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O n the com plexity of m ultivariate blockwise p olynom ial multiplication *
"... ABSTRACT In this article, we study the problem of multiplying two multivariate polynomials which are somewhat but not too sparse, typically like polynomials with convex supports. We design and analyze an algorithm which is based on blockwise decomposition of the input polynomials, and which perform ..."
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ABSTRACT In this article, we study the problem of multiplying two multivariate polynomials which are somewhat but not too sparse, typically like polynomials with convex supports. We design and analyze an algorithm which is based on blockwise decomposition of the input polynomials, and which performs the actual multiplication in an FFT model or some other more general so called "evaluated model". If the input polynomials have total degrees at most d, then, under mild assumptions on the coefficient ring, we show that their product can be computed with O(s 1.5337 ) ring operations, where s denotes the number of all the monomials of total degree at most 2 d.
From implicit to recursive equations *
"... The technique of relaxed power series expansion provides an efficient way to solve so called recursive equations of the form F = Φ(F ), where the unknown F is a vector of power series, and where the solution can be obtained as the limit of the sequence 0, Φ(0), Φ(Φ(0)), . With respect to other tech ..."
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The technique of relaxed power series expansion provides an efficient way to solve so called recursive equations of the form F = Φ(F ), where the unknown F is a vector of power series, and where the solution can be obtained as the limit of the sequence 0, Φ(0), Φ(Φ(0)), . With respect to other techniques, such as Newton's method, two major advantages are its generality and the fact that it takes advantage of possible sparseness of Φ. In this paper, we consider more general implicit equations of the form Φ(F ) = 0. Under mild assumptions on such an equation, we will show that it can be rewritten as a recursive equation. If we are actually computing with analytic functions, then recursive equations also provide a systematic device for the computation of verified error bounds. We will show how to apply our results in this context.
Homotopy techniques for multiplication modulo triangular sets
, 2009
"... We study the cost of multiplication modulo triangular families of polynomials. Following previous work by Li, Moreno Maza and Schost, we propose an algorithm that relies on homotopy and fast evaluation-interpolation techniques. We obtain a quasi-linear time complexity for substantial families of exa ..."
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We study the cost of multiplication modulo triangular families of polynomials. Following previous work by Li, Moreno Maza and Schost, we propose an algorithm that relies on homotopy and fast evaluation-interpolation techniques. We obtain a quasi-linear time complexity for substantial families of examples, for which no such result was known before. Applications are given to notably addition of algebraic numbers in small characteristic.
Short Division of Long Integers
"... Abstract-We consider the problem of short divisioni.e., approximate quotient -of multiple-precision integers. We present ready-to-implement algorithms that yield an approximation of the quotient, with tight and rigorous error bounds. We exhibit speedups of up to 30% with respect to GMP division wit ..."
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Abstract-We consider the problem of short divisioni.e., approximate quotient -of multiple-precision integers. We present ready-to-implement algorithms that yield an approximation of the quotient, with tight and rigorous error bounds. We exhibit speedups of up to 30% with respect to GMP division with remainder, and up to 10% with respect to GMP short division, with room for further improvements. This work enables one to implement fast correctly rounded division routines in multiple-precision software tools.
