B. Salvy. Fonctions g'en'eratrices et asymptotique automatique. Research Report 967, Institut National de Recherche en Informatique et en Automatique, 1989.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....series and describes a new algorithm for this task involving Grae e s root squaring steps. 1. Introduction By means of Newton s iteration, reciprocals of power series modulo x n 1 can be computed with complexity O M(n) where M(n) denotes the complexity of multiplication (see, e.g. [6] for a survey) However, the Bachmann Landau O notation hides a multiplicative constant, which needs to be investigated, for instance in order to determine cross over points when a collection of algorithms is available. Section 2 sets the required background by recalling a few de nitions from ....

Salvy (Bruno). { Asymptotique automatique. { Research Report n 3707, Institut National de Recherche en Informatique et en Automatique, 1999. 20 pages.

....of various types of algorithms. It is also meant to experiment with the descriptive power of the theories developed here. The system itself is described elsewhere by its two implementers: P. Zimmermann is responsible for the algebraic analyzer [89] and B. Salvy has designed the asymptotic analyzer [72]. The algebraic analyzer compiles data type and procedure specifications into equations over generating functions (that are counting generating functions or complexity descriptors) It is implemented in CAML. The analytic analyzer is an extensive collection of routines that manipulate generalized ....

Salvy, B. Fonctions g'en'eratrices et asymptotique automatique. Research Report 967, Institut National de Recherche en Informatique et en Automatique, 1989. 118 pages.

....(z)e W (z) z. taunumberofcomponents(z) exp(L( z W( z) L( z W( z) exp(L( z W( z) z W( z) taunumberofcyclicpoints(z) 1 z W( z) At this stage, an Analytic Analyzer with extensive asymptotic capabilities, takes control of the asymptotic analysis [40]. It is built on a large library of Maple programmes (currently about 7000 lines) and on this problem, it selects a strategy based on singularity analysis. The number of special functional graphs (Sfungraph) of size n then appears in its raw form produced by the system as: n times: 3 2 1 2 ....

....is 1:45; ii) the expected number of points lying on cycles is 2:16. This example demonstrates an unusual case of model sensitivity (compare with the corresponding values of O(log n) and O( p n) for unconstrained random mappings) The precise capabilities of the system are described in [14, 15, 40, 46]. 6 Conclusions We have seen a systematic approach to the analysis of a large number of parameters of random mappings (or functional graphs) using a coherent generating function framework. In a random mapping of size n, cycles presents themselves after about p n iteration steps (Section 2) and ....

B. Salvy. Fonctions g'en'eratrices et asymptotique automatique. Research Report 967, Institut National de Recherche en Informatique et en Automatique, 1989.

.... the time when this report is written, a striking feature of these examples is that none of them can be computed directly with any of today s most widespread symbolic computation systems (Macsyma 1 , Mathematica 2 , Maple 3 or Scratchpad II 4 ) This document is part of a technical report [4] available from INRIA. Introduction Current symbolic computation systems generally lack facilities for manipulating asymptotic expansion computations of a form more complex than the first terms of a Taylor series or a Puiseux expansion (involving fractional powers) We introduce a set of programs ....

....Pi ln(x) 5 1 (O( 5 2 ln(x) With this first section, we have given a general view of the capabilities of our library in the case of expressions in closed form. This in turn will become the basic engine for specific applications such as the ones we describe in the next two sections of [4]. Conclusion Non trivial automatic asymptotic expansions can be derived, provided asymptotic scales are carefully handled. Once a good engine for performing expansions in large scales is implemented, expansions which were not even expressible in symbolic computation systems previously can be ....

Salvy, B. Examples of automatic asymptotic expansions. Technical Report 114, Institut National de Recherche en Informatique et en Automatique, 1989.

....a sequence of nested forms; this sequence is called a nested expansion. The field H(x) of exp log functions of a single variable, x, is formed of expressions built from x and real constants by means of arithmetic operations and the operations: f 7 exp(f) f 7 log jf j: 1) In previous works [21, 22, 24, 25, 6, 13], an algorithmic treatment in terms of nested expansions was given to the asymptotics of i) exp log functions; ii) Liouvillian functions and iii) Hardy field solutions of algebraic differential equations. All these algorithms require the use of a method for deciding zero equivalence in the class ....

....use the conditions associated with the question mark estimates to compute the nested forms of solutions of bivariate exp log equations. We also show there how one can refine a question mark estimate. The algorithm in this section can be viewed as a generalization of the univariate algorithm from [13] (itself a descendant of [21] 2.1. Sketch of the method. We assume that a function h(x; y) 2 H(x; y) is given by an expression tree E in which the leaves are either constants or one of the variables, x, y, and the nodes are operations which are either arithmetic or an application of the ....

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Richardson, D., Salvy, B., Shackell, J., and Van der Hoeven, J. Asymptotic expansions of explog functions. Research Report 2859, Institut National de Recherche en Informatique et en Automatique, 1996. To appear in the Proceedings of ISSAC'96.

....computation systems previously can be obtained quite easily. Work is in progress to take benefit of this implementation of asymptotic scales to program various asymptotic methods in a general setting. The nuts and bolts of our implementation of asymptotic scales will be described elsewhere (see [14] for a preliminary report) References [1] Abramowitz, M. and Stegun, I. A. Handbook of Mathematical Functions. Dover, 1973. 2] Bleistein, N. and Handelsman, R. A. Asymptotic Expansions of Integrals, 2nd ed. Holt, Rinehart and Winston, New York, 1975. Reprinted by Dover, 1986. 3] Darboux, G. ....

Salvy, B. Fonctions g'en'eratrices et asymptotique automatique. Research Report 967, Institut National de Recherche en Informatique et en Automatique, 1989.

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B. Salvy. Fonctions g'en'eratrices et asymptotique automatique. Research Report 967, Institut National de Recherche en Informatique et en Automatique, 1989.

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