The Unifying View
on Ordinary Differential Equations and Automatic Differentiation
basic terminology. The seed of the theory discussed here
originates in one chapter of the book by Ramon Moore "Interval
Analysis" (1966) where he linked the optimized n-order differentiation
of general expressions with the elementary functions. Though he didn't
offer his own term, in most straightforward way his idea must be called
the Optimized Taylor Method.
Later the practitioners attached to this method the name Automatic Differentiation (AD)
which relies on the formulas of the Optimized n-order differentiation
of general expressions by Moore using them for automation in converting
the code computing the functions into a code computing the n-order
derivatives of those functions. In the frame of the "Unifying View on ODEs and AD" we
use intermittently either the terms Optimized Taylor method or AD (yet
in the sense of the optimized formulas only). We strongly suggest to
stay with these two terms not adding any alternatives like "Power Series Method" or "Differential Transform Method",
because it all in essence is the Optimized
Taylor Method framed in the "Unifying
View on ODEs and AD".
Here we discuss and deal with ODEs which are represented by
holomorphic functions and whose solutions are therefore holomorphic
function in a particular domain. That means that the right hand sides
of the ODEs are expendable into a multi-variate Taylor series with
non-zero convergence radii for each variable at every point of the
derivatives therefore are understood as complex derivatives (even if
the integration occurs along the real axis only).
The main formulas for n-order
derivatives including what was later named the Faa di-Bruno formula
were introduced by Antoine Arbogast as early as in 1800 (see the Reference compiled by the late Harley
Flanders). Brook Taylor suggested and studied the expansions of
(holomorphic) functions into power series (the Taylor expansions) even
During those two centuries that the idea of the
Taylor expansion and the classical Taylor method for integration of
ODEs were around, the possibility of computing n-order derivatives of functions
and expressions over them was rather taken for granted. It appears
however that the algorithms of n-order
practically applicable only to a particular sub-class of holomorphic
derived from the so called general
elementary functions. The Taylor
method exists in a frame of the concept called the Unifying view on
Ordinary Differential Equations and Automatic Differentiation outlined
in the Preface of the article
same title .
The central concept of the Unifying view is the definition of the
general elementary vector-functions (widening the class of conventional
elementary functions as defined by Liouville). Since Ramon Moore in the
1960s, the general elementary functions were defined as
vector-solutions of systems of m
rational ODEs of order 1. It is also possible to introduce
competing definition for a stand alone elementary function as a
solution of one n-order
rational ODE. However the issue of equivalency of both definitions
remains open depending on the not yet resolved Conjecture.
First this Conjecture was published in 2008 , then in 2009 . In
the most comprehensive form and with the necessary context it is posted
(and pdf) .
Integration of holomorphic ODEs in fact is a process of analytical
continuation which allows to reach any point of the domain at which the
right hand sides of the ODEs are regular (holomorphic). But what if the
ODEs happen to be singular at a particular point of the phase space?
Even if the given ODEs are singular at a particular point, their
solution may exist and be holomorphic at this very point  such as
= 2x/t, x|t=0 = 0
whose solution is x(t)=t2
satisfying also a regular ODE x' = 2t.
Is such a
replacement of a singular rational ODE with a regular one always
possible? As it
was shown in , there exist such special points in holomorphic
functions at which the function can satisfy no regular rational ODE of
any order at these points.
Though such a special point in a function is a point of holomorphy, its
specialty is in that it cannot be removed by choosing some regular
satisfied by this function. This specialty is unremovable and intrinsic
for the function, such as the point t=0
for x(t)=sin(t)/t. However in
order to classify such a point as a point where elementariness of the
function x(t) is
violated, we need to prove the Conjecture.
Though the Conjecture looks like a small gap in the Table
on slide 4 (and pdf)
, its role in this theory is absolutely
crucial. Everybody are welcomed to brainstorm this challenge.
"removable" singularities. Complex Variables and Elliptic Equations, Vol. 53, No.
2008, pp. 633-642.
ordinary differential equations and automatic differentiation unified.
Complex Variables and Elliptic Equations, Vol. 54, No. 9, September
Unifying View on ODEs and AD yet
with a gap to fill (and pdf) 2012 - the most brief presentation
of the gap (slide 4), the Conjecture (slides 6-8) and its context.