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G e n e r a L p r o b L e m t h e o r y

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2000 MSC: prim. 00A69; sec. 35A35, 39B05, 39B72, 58J70


L.G.Gelimson, Ph.D., D.Sc., RUAG Munich


For many problems, e.g., in information processing, 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 with contradictoriness 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 especially 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 any further transformations. The unique exception is changing the signs of expressions by moving them to the other sides of the same equations. The form of the quantiset becomes

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 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 general 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 general relations containing both known elements and unknown ones, which can be naturally regarded as values and variables, respectively.

A pseudosolution to a problem is a quantisystem of such values of all variables that, after substituting all values, the problem quantisystem contains no unknown elements, and each relation has certain sense and is fully determinable (i.e., true or false).

An autoerror irreproachably corrects the relative error and generalizes it possibly for any conceivable applicability range. For a formal equality (true or not true) a =? b, an autoerror is

ea =? b = |a - b|/(|a| + |b|) if |a| + |b| > 0, ea =? b = 0 by a = b = 0,

or, due to introducing the following extended division

a//b = a/b by nonzero a, a//b = 0 by a = 0 and any, e.g. zero, b,

ea =? b = |a - b|//(|a| + |b|).

Instead of the linear estimation fraction, the quadratic one

2ea =? b = |a - b|//[2(|a|2 +|b|2)]1/2 with any real numbers a and b and values in [0, 1], too, is also suitable and can be simply extended.

A reserve R with values in [-1, 1] extends the autoerror e. For an inexact object I, R(I) = -e(I). For an exact object E, map it at its exactness boundary and take the autoerror. For inequalities, negate inequality relations and conserve equality ones, also with natural extending to any functions and distributions.

Autoerrors and reserves bring reliable estimations of approximation quality and exactness confidence and lead to natural definitions of quasi-, anti-, and supersolutions to general problems. Using them all unlike the least-square method, iteration methods of the least normed powers, of autoerror and reserve equalization, and of a direct solution give quasi-, anti-, and supersolutions to any general problems including overdetermined ones with contradictoriness and its invariant measure. They all are very effective by setting and solving urgent problems in science and life, e.g. coding ones.

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