G.H. Gonnet, D. Gruntz. Limit computation in computer algebra. Technical report 187, ETH. Zurich.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....approximation, algorithm. Ecole Polytechnique 91128, Palaiseau France Email: vdhoeven lix.polytechnique.fr Web : http: lix.polytechnique.fr:80 vdhoeven February 15, 1996 1 Introduction In the last years, several asymptotic expansion algorithms have appeared (see [Sh 90] Sh 91] GoGr 92] RSSV 96] VdH 96b] etc. These algorithms are have the property that they can deal with very general types of singularities, such as singularities arising in the study of certain algebraic differential equations. However, attention has been restricted so far to functions with strongly ....

G.H. Gonnet, D. Gruntz. Limit computation in computer algebra. Technical report 187, ETH. Zurich.

....but which do not allow to derive complete asymptotic expansions. This drawback is removed in [30] where Shackell gives a complete and natural asymptotic expansion algorithm. A weaker version of this algorithm, which only computes limits of exp log functions was discovered independently in [10], and is currently incorporated in MAPLE V.3. The author generalized this limit computation algorithm and obtained variants of Shackell s algorithm in [14, 13] An elegant synthesis of these algorithms appeared in [25] However, several related problems were overlooked up till now. First, can we ....

G.H. Gonnet and D. Gruntz. Limit computation in computer algebra. Technical Report 187, ETH, Zurich, 1992.

....ae(x) x, ae(x) x or ae(x) x in a small neighbourhood of the limit cycle. In other words, either all or no orbits are periodic in this neighbourhood. Introduction 1 Eoeective asymptotic analysis. Transseries also implicitly appeared during the research of algorithms for doing asymptotic analysis [Sha90, Sal91, GG92]. In the formal context of transseries, we were able to do such eoeective computations in a more systematic way [vdH97] There is no doubt that the combination of the techniques from these three dioeerent areas will lead to an extremely powerful theory, whose development is far from nished. A ....

G.H. Gonnet and D. Gruntz. Limit computation in computer algebra. Technical Report 187, ETH, Z#rich, 1992.

....1 is to show that in such cases it is possible to combine floating point methods with asymptotic methods to yield an efficient algorithm. More precisely, we will adapt the asymptotic expansion algorithm for exp log functions from [VdH 94a] which generalizes Gonnet and Gruntz algorithm (see [GoGr 92] See also [GeGo 88] Sh 90] We will briefly recall this algorithm in section 2. We discuss how to adapt the symbolic algorithm to the numerical case in section 3. The numerical expansion algorithm for exp log constants is given in section 4. In fact, numerical expansions contain much more ....

.... logarithm (of positive elements) Modulo Schanuel s conjecture (see section 5) all the above operations, as well as zero equivalence testing, can be performed algorithmically in T (see [Rich 94] Sh 89] In [VdH 94a] we generalized the limit computation algorithm due to Gonnet and Gruntz (see [GoGr 92] to yield a symbolic expansion algorithm. We will briefly recall this algorithm, without going into details. All expansions will be done in a neighbourhood of 1. An example of how the algorithm works is given at the end of this section. We start with some definitions. Let A be some abelian ....

G.H. Gonnet, D. Gruntz. Limit computation in computer algebra. Report technique 187 du ETH, Zurich.

....simpler and we think that the obtained expansions are a bit more natural than nested expansions. In section 2, we consider solutions to polynomial equations and we use the standard technique to expand them. In section 3, we consider derivatives and integrals. The upward movement technique (see [GoGr 92] will be used to get rid of logarithms in the expansions. Then derivatives and integrals of standard expansions can easily be computed. Results are converted back by using downward movement . Here we essentially use the fact that we expand w.r.t. normal and not w.r.t. weakly Common operations 2 ....

G.H. Gonnet, D. Gruntz. Limit computation in computer algebra. Technical report 187, ETH. Zurich.

....of analyzing algorithms that often involve classical combinatorial structures like strings, trees, graphs, permutations. In [1] an algorithm for random generation of sets of (i.e. without repetition) of various combinatorial objects is described. Aspects of the study of RNA are the subject of [2]. Automatic sequences and their complexity are discussed in [3] Some double sequences and their asymptotic properties are studied in [4] Finally, 5] gives an arithmetic interpretation to simple operations on binary trees. 1 This work was supported in part by the ESPRIT III Basic Research ....

....Finally, 5] gives an arithmetic interpretation to simple operations on binary trees. 1 This work was supported in part by the ESPRIT III Basic Research Action Programme of the E.C. under contract ALCOM II (#7141) i [1] Uniform Random Generation for the Powerset Construction. Paul Zimmermann [2] An Efficient Parser Well Suited to RNA Folding. Fabrice Lefebvre [3] Pascal s Triangle, Automata, and Music. Jean Paul Allouche [4] Riordan Arrays and their Applications. Donatella Merlini [5] Structured Numbers. Vincent Blondel PART II. SYMBOLIC COMPUTATION Most exactly solvable models of ....

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Gonnet (Gaston H.) and Gruntz (Dominik). -- Limit Computation in Computer Algebra. -- Technical Report n 187, ETH, Zurich, November 1992.

....1991b) This program does not completely solve the problem of finding local expansions for the class of explog functions. A general algorithm for doing so was given by Shackell (1990) but no implementation of this algorithm is yet available. An approach similar to Shackell s has been developed by Gonnet Gruntz (1992). This may soon provide Maple with the best asymptotic expander of all existing computer algebra systems (Gruntz 1995) 2.3. Saddle point method Not all combinatorial generating functions have an algebraic logarithmic singularity. In many cases, the function is entire or has an essential ....

Gonnet, G. H. and D. Gruntz (1992, November). Limit computation in computer algebra. Technical Report 187, ETH, Zurich.

....problem of expanding exp log functions has been considered by several authors and goes back to Hardy s work on L functions [Har 11] However, the computational version of the problem has been settled only recently. Shackell [Sh 90] was the first to give an explicit algorithm and Gonnet and Gruntz [GoGr 92] were the first to give one, which has been implemented in practice. A variant of this latter algorithm was rediscovered independently by the author [VdH 94a] We remark that all algorithms make the implicit hypothesis that one can decide whether an exp log constant is zero (see [Sh 90] for a ....

....Our definition will be purely algebraic and we proceed by successive closures. Contrary to Ecalle, we will start by closure w.r.t. logarithm and then perform closure w.r.t. exponentiation. This approach has the computational advantage that we can often avoid the use of upward movements (see [GoGr 92] We remark that the construction can easily be extended to the case of transseries over an arbitrary totally ordered field, stable by exponentiation. The different types of closure we use presuppose that we dispose of a field K = R[ X] where X is a totally ordered commutative multiplicative ....

G.H. Gonnet, D. Gruntz. Limit computation in computer algebra. Technical report 187, ETH. Zurich.

....For the case of asymptotic series, one sided limits are computed and for all other cases complex ones. The power of the procedure for computing LPS stands and falls with the capabilities of the tool for computing limits. The algorithms which are used in Maple to compute limits are described in [4, 5]. Further, in cases when the resulting RE cannot be solved explicitly, it can, in principle be used to calculate the coefficients iteratively in a lazy evaluation scheme. This is particularly efficient as the resulting RE always is homogeneous and linear, so that each coefficient can be calculated ....

G.H. Gonnet and D. Gruntz, Limit Computation in Computer Algebra, Technical Report 187, Department for Computer Science, ETH Zurich, 1992, (submitted to the JSC).

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