Meta-expansion of transseries

The asymptotic behaviour of many univariate functions can only be expressed in generalized asymptotic scales, which are not merely formed of powers of a single variable.

It was already noticed by Hardy [Har10, Har11] that many interesting functions arising in combinatorics, number theory or physics can be expanded w.r.t. scales formed by so called exp-log functions or L-functions. An exp-log function is constructed from an indeterminate

and the real numbers using the field operations, exponentiation and logarithm. An L-function is defined similarly, by adding algebraic functions to our set of building blocks.

However, the class of functions which can be expanded with respect to a scale formed by exp-log functions (or L-functions) is not stable under several simple operations such as integration or functional inversion [Sha93, vdH97, MMvdD97]. More recently, exp-log functions have been generalized so as to allow for expressions with infinite sums, giving rise to the notion of transseries [DG86, É92]. In section 2, we briefly recall some of the most important definitions and properties. For more details, we refer to [vdH06c].

Given an explicit expression (such as an exp-log function), or the solution to an implicit equation, an interesting question is how to find its asymptotic expansion automatically. When working with respect to a generalized asymptotic scale, such as

at infinity (

), even simple expressions can lead to infinite cancellations:

In many cases, the detection of infinite cancellations can be reduced to the zero-test problem in a suitable class of functions [Sha90, GG92, RSSvdH96, Sal91, vdH97].

However, zero-test problems are often very hard. In the case of exp-log functions, a complete zero-test is only known if Schanuel's conjecture holds [Ric97, vdH98, vdHS06, Ric07]. If we want to expand more general expressions or more general solutions to differential or functional equations, the corresponding zero-test problem tends to get even harder. Consequently, the zero-testing tends to complicate the design of mathematically correct algorithms for more general asymptotic expansions. From the practical point of view, the implementation of robust zero-tests also requires a lot of work. Moreover, mathematically correct zero-tests tend to monopolize the execution time.

In this paper, we will investigate an alternative, more heuristic approach for the computation of asymptotic expansions. We will adopt a similar point of view as a numerical analyst who conceives a real number as the limit of a sequence of better and better approximations. In our case, the asymptotic expansion will be the limit of more and more precise polynomial approximations, where the monomials are taken in the asymptotic scale. As is often the case in numerical analysis, our approach will only be justified by informal arguments and the fact that it seems to work very well in practice.

Besides the analogy with numerical analysis there are a few additional interesting points which deserve to be mentioned. First of all, a finite sequence

of better and better approximations gives rise to a second sequence

, which can itself be encoded by a generating series

The computation of the expansion of

can thus be re-interpreted as the computation of the expansion of

, which we therefore regard as the “meta-expansion” of

. This technique will be detailed in section 3. Additional complications arise in the context of transseries, because the elements of the asymptotic scale are themselves exponentials of other transseries. The computation of meta-expansions for transseries will be detailed in sections 4 and 5.

A second interesting aspect of meta-expansions is that we may operate on the meta-expansion without changing the underlying expansion. In a complex computation involving lots of auxiliary series, this provides some meta-control to the user. For instance, some subexpressions can be computed with more accuracy (or less accuracy) and one can focus on a specific range of terms. Techniques for the acceleration of convergence play a similar role in numerical analysis [PTVF07, Section 5.3]. Another operation, called “stabilization”, removes those terms in the expansions

which change at every few steps. After stabilization, we tend to compute only terms which occur in the final expansion of

, even though they usually appear in a different order. In particular, stabilization gives rise to a heuristic zero-test. Meta-operations on meta-expansions will be discussed in section 6.

One motivation behind the present paper was its application to the asymptotic extrapolation of sequences by transseries [vdH06a]. This application requires the computation of discrete sums and products of transseries. In section 7, we have included a small demonstration of our current implementation in the Mathemagix system [vdH02b].

For the purpose of this application, we have mainly considered univariate transseries expansions so far. Of course, the approach of our paper generalizes to expansions in several variables. A natural next step for future developments would be to implement the Newton polygon method for rather general functional equations. Another interesting question is how to re-incorporate theoretically correct zero-tests in our mechanism and how much we really sacrifice when using our heuristic substitute. A few ideas in these directions will be given in section 8, some of which actually go back to [vdH94].

2.Transseries

In this section, we briefly survey some basic properties of transseries. For more details, we refer to [vdH06c, vdH97, É92, MMvdD97, MMvdD99].

Let

be a ring and

a commutative monomial monoid which is partially ordered by an asymptotic dominance relation

. A subset

is said to be well-based if it well-quasi-ordered [Pou85, Mil85] for the opposite ordering of

and grid-based if it satisfies a bound of the form

A well-based power series is a formal sum

, whose support

is well-based. It is classical [Hah07, Hig52] that the set

of well-based power series forms a ring. The subset

of grid-based power series (i.e. with grid-based support) forms a subring of

with

for

(i.e.

). Then the first series is grid-based and the second one only well-based.

A family

of series in

is said to be well-based if

is well-based and

is finite for every

. In that case, the sum

with

is again in

. A linear mapping

is said to be strong it preserves well-based summation. Grid-based families and the corresponding notion of strong linearity are defined similarly.

In the case when

is a field and

is totally ordered, then

and

are also fields. Furthermore, any non-zero

admits a unique dominant monomial

with corresponding dominant coefficient

and relative remainder

such that

. We call

the dominant term of

and define

in the case when

. The series

also admits a canonical decomposition

Here

just means that

; more generally,

. If

is grid-based, then so are

and

. If

is actually an ordered field, then so are

and

, by taking

for all

The field

of grid-based transseries is a field of the form

with additional operators

and

. The set of transmonomials

coincides with the set

of exponentials of transseries

with

. More generally, we have

The construction of

is detailed in [vdH06c, Chapter 4]. The construction of fields of well-based transseries is a bit more delicate [DG86, vdH97, Sch01], because one cannot simultaneously ensure stability under exponentiation and infinite summation. However, there is a smallest such field

, if we exclude transseries with arbitrarily nested logarithms or exponentials, such as

Furthermore, there exist very general implicit function theorems [vdH01, vdH06c], which can be used in order to solve functional equations, when put in a suitable normal form.

The field

is highly non-Archimedean. For what follows it will be introduce the asymptotic flatness relation

3.Meta-expansions

The intuitive idea behind meta-expansion is that, from the computational point of view, series usually arise as a sequence of successive approximations of an interesting object. We will make this simple idea operational by taking the approximations to be polynomials. In order to avoid repetitions, we will systematically work in the well-based setting; it is easy to adapt the definitions to the grid-based case.

Let

be an effective ring and

an effective monomial monoid. Recall that

stands for the ring of well-based generalized power series and let

be the corresponding set of polynomials (i.e. series with finite support). We define an expander to be a computable well-based sequence

of polynomials. Its sum

will be called the result of the expander and we say that

is an expander for

. We also define an approximator to be a computable sequence

, such that

is well-based and such that for each

, there exists an

for which

is constant for

. In that case, the limit

is called the result of the approximator.

Clearly, the notions of expander and approximator are variants of another: if

is an expander, then

with

defines an approximator with the same result. Similarly, if

is an approximator, then

with

(

) defines an expander with the same result. However, for certain purposes, expanders are more suitable, because an expander

can be manipulated via its generating series

For other purposes though, approximators are the more natural choice. As far as notations are concerned, it is convenient to pass from expanders

to approximators

(and vice versa) by prefixing the index by a semicolon (resp. removing the semicolon).

We will denote by

the set of approximable series in

which admit an expander (or approximator). The corresponding set of expanders will be denoted by

. Given

, we use the notation

to indicate that

represents

. Given

, we will also use the notation

to indicate that

is a representation for

. For more details on this convention, see [vdH07, Section 2.1].

In practice, expanders and approximators are usually implemented by pointers to an abstract class with a method to compute its coefficients. For more details on how to do this, we refer to [vdH02a]. Let us now show how to implement expanders and approximators for basic operations in

Constructor

Addition

Multiplication

However, a subtlety occurs here. Consider for instance the case when

and

, so that

and

. Contrary to what happened in the case of addition, the definitions (6) and (7) do not coincide in the sense that

. Indeed, we respectively find

As a general rule, the manipulation of expanders tends to be a bit more economic from the computational point of view, its coefficients being smaller in size.

Left composition with power series

Notice that the individual coefficients of

need not be infinitesimal. Besides, the composition

is usually not a polynomial. Therefore, we have forced

to become infinitesimal using a multiplication by

. This is justified by the fact that

Inversion

Unfortunately, there exists no general algorithm for the computation of the dominant term

. We will therefore assume the existence of an oracle for this purpose. In section 6, we will present a heuristic algorithm which can be used in practice.

Fixed points

and apply a fixed-point theorem. In our case, this requires a well-based operator

for which we can prove that

admits a well-based limit in

. The obvious expander for

is given by

For some general fixed-point theorems for sequences

of this kind, we refer to [vdH01] and [vdH06c, Chapter 6].

4.Meta-expansion of transseries

In order to compute with well-based transseries in

, we take our coefficients in an effective subfield

and our monomials in an effective subgroup

. Moreover, the monomials in

which are not iterated logarithms are themselves exponentials of approximable transseries.

More precisely, elements in

are represented by monomials

which are of one of the following forms:

In the first case, the exponential height

is defined to be zero and in the second case, we set

Elements

are multiplied using

and inverted using

. Here

: if

, then

; if

, then

. The asymptotic ordering

is implemented using

and therefore relies on an oracle for the computation of

. Setting

, we notice that heuristic algorithms for the computation of this dominant term recursively need to compare elements

with respect to

. The termination of the recursion is based on the fact that

The main additional operations for the manipulation of transseries are exponentiation and logarithm. Since exponentiation relies on canonical decompositions, we start with the general operation of restriction of support.

Restriction of support

If the subset

is a computable, i.e.

admits a computable membership test, then the mapping

is computable. In particular, given

, we may compute

using

Now making continued use of our oracle for the computation of dominant terms, the sets

and

are computable. Consequently, we have algorithms to compute

and

Logarithm and exponentiation

Remark 3. In practice, it is useful to have efficient algorithms for the manipulation of transmonomials. In a similar way as in [RSSvdH96, vdH97, vdH06c], we therefore write transmonomials as power products

with respect to a transbasis

which is constructed incrementally during the computations. In our well-based setting, we merely require that the

satisfy the hypotheses

In the grid-based setting TB2 may be replaced by the stronger requirement that

can be expanded w.r.t.

for all

5.Other operations on transseries

Let us now come to the more interesting operations on transseries. Differentiation and composition rely on the general principle of extension by strong linearity.

Extension by strong linearity

Differentiation

Composition

It can be shown [vdH06c, vdH97] that the derivation w.r.t.

admits a unique strongly linear right inverse

with the “distinguished property” that

for all

. One way to construct

is to first compute its “trace”

which is the unique strongly linear operator with

. We then compute

by solving the implicit equation

One may either apply (10) for monomials and extend by strong linearity, or apply it directly for arbitrary transseries

Trace of the distinguished integration

Distinguished integration

Then distinguished integration can be regarded as the application of this operator expander to another expander

Adapting our general mechanism for the computation of fixed points for operators instead of series, we find

as the natural expander of

Functional inversion of transseries can be done using formulas in [vdH06c, Section 5.4] and we will not detail this operation here. Two other interesting operations are finite differences and discrete summation:

We implemented these operations because they are critically needed in an algorithm for the asymptotic extrapolation of sequences [vdH06a]. Modulo compositions with

and

, they are related to finite quotients and discrete products:

Our algorithm for right-composition clearly yields a way to compute

for

. The distinguished summation

is the unique distinguished strongly linear right-inverse of

, i.e.

and

for all

. It therefore suffices to show how to compute

for monomials

. Three different cases need to be distinguished :

Flat discrete summation

is computed in a similar way as in the case of distinguished integration. In fact, the expander

can directly be applied to expanders

with

for all

. Moreover, this application preserves grid-basedness.

Moderate discrete summation

and we invert the operator

as in the flat case. No integration is needed this time, since

. Again, the grid-based property is preserved by moderate discrete summation.

Steep discrete summation

It can be shown that

is grid-based if

is grid-based and there exists a

such that

for all

6.Meta-operations on expanders

So far, we have not really exploited the extra level of abstraction provided by expanders. In this section, we will describe several “meta-operations” on expanders. These operations do not affect the series being represented, but rather concern qualitative aspects of the approximation process: they guide the rate of convergence, the terms which appear first, etc. Based on the process of “stabilization”, we will also describe a heuristic zero-test and a heuristic method for the computation of dominant terms.

Shortening

The shortening operator is typically used for the expansion of expressions which involve an expander

, such that the expression size of

tends to grow very rapidly with

. For instance, we may prefer to compute a sum

using the expander

instead of

Lengthening

During the expansion of an expression, the lengthening operator may for instance be used in order to boost the precision of a subexpression. We may also use it as a substitute for the order parameter of a typical expansion command. E.g., we would simply display

in order to show an additional

terms of

Stabilization

The stabilization operator removes all terms from the expansion

which are still subject to changes during the next

approximations. Even for small values of

, such as

, we usually have

In particular, the successive approximations

usually only contain terms which occur in the final result

Example 4. Let us reconsider the function

from example 2. When approximating

using

instead of

, we get:

Example 5. An example for which (12) is not satisfied for any finite

is the expander

In this kind of situations, it may be necessary to consider more powerful stabilizations of the form

Dominant terms

for a sufficiently large value of

. When taking

and

fixed, the formula (13) also provides us with a reasonable heuristic for the computation of

(which implies a zero-test for

). Of course, the values

and

should not be taken too small, so as to provide sufficient robustness. On the other hand, large values of

and

may lead to unacceptable computation times. Our current compromise

has worked for all practical examples we have tried so far.

Remark 6. Our claim that relatively small values of

and

provide sufficient robustness may seem very surprising at first sight and is indeed the remarkable feature which make meta-expansions so useful in our opinion. The intuitive justification lies in the fact that we expand in a really massive way all our operations and all our parameters. On the one hand, canceling terms usually change after every step before they vanish, and are thereby “stabilized out”. On the other hand, deeper combinations of parameters which lead to a genuine non-canceling contribution can usually be detected after a few steps. In particular, small power series expressions with large valuations [vdH06b] tend to be less harmful in our context.

Remark 7. Let

be the class of expanders which are obtained by applying our expansion algorithms to exp-log expressions. From a theoretical point of view, it might be interesting to investigate the existence of a simple computable function

such that, for any

, there exists a

with

Printing

with

and

. In (14), we now replace each non-zero

by the expression

, and print the result. For instance, if

An interesting feature of this way of printing is that it allows us to see some of the remaining terms after the first

leading terms. In certain cases, such as

one might prefer to suppress some of these extra terms. One criterion for suppression could be the following: given the last term

of some

and any term

, suppress all terms

with

for some

Dominant bias

We call

a dominant bias operator. We may typically apply it on auxiliary series

with a sharp increase of the number of terms of

with

. More generally, it is possible to define operators which favour terms at the tail, in the middle, or close to a specified monomial. However, these generalization do not seem to have any practical applications, at first sight.

7.Expansion gallery

Most of the algorithms described in this paper have been implemented inside the Mathemagix system [vdH02b]. Below, we illustrate the implementation with a sample session.

7.1.Exp-log functions

Mmx]

use "numerix"; use "algebramix"; use "multimix"; use "symbolix";

Mmx]

x == infinity ('x);

Mmx]

1 / (x + log x + log log x)

Mmx]

1 / (1 + 1/x + 1/exp x)

Mmx]

1 / (1 + 1/x + 1/exp x) - 1 / (1 + 1/x)

Mmx]

exp (x + exp (-exp x)) - exp (x)

Mmx]

exp (exp (x) / (x + 1))

7.2.Calculus

Mmx]

derive (exp (exp (x) / (x + 1)), x)

Mmx]

integrate (exp (x^2), x)

Mmx]

integrate (x^x, x)

Mmx]

product (x, x)

Mmx]

lengthen (product (x, x), 8)

Mmx]

product (log x, x)

7.3.Functional equations

Mmx]

fixed_point (f :-> log x + f @ (log x))

Mmx]

la == derive (fixed_point (f :-> log x + f @ (log x)), x)

Mmx]

mu == la * la + 2 * derive (la, x)

Mmx]

fixed_point (f :-> 1/x + f @ (x^2) + f @ (x^x))

8.Towards more robustness

Even though the expansion algorithms developed so far are usually sufficient for applications, they lack robustness in several ways. First of all, we have used heuristic algorithms for zero-testing and the computation of dominant terms. Some of our algorithms crucially depend on the correctness of these heuristic algorithms. For instance, our algorithm for the computation of an inverse

yields an erroneous result if

is computed incorrectly. Finally, expanders

only asymptotically tend to

. Even if we know that a given monomial

is in the support of

, we do not know how large

should be in order to guarantee that

. In this section, we describe a few ideas which may be used to increase the robustness of our algorithms.

Auto-correction

In order to have

, it now suffices to have

(

), instead of

. Of course, in nasty cases, it might still happen that

for all

. In other words, the strategy of auto-correction produces no miracles and does not substitute for a genuine zero-test. Even in the case when the stabilization operator

is sufficiently powerful to guarantee

for a certain class of expanders, one should keep in mind that the result only becomes correct at the limit; we still don't know how many terms need to be computed.

From a practical point, the strategy of auto-correction is easy to implement on the series level from section 3: in our example of inversion, the expander

may simply keep

in memory for the largest

considered so far and only update its value when

changes. Implementations become more involved when considering recursive transseries expansions, as in section 4. Indeed, in this more general setting, we also need to correct erroneous outcomes for the asymptotic ordering

, which recursively relies on the orderings

and

for expanders of lower exponential height. In order to generalize the idea, one thus has to define sequences of approximations

and

for

and

, and systematically work with the relations

and

when making decisions for expansions at stage

Reduction to zero-testing

An interesting question is whether we can use a zero-test in

in order to design a non-heuristic algorithm for the computation of dominant terms. In order to make this work, we have to be able to detect infinite cancellations of terms which occur in expressions such as

. A general mechanism for doing this is to refine the mechanism of expanders by indexing over a suitable well-based set.

More precisely, given an abstract well-based set

(i.e. a set

which is well-quasi-ordered for the opposite ordering of

), we define a sequence

of finite subsets of

. In general, the sequence

is transfinite, but if we have

, then we say that

is accessible. We say that

is computable, if for any finite subset

, we can compute

. In particular, this implies the sequence

to be computable.

A well-based expander is a computable well-based family

, indexed by a computable accessible well-based set

. A well-based expander is the natural refinement of an expander

in the usual sense, by regrouping terms

. We say that

is a termwise well-based expander if each

is of the form

and the mapping

is increasing. Notice that

is automatically well-based if

is increasing.

Recall that the initial segment generated by a finite subset

is defined by

. Now consider a termwise well-based expander

such that

for any finite subset

. If the mapping

is effective, then we call

an expander over . For the operations we have considered in this paper, it should be possible to replace the usual notion of expander by expanders of

(assuming that

is stable under the operation). This was already shown in [vdH94] for the basic operations from section 3 and still waits to be worked out for the other ones.

Given an expander

over

, the zero-test in

may now be used in order to compute the set

of dominant monomials of

. The algorithm again goes back to [vdH94]:

Computable well-based series

There are really two aspects to a computable well-based series

. On the one hand, we should be able to compute its coefficient

for any monomial

. On the other hand, its support should be sufficiently effective. In the case of ordinary power series

, the second issue does not really arise, because the support is necessarily included in the well-based set

. In general, one might require that

is given by a termwise well-based expander, which yields quite a lot of information about

As to the computation of coefficients

, consider the case of a product

, where

is totally ordered and

and

are given by

and

with

and

. Given

, we hit the problem that we don't have any a priori information on the asymptotic behaviour of the

and the

. In order to design an algorithm for the computation of

, we need more control over this asymptotic behaviour.

In the grid-based setting, we are really computing with multivariate power series, and no real difficulties arise. In the well-based setting, things get more involved. When we restrict our attention to transseries, there are ways to represent and compute with monomial cuts [vdH06c, Chapter 9]. A monomial cut is the analogue of a Dedekind cut for the set of transmonomials instead of

. Given a termwise well-based expander

, any initial segment

naturally induces a transseries

and a monomial cut, called the width of

. For a fully satisfactory definition of computable well-based series, one should be able to compute these widths. However, we have not investigated this matter in detail yet.

Bibliography

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			(2)
			(3)

			(4)
			(5)

1.Introduction

2.Transseries

3.Meta-expansions

Constructor

Addition

Multiplication

Left composition with power series

Inversion

Fixed points

4.Meta-expansion of transseries

Restriction of support

Logarithm and exponentiation

5.Other operations on transseries

Extension by strong linearity

Differentiation

Composition

Trace of the distinguished integration

Distinguished integration

Flat discrete summation

Moderate discrete summation

Steep discrete summation

6.Meta-operations on expanders

Shortening

Lengthening

Stabilization

Dominant terms

Printing

Dominant bias

7.Expansion gallery

7.1.Exp-log functions

7.2.Calculus

7.3.Functional equations

8.Towards more robustness

Auto-correction

Reduction to zero-testing

Computable well-based series

Bibliography