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Algorithms for asymptotic interpolation *Joris* *van* *der* *Hoeven*

, 2006

"... Consider a power series f ∈ R[[z]], which is obtained by a precise mathematical construction. For instance, f might be the solution to some differential or functional initial value problem or the diagonal of the solution to a partial differential equation. In cases when no suitable method is beforeh ..."

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Consider a power series f ∈ R[[z]], which is obtained by a precise mathematical construction. For instance, f might be the solution to some differential or functional initial value problem or the diagonal of the solution to a partial differential equation. In cases when no suitable method is beforehand for determining the asymptotics of the coefficients fn, but when many such coefficients can be computed with high accuracy, it would be useful if a plausible asymptotic expansion for fn could be guessed automatically. In this paper, we will present a few such “asymptotic interpolation algorithms”. Roughly speaking, using discrete differentiation and techniques from automatic asymptotics, we strip off the terms of the asymptotic expansion one by one. The knowledge of more terms of the asymptotic expansion will then allow us to approx-imate the coefficients in the expansion with high accuracy.

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Counterexamples to Witness Conjectures *Joris* *van* *der* *Hoeven*

"... Consider the class of exp-log constants, which is constructed from the integers using the field operations, exponentiation and logarithm. Let z be such an exp-log constant and let n be its size as an expression. Witness conjectures attempt to give bounds $(n) for the number of decimal digits which n ..."

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Consider the class of exp-log constants, which is constructed from the integers using the field operations, exponentiation and logarithm. Let z be such an exp-log constant and let n be its size as an expression. Witness conjectures attempt to give bounds $(n) for the number of decimal digits which need to be evaluated in order to test whether z equals zero. For this purpose, it is convenient to assume that exponentials are only applied to arguments with absolute values bounded by 1. In that context, several witness conjectures have appeared in the literature and the strongest one states that it is possible to choose $(n) = O(n). In this paper we give a counterexample to this conjecture. We also extend it so as to cover similar, polynomial witness conjectures. 1.

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Uniformization of multivariate power series ∗ *Joris* *van* *der* *Hoeven*

, 2009

"... In this paper, we describe an algorithm for the “uniformization ” of a multivariate power series. Let K[[T]] be the field of “grid-based power series ” over a sufficiently α1 αn large non archimedean “monomial group ” (or value group) T, such as T = {t1 tn: α1, , αn ∈ R} with the lexicographical ord ..."

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In this paper, we describe an algorithm for the “uniformization ” of a multivariate power series. Let K[[T]] be the field of “grid-based power series ” over a sufficiently α1 αn large non archimedean “monomial group ” (or value group) T, such as T = {t1 tn: α1, , αn ∈ R} with the lexicographical ordering on α1, , αn. We interpret power series f ∈ K[[x1, , xn]] as functions K[[T ≺ ] n → K[[T �]]. On certain “regions ” R of the space K[[T ≺]] n, it may happen that the valuation of f can be read off from the valuations of the xi. In that case, f is said to be “uniform ” on R. We will describe an algorithm for cutting K[[T ≺]] n into a finite number of regions, each on which f is uniform for a suitable change of coordinates, which preserves the elimination ordering on x1,,xn. The algorithm can probably be seen as an effective counterpart of local uniformization in the sense of Zariski, even though this connection remains to be established in detail. 1.

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Operators on generalized power series by *Joris* *van* *der* *Hoeven*

, 2008

"... Given a ring C and a totally (resp. partially) ordered set of “monomials ” M, Hahn (resp. Higman) defined the set of power series C[[M]] with well-ordered (resp. Noethe-rian or well-quasi-ordered) support in M. This set C[[M]] can usually be given a lot of additional structure: if C is a field and M ..."

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Given a ring C and a totally (resp. partially) ordered set of “monomials ” M, Hahn (resp. Higman) defined the set of power series C[[M]] with well-ordered (resp. Noethe-rian or well-quasi-ordered) support in M. This set C[[M]] can usually be given a lot of additional structure: if C is a field and M a totally ordered group, then Hahn proved that C[[M]] is a field. More recently, we have constructed fields of “transseries ” of the form C[[M]] on which we defined natural derivations and compositions. In this paper we develop an operator theory for generalized power series of the above form. We first study linear and multilinear operators. We next isolate a big class of so-called Noetherian operators Φ: C[[M]] → C[[N]], which include (when defined) summation, multiplication, differentiation, composition, etc. Our main result is the proof of an implicit function theorem for Noetherian operators. This theorem may be used to explicitly solve very general types of functional equations in generalized power series. 1

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A differential intermediate value theorem by *Joris* *van* *der* *Hoeven*

, 2008

"... In this survey paper, we outline the proof of a recent differential intermediate value theorem for transseries. Transseries are a generalization of power series with real coefficients, in which one allows the recursive appearance of exponentials and loga-rithms. Denoting by T the field of transserie ..."

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In this survey paper, we outline the proof of a recent differential intermediate value theorem for transseries. Transseries are a generalization of power series with real coefficients, in which one allows the recursive appearance of exponentials and loga-rithms. Denoting by T the field of transseries, the intermediate value theorem states that for any differential polynomials P with coefficients in T and f < g in T with P (f)P (g) < 0, there exists a solution h∈T to P (h) = 0 with f <h < g. 1

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Relaxed Multiplication Using the Middle Product *Joris* *van* *der* *Hoeven*

"... In previous work, we have introduced the technique of re-laxed power series computations. With this technique, it is possible to solve implicit equations almost as quickly as doing the operations which occur in the implicit equation. In this paper, we present a new relaxed multiplication algo-rithm ..."

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In previous work, we have introduced the technique of re-laxed power series computations. With this technique, it is possible to solve implicit equations almost as quickly as doing the operations which occur in the implicit equation. In this paper, we present a new relaxed multiplication algo-rithm for the resolution of linear equations. The algorithm has the same asymptotic time complexity as our previous al-gorithms, but we improve the space overhead in the divide and conquer model and the constant factor in the F.F.T. model.

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Complexity bounds for zero-test algorithms *Joris* *van* *der* *Hoeven*

"... In this paper, we analyze the complexity of a zero test for expressions built from formal power series solutions of first order differential equations with non degenerate initial conditions. We will prove a doubly exponential complexity bound. This bound establishes a power series analogue for “witn ..."

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In this paper, we analyze the complexity of a zero test for expressions built from formal power series solutions of first order differential equations with non degenerate initial conditions. We will prove a doubly exponential complexity bound. This bound establishes a power series analogue for “witness conjectures”. 1.

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Effective real numbers in Mmxlib *Joris* *van* *der* *Hoeven*

"... Until now, the area of symbolic computation has mainly fo-cused on the manipulation of algebraic expressions. Based on earlier, theoretical work, the author has started to de-velop a systematic C++ library Mmxlib for mathemati-cally correct computations with more analytic objects, like complex numbe ..."

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Until now, the area of symbolic computation has mainly fo-cused on the manipulation of algebraic expressions. Based on earlier, theoretical work, the author has started to de-velop a systematic C++ library Mmxlib for mathemati-cally correct computations with more analytic objects, like complex numbers and analytic functions. While implement-ing the library, we found that several of our theoretical ideas had to be further improved or adapted. In this paper, we re-port on the current implementation, we present several new results and suggest directions for future improvements. Categories and Subject Descriptors F.2.1 [Theory of Computation]: Analysis of algorithms and problem complexity—Numerical algorithms and prob-lems

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Majorants for formal power series *Joris* *van* *der* *Hoeven*

, 2008

"... In previous papers, we have started to develop a fully effective 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 ..."

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In previous papers, we have started to develop a fully effective 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 computations are an important part of this program. In this paper we will recall or show how the classical majorant technique can be used in order to obtain many such bounds.