2000 MSC: prim. 00A69; sec. 35A35, 39B05, 39B72, 58J70
by
© Ph. D. & Dr. Sc. Lev Gelimson
Academic Institute for Creating Fundamental Sciences (Munich, Germany)
RUAG Aerospace Services GmbH, Germany
Mathematical Journal
of the "Collegium" All World Academy of Sciences
Munich (Germany)
3 (2003), 1
For many typical problems, there are no concepts and methods adequate and
general enough. The absolute error alone is not sufficient for approximation
quality estimation. The relative error is uncertain in principle and has a very
restricted applicability domain. The unique known method applicable to
overdetermined problems usual in data processing is the least-square method with
narrow applicability and adequacy domains and many fundamental defects. No known
proposition applies to estimating the quality of approximations to functions and
distributions.
A general equation problem generalizes sets of equations of any types including
initial and boundary value problems, etc. Consider a quantiset of any equations
over indexed functions (dependent variables) fg of indexed
independent variables zw, all of them belonging to their
individual vector spaces. Gather all available functions in the left-hand sides
of the equations without further transformations. The unique exception is
changing the signs of expressions by moving them to the other sides of the same
equations. We receive
w(l)(Kl[g
belongs to G fg[w belongs
to W zw]]
= 0) (l belongs to L)
where Kl is an operator with index l; L, G,
W are index sets; [w belongs to W
zw] is a set of indexed elements; w(l) is the
quantity as a weight of the lth equation. When replacing all the unknowns
(unknown functions) with their possible ”values” (some known functions), the
quantiset is transformed into the corresponding quantiset of formal functional
equalities. To conserve the quantiset form, for the known functions also use the
same designations fg.
A general relation problem is a quantiset of relations
Rl:
w(l)Rl[g
belongs to G fg[w belongs
to W zw]] (l belongs to L).
A general problem is a quantisystem of relations containing both known elements
and unknown ones, which can be regarded as values and variables, respectively.
A pseudosolution to a problem is a quantisystem of such values of all the
variables that, after replacing each variable with its value, the problem
quantisystem contains no unknown elements, and each of its relations has certain
sense and is determinable (i.e., true or false).
A unierror irreproachably corrects the relative error and generalizes it
possibly for any conceivable range of applicability. For an equality a =?
b (true or not), a unierror can be represented as ea
=? b = |a - b|/(|a| + |b|) if |a| + |b|
> 0, ea =? b = 0 by a = b = 0,
or, by introducing extended division a//b = a/b by
nonzero a, a//b = 0 by a = 0 and any (even zero)
b, ea =? b = |a - b|//(|a|
+ |b|). Another possibility is using, instead of the linear estimation
fraction, the quadratic one 2ea =? b
= |a - b|//[2(|a|2 +|b|2)]1/2
with values in [0, 1], too.
A reserve R with values in [-1, 1] extends the unierror e. For an
inexact object I, R(I) = -e(I). For an exact
object E, map it at its exactness boundary and take the unierror. For
inequalities, negate inequality relations and conserve equality ones, also with
natural extending to any functions.
Unierrors and reserves bring reliable estimations of approximation quality and exactness confidence. Using them unlike the least-square method, iteration methods of the least normed powers, of unierror and reserve equalization, and of a direct solution give both quasisolutions and its invariant measure. They all are very effective by setting and solving many urgent general problems in science, engineering, and life, e.g. coding ones.