that of finding a very smal ..."
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Cited by 11 (4 self)
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We present a new iterative algorithm for solving large sparse systems of linear Diophantine equations which is fast, provably exploits sparsity, and allows an efficient parallel implementation. This is accomplished by reducing the problem of finding an integer solution to that of finding a very small number of rational solutions of random Toeplitz preconditionings of the original system. We then employ the BlockWiedemann algorithm to solve these preconditioned systems efficiently in parallel. Solutions produced are small and space required is essentially linear in the output size.
On the Genericity of the Modular Polynomial GCD Algorithm
, 1999
"... In this paper we study the generic setting of the modular GCD algorithm. We develop the algorithm for multivariate polynomials over Euclidean domains which have a special kind of remainder function. Details for the parameterization and generic Maple code are given. Applying this generic algorithm to ..."
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Cited by 6 (2 self)
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In this paper we study the generic setting of the modular GCD algorithm. We develop the algorithm for multivariate polynomials over Euclidean domains which have a special kind of remainder function. Details for the parameterization and generic Maple code are given. Applying this generic algorithm to a GCD problem in Z=(p)[t][x] where p is small yields an improved asymptotic performance over the usual approach, and a very practical algorithm for polynomials over small finite fields.
A Modular Method to Compute the Rational Univariate Representation of ZeroDimensional Ideals
 J. Symb. Comp
, 1999
"... To give an efficiently computable representation of the zeros of a zerodimensional ideal I, Rouillier (1996) introduced the rational univariate representation (RUR) as an extension of the generalized shape lemma (GSL) proposed by Alonso et al. (1996). In this paper, we propose a new method to compu ..."
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Cited by 6 (5 self)
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To give an efficiently computable representation of the zeros of a zerodimensional ideal I, Rouillier (1996) introduced the rational univariate representation (RUR) as an extension of the generalized shape lemma (GSL) proposed by Alonso et al. (1996). In this paper, we propose a new method to compute the RUR of the radical of I, and report on its practical implementation. In the new method, the RUR of the radical of I is computed efficiently by applying modular techniques to solving the systems of linear equations. The performance of the method is examined by practical experiments. We also discuss its theoretical efficiency. (~) 1999 Academic Press 1.
Spaceefficient evaluation of hypergeometric series
 SIGSAM Bulletin, Communications in Computer Algebra
"... Many important constants, such as e and Apéry’s constant ζ(3), can be approximated by a truncated hypergeometric series. The evaluation of such series to high precision has traditionally been done by binary splitting followed by fixedpoint division. However, the numerator and the denominator comput ..."
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Cited by 4 (4 self)
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Many important constants, such as e and Apéry’s constant ζ(3), can be approximated by a truncated hypergeometric series. The evaluation of such series to high precision has traditionally been done by binary splitting followed by fixedpoint division. However, the numerator and the denominator computed by binary splitting usually contain a very large common factor. In this paper, we apply standard computer algebra techniques including modular computation and rational reconstruction to overcome the shortcomings of the binary splitting method. The space complexity of our algorithm is the same as a bound on the size of the reduced numerator and denominator of the series approximation. Moreover, if the predicted bound is small, the time complexity is better than the standard binary splitting approach. Our approach allows a series to be evaluated to a higher precision without additional memory. We show that when our algorithm is applied to compute ζ(3), the memory requirement is significantly reduced compared to the binary splitting approach. 1
On Computing Univariate GCDs over Number Fields.
, 1998
"... We compare the two main competing methods for fast univariate polynomial GCD computation over an algebraic number field, namely, the modular method of Langymyr et al (1987), and the heuristic method of Smedley et al (1988). Because of recent improvements to the modular method by Encarnacion (1994), ..."
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We compare the two main competing methods for fast univariate polynomial GCD computation over an algebraic number field, namely, the modular method of Langymyr et al (1987), and the heuristic method of Smedley et al (1988). Because of recent improvements to the modular method by Encarnacion (1994), we expected that the modular method, if implemented "properly ", would now be the method of choice in Maple. This turned out to be the case for several kinds of GCD problems. As an exercise, to complete the comparison, we implemented also a Hensel based method. We then realized that Hensel lifting is "pointless" when applied to univariate GCD computation and implemented a more direct method that we call the primepower method. It turns out that not only is the primepower method simple to implement, it is also better than the heuristic method. Due to the large effort required to implement the modular method "properly", we recommend that the primepower method to systems' implementors as a ve...