fuzzy logarithms

["Steven M. Belknap" <sbelknap@UIC.EDU>]

>> la xorxes cusku di'e
>> >There are some (e.g. dugri, tenfa, sinso, tanjo) for which it wouldn't be
>> >very easy to consider them fuzzy.
>> >
>> No dice. Fuzzy numbers and fuzzy arithmetic can be easily extended to
>> logarithms, exponentials, and trigonometric functions by expressing the
>> Reimann remainder term at the end of the Taylor Series expansion as a fuzzy
>> interval. ...
>
la dilyn cusku di'e
>Certainly operations on fuzzy numbers are well-defined (and quite useful
>for, e.g., doing mathematical proofs by computer and keeping track of the
>possible error).  But I don't see the relevance.  When I say '2 is the
>log base 2 of 4', I'm making an exact statement.  There's no fuzziness
>whatsoever about it.  It would be absolutely false to say '2.00000000001
>is the log base 2 of 4.'  So the fuzziness you refer to is not due to the
>predicate 'dugri', but to the fuzziness of its arguments in some specific
>application.

The natural logarithm function can be defined in pure mathematical terms as
the ln[t] <-> Integral[1/x dx, 1,t]. This is the definition of logarithm to
which Dylan is referring.

There is a related function which is defined as an infinite sum. This
function has been constructed to approximate the logarithm function to
arbitrary precision, but once defined it exists *independently* of the pure
mathematical function. Several definitions are possible. Some rather
complex numerical functions converge with amazing rapidity. When you
calculate one of these sums to a given number of terms for a given value of
<t>, you calculate a numerical approximation of the pure logarithm
function. The rest of the infinite sum is lumped into a term which is
usually designated as R(n), and represents the unevaluated higher order
terms of the Taylor Series expansion. It is this numerical type of
function, which is used to make numerical approximations, which can be
thought of as "fuzzy." It is possible to estimate the accuracy and
precision of the numerical estimate, and to consider the plus or minus term
as an indicator of how fuzzy the calculation is.

co'omi'e la stivn


Steven M. Belknap, M.D.
Assistant Professor of Clinical Pharmacology and Medicine
University of Illinois College of Medicine at Peoria

email: sbelknap@uic.edu
Voice: 309/671-3403
Fax:   309/671-8413