B. Salvy, J. Shackell. Asymptotic expansions of functional inverses. Technical report nr. 1673, INRIA, France.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....in a theoretical perspective. In this paper we will show that the technique used naturally extends to obtain automatic expansions of more general types of functions. Our results slightly generalize those of Shackell, which were obtained by his technique of nested expansions (see [Sh 91] SalSh 92] Furthermore, our approach is expected to be a little bit more efficient (at least, a constant factor should be gained) Before we run actual benchmarks, we think that in any case our algorithms are a bit simpler and we think that the obtained expansions are a bit more natural than nested ....

B. Salvy, J. Shackell. Asymptotic expansions of functional inverses. Technical report nr. 1673, INRIA, France.

....of linear differential equations. Alternatively, the coefficients of these generating functions can be given by a linear recurrence. Any kind of solution to these equations can be used to help the analysis. Polynomial solutions of very general linear equations can be found algorithmically [7]. Divergent series often occur as solutions to linear differential equations, but several algorithms make it possible to deal with them [8] A general framework for the manipulation of linear operators and proof of combinatorial identities is the subject of [9] In [10] a remark about the use of ....

....with singleprecision arithmetic [6] fast computation on matrices over a finite field [13] factorization over finite fields [14] efficient computations with algebraic curves [15] and integration of hyperelliptic functions [16] 6] Evaluating Signs of Determinants. Jean Daniel Boissonnat [7] Polynomial Solutions of Linear Operator Equations. Marko Petkovsek [8] Symbolic and Numerical Manipulations of Divergent Power Series. Jean Thomann [9] Holonomic Systems and Automatic Proofs of Identities. Fr ed eric Chyzak [10] Short and Easy Computer Proofs of Partition and q Identities. ....

[Article contains additional citation context not shown here]

Salvy (Bruno) and Shackell (John). -- Asymptotic expansions of functional inverses. In Wang (Paul S.) (editor), Symbolic and Algebraic Computation. pp. 130--137. -- ACM Press, 1992. Proceedings of ISSAC '92, Berkeley, July 1992.

....The first concrete algorithm was given by J. Shackell in [15] D. Gruntz made a number of improvements, and also implemented the method, 5] Since the appearance of [15] many problems of asymptotics have been treated (at least partially) from the effective point of view: functional inversion [14, 20], Liouvillian functions [16] composition [17, 20] algebraic differential equations [18] However, implementations have not followed these recent advances. The first implementation of a non trivial limit computation was done in [4] A program to perform asymptotic expansions in general scales was ....

Salvy, B., and Shackell, J. Asymptotic expansions of functional inverses. In Symbolic and Algebraic Computation (1992), P. S. Wang, Ed., ACM Press, pp. 130-- 137. Proceedings of ISSAC'92, Berkeley, July 1992.

....This brings particular difficulties regarding constants. As is the normal practice in this area we shall use an oracle for the determination of signs of these. We discuss this matter more fully in Section 1.4. Inverse functions have long been problematic in asymptotics [5, 8] However in [18], the authors gave an algorithm for inverting nested forms which solves the problem of expressing the asymptotic behaviour of inverse functions. In the present paper we treat the more general problem of implicit 2 BRUNO SALVY AND JOHN SHACKELL functions. More precisely, let H(x; y) denote the ....

Salvy, B., and Shackell, J. Asymptotic expansions of functional inverses. In Symbolic and Algebraic Computation (1992), P. S. Wang, Ed., ACM Press, pp. 130--137. Proceedings of ISSAC'92, Berkeley, July 1992.

.... In an example like this, Equation (2.2) admits a simple closed form solution. In general though, no such closed form exists and it is necessary to find an asymptotic expansion of the saddle point location in terms of n. A general procedure for doing so when f is any exp log function was given by Salvy Shackell (1992) and a fast algorithm in a special but frequent case was given in (Salvy 1994) Knowing the asymptotic expansion of the location of the saddle point is not always sufficient to complete the asymptotic expansion of the coefficients, since in some cases substituting the expansion of Rn in (2.3) does ....

Salvy, B. and J. Shackell (1992). Asymptotic expansions of functional inverses. In P. S. Wang (Ed.), Symbolic and Algebraic Computation, pp. 130--137. ACM Press. Proceedings of ISSAC'92, Berkeley, July 1992.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC