Hello.
About the automatic differentiation /encore/
Le 20/04/2016 18:02, Hendrik Boom a écrit :
Instead of eveluating the function at x = 1,
you essentially symbolically evaluate it at 1 + dx. The key here is
that you use the algebra of differentials instead of the arithmetic of
numbers.
Yes, more or less, although this is one of numerous interpretations of
the process. You may map your numerical domain into the dual numbers of
Clifford (Wipedia duly mentions the relation to auto. diff.) or do some
specific processing of the original source code, which is not easy to
interpret mathematically.
Some hundreds years ago I implemented the lazy generalization of this "x
+ dx" idea, constructing in Haskell the closed differential algebra
based on "infinite" lists, permitting (in principle) to compute ALL the
derivatives at a given point.
The complexity of higher derivatives could be quite nasty, although the
co-recursive algorithmization of the technique was extremely compact,
and, I think, elegant...
==
And educated gentlemen should know also a bit something about the
*reverse* automatic differentiation, which is algorithmically very
different from the standard "forward" one. The reverse technique is a
kind of "time vehicle", and it is a fascinating subject.
Jerzy Karczmarczuk
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