Joris van der Hoeven. Relax, but don't be too lazy. JSC , 34:479542, 2002.  Home/Search   Document Not in Database   Summary   Related Articles   Check

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Joris van der Hoeven. Relax, but don't be too lazy. JSC , 34:479542, 2002.

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Joris van der Hoeven. Relax, but don't be too lazy. JSC , 34:479542, 2002.

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Joris van der Hoeven. Relax, but don't be too lazy. JSC , 34:479542, 2002.

....= fg = h0 h1z . If the first n coe#cients of f and g are known beforehand, then we may use any fast multiplication for polynomials in order to achieve this goal, such as divide and conquer multiplication [6, 7] which has a time complexity K(n) O(n log 3 log 2 ) or F.F.T. multiplication [2, 9, 1, 11], which has a time complexity M(n) O(n log n log log n) For simplicity, time complexity stands for the required number of operations in R. Similarly, space complexity will stand for the number of elements of which need to This paper is in the public domain. Permission is granted to ....

....is interesting to have so called relaxed algorithms which output the first i coe#cients of h as soon as the first i coe#cients of f and g are known for each i 6 n. This allows for instance the computation of the exponential g = exp f of a series f with f0 = 0 using the formula f # g. 1) In [10, 11], we proved the following two theorems: Theorem 1. There exists a relaxed multiplication algorithm of time complexity K(n) and space complexity O(log n) and which uses K(n) multiplications. Theorem 2. There exists a relaxed multiplication algorithm of time complexity O(M(n) log n) and space ....

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van der Hoeven, J. Relax, but don't be too lazy. JSC 34 (2002), 479--542.

....classical majorant technique can be used in order to obtain many such bounds. Keywords: majorant equations, power series, computer algebra, partial dioeerential equations, singular dioeerential equations, convolution products A.M.S. subject classification: 35A10, 13F25, 44A35 1. Introduction In [vdH03, vdH99, vdH01a, vdH02], we have started to develop a fully eoeective complex analysis. The aim of this theory is to evaluate constructible analytic functions to any desired precision and to continue such functions analytically whenever possible. In order to guarantee that the desired precision is indeed obtained, bound ....

Joris van der Hoeven. Relax, but don't be too lazy. JSC , 34:479542, 2002.

....given reasonably good uniform bounds (of the form O(2 ) where s denotes the input size) for the number of zeros for systems of real Pfaan functions [6] These bounds may be adapted to the power series context. This approach is interesting because it only requires fast power series expansions [3, 14] for implementing a zerotest. However, such a zero test might be slow for expressions which can be quickly rewritten to zero (like x x, where x is a complicated expression) Also, if we want the approach to be ecient, good bounds (such as the ones predicted by witness conjectures [17, 13, 16, 8] ....

....Newton degree deg z k P is the minimal degree of a term P z k ;i f in P z k with dP z k ;i = dP z k . In particular, the minimal k in step 3 can be found by expanding the power series coecients H i (f) of H in z using any fast expansion algorithm for solutions to di erential equations [3, 14]. 5.2 Correctness and termination proof Theorem 2. The above algorithm for testing whether P 0 terminates and is correct. Proof. In the loop in step 2, we notice that the rank of R strictly decreases at each iteration. Also, the rank of IR (or SR) in each recursive call of the zero test is ....

van der Hoeven, J. Relax, but don't be too lazy. Tech. Rep. 78, Prepublications d'Orsay, 1999. Submitted to JSC.

....f at order n as a function of n2N. The asymptotic complexity of f is the asymptotic complexity of this expansion algorithm. In particular, we have an algorithm to compute the n th coeOEcient f n of an eoeective series. A survey of eOEcient methods to compute with eoeective series can be found in [vdH02c]. When f is an eoeective series in Series(Complex) then we denote by f 2C[ z] the actual series which is represented by f . If f is the germ of analytic function, then we will denote by ae(f ) the radius of convergence of f and by jf j r the maximum of jf j on the closed ....

....in Complex and the identity function z, using the operations ; Gamma ; Delta ; exp and log. In our specications of the corresponding concrete data types which inherit from AnFunc, we will omit the algorithms for computing the coeOEcients of the series expansions, and refer to [vdH02c] for a detailed treatment of this matter. 3.1. Basic eoeective analytic functions Constant eoeective analytic functions are implemented by the following concrete type ConstantAnFunc which derives from AnFunc (this is reAEected through the B symbol below) Class ConstantAnFunc BAnFunc ffl ....

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Joris van der Hoeven. Relax, but don't be too lazy. JSC , 34:479542, 2002.

.... i ( k Xi i 1 ff i 1 ) for all 2R and ff k ; ff i 1 2N. More generally, if P 2S , then we may formally represent the coeOEcient ff k ; ff i 1 of = ae(P ) by polynomials in i (S ) Such representations are best derived through relaxed evaluation of formal power series [vdH99], by using the partial dioeerential equations satised by f . 3 The Bareiss method and g.c.d. computations 3.1 Pseudo norms Let R be an eoeective integral domain. In what follows, we will describe algorithms to triangulate matrices with entries in R and compute g.c.d.s of polynomials with ....

....1. trivial case] If P = 0 then return true. Step 2. g.c.d. computations] Replace P sqfree(P ) Let G gcd(P ; d 1 P ; d k P ) Step 3. compute the valuation of G] Denote G=G q F q Xi G 0 . For i = k; 1 do Expand G 0;ff k ; ff i 1 ; G q;ff k ; ff i 1 in a relaxed way [vdH99] w.r.t. z i . Stop at the least ff i , such that there exists a p with G p;ff k ; ff i Delta 0. Step 4. evaluate and conclude] Return G ff ( f ) 0. 5.2 Complexity bounds In order to derive complexity bounds, we will to assume that we have a pseudo norm on R and that there exists a ....

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J. van der Hoeven. Relax, but don't be too lazy. Technical Report 78, Pr#publications d'Orsay, 1999. Submitted to JSC.

....Joris van der Hoeven calls the relaxed approach, intermediate between the zealous approach and the lazy approach. This relaxed approach was invented in 1997, with the presentation of two relaxed algorithms for the multiplication of formal power series at the ISSAC 97 conference [8] The report [9] details these algorithms and their implantation, presents some other multiplication algorithms, shows how the relaxed approach extends naturally to other operations on formal power series, and nally o ers several experimental comparisons between classical and relaxed algorithms. 1. The Zealous ....

....destroyed , nally which have to be saved for latter use. Another algorithm proposed by Joris van der Hoeven consists in tiling the square n n by a sequence of L shapes of increasing width. That leads to a relaxed multiplication in O M(n) log n . Several other alternatives are proposed in [9], both for complete products (polynomials) and truncated products (formal power series) The other operations (division, composition) are also essentially relaxed. Finally we obtain the following complexities for the relaxed alternatives of the operations on formal power series: Algorithm Times ....

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van der Hoeven (Joris). { Relax, but don't be too lazy. { Technical Report n 78, Universite de Paris-Sud, Mathematiques, B^atiment 425, F-91405 Orsay, 1999. Submitted to the Journal of Symbolic Computation. Available from http://www.math.u-psud.fr/~vdhoeven/.

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