Ph. D. & Dr. Sc. Lev Gelimson's Single-Source Iteration Theory in Fundamental Science on General Problem Iteration

Single-Source Iteration Theory in Fundamental Science on General Problem Iteration

© Ph. D. & Dr. Sc. Lev Gelimson

Academic Institute for Creating Fundamental Sciences (Munich, Germany)

Mathematical Journal

of the "Collegium" All World Academy of Sciences

Munich (Germany)

11 (2011), 40

Introduction

In classical mathematics [1], iteration methods are very efficient by solving usual noncontradictory problems which are not sufficiently general.

Single-source iteration theory (SSIT) in fundamental science on general problem iteration applies known iteration approaches to any urgent general (including contradictory) problems and creates new methods and know-how with managing and improving computer aided calculation to provide its practical real-time feasibility and acceleration.

General Problem

General problem type and setting theory (GPTST) in fundamental science on general problem essence [5] defines a general quantitative mathematical problem, or simply a general problem, to be a quantisystem [2-5] (former hypersystem) P which includes unknown quantisubsystems and possibly includes its general subproblems.

In particular, a general problem can be a quantiset

_q(λ)R_λ{_φ∈Φ _r(φ)f_φ[_ω∈Ω _s(ω)z_ω]} (λ∈Λ)

of indexed known quantirelations _q(λ)R_λ (with their own, or individual, quantities q(λ)) [2-5] over indexed unknown quantifunctions (dependent variables), or simply unknowns (with their own, or individual, quantities r(φ)), _r(φ)f_φ , of indexed independent known quantivariables _s(ω)z_ω (with their own, or individual, quantities s(ω), all of them belonging to their possibly individual vector spaces, in their initial forms without any further transformations

where

R_λ is a known relation with index λ from an index set Λ ;

f_φ is an unknown function (dependent variable) with index φ from an index set Φ ;

z_ω is a known independent variable with index ω from an index set Ω ;

[_ω∈Ω _s(ω)z_ω]

is a quantiset of indexed quantielements _s(ω)z_ω .

Nota bene: Own, or individual, quantities play the roles of the weights of quantiset elements. In particular, q(λ) is the own, or individual, quantity (former hyperquantity) [2-5] as a weight of relation R_λwith index λ in a quantiset

_q(λ)R_λ{_φ∈Φ _r(φ)f_φ[_ω∈Ω _s(ω)z_ω]} (λ∈Λ).

Replace all the unknown quantisubsystems, or simply unknown variables, or unknowns (in the above particular case, unknown quantifunctions), with their possible ”values” which are known quantisubsystems (in the above particular case, some known quantifunctions). Then a general problem which is a quantisystem [2-5] (former hypersystem) P including unknown quantisubsystems becomes the corresponding known quantisystem without any unknowns. To conserve the quantisystem form, let us use the same designations for these known quantisystem and quantisubsystems, too. This known quantisystem can be further estimated qualitatively and (or) quantitatively discretely (e.g., either true or false) or continuously (e.g., via absolute and relative errors [1], unierrors, reserves, reliabilities, and risks [2-5]). If a quantisystem of pseudosolutions to a general problem transforms this problem into a known true quantisystem, then this pseudosolutions quantisystem (or simply a pseudosolution by obviously using the system meta-level) is a solutions quantisystem (or simply a solution by obviously using the system meta-level) to this general problem.

In quantitative mathematical problems, namely equations and inequations are the most typical relations.

Further general problem type and setting theory (GPTST) in fundamental science on general problem essence [5] naturally defines a general pure equation problem and a general pure inequation problem.

General Pure Equation Problem

General problem type and setting theory (GPTST) in fundamental science on general problem essence [5] defines a general quantitative mathematical pure equation problem, or simply a general pure equation problem, to be a general problem that can be represented in a form in which all relations are namely equality relations.

In the left-hand sides of all the equations in a general pure equation problem, gather all the expressions available namely in the initial forms of these equations without any further transformations. The unique natural exception is changing the signs of the expressions by moving them to the other sides of the same equations. Then a general pure equation problem can be represented, in particular, as a quantiset

_q(λ){L_λ{_φ∈Φ _r(φ)f_φ[_ω∈Ω _s(ω)z_ω]} = 0} (λ∈Λ)

of indexed known quantiequations (with their own, or individual, quantities q(λ)) [2-5] in a form of vanishing operators L_λ over indexed unknown quantifunctions (dependent variables), or simply unknowns (with their own, or individual, quantities r(φ)), _r(φ)f_φ , of indexed independent known quantivariables _s(ω)z_ω (with their own, or individual, quantities s(ω), all of them belonging to their possibly individual vector spaces, in their initial forms without any further transformations

where

L_λ is a known operator with index λ from an index set Λ ;

f_φ is an unknown function (dependent variable) with index φ from an index set Φ ;

z_ω is a known independent variable with index ω from an index set Ω ;

[_ω∈Ω _s(ω)z_ω]

is a quantiset of indexed quantielements _s(ω)z_ω .

_q(λ){L_λ{_φ∈Φ _r(φ)f_φ[_ω∈Ω _s(ω)z_ω]} = 0} (λ∈Λ).

When replacing all the unknowns (unknown functions) with their possible ”values” (some known functions), the above quantiset of quantiequations is transformed into the corresponding quantiset of formal functional quantiequalities. To conserve the quantiset form, let us use for these known functions the same designations f_φ .

General Pure Inequation Problem

General problem type and setting theory (GPTST) in fundamental science on general problem essence [5] defines a general quantitative mathematical pure inequation problem, or simply a general pure inequation problem, to be a general problem that can be represented in a form in which all relations are namely inequality relations.

In the left-hand sides of all the inequations in a general pure inequation problem, gather all the expressions available namely in the initial forms of these inequations without any further transformations. The unique natural exception is changing the signs of the expressions by moving them to the other sides of the same inequations. Then a general pure inequation problem can be represented, in particular, as a quantiset

_q(λ){L_λ{_φ∈Φ _r(φ)f_φ[_ω∈Ω _s(ω)z_ω]} R_λ 0} (λ∈Λ)

of indexed known inequality quantirelations (with their own, or individual, quantities q(λ)) [2-5] in a form of the comparison with zero of the values of operators L_λ over indexed unknown quantifunctions (dependent variables), or simply unknowns (with their own, or individual, quantities r(φ)), _r(φ)f_φ , of indexed independent known quantivariables _s(ω)z_ω (with their own, or individual, quantities s(ω), all of them belonging to their possibly individual vector spaces, in their initial forms without any further transformations

where

L_λ is a known operator with index λ from an index set Λ ;

R_λ is an inequality relation (e.g., ≈ , ∼ , ≠ , < , > , ≤ , ≥) with index λ from an index set Λ ;

f_φ is an unknown function (dependent variable) with index φ from an index set Φ ;

z_ω is a known independent variable with index ω from an index set Ω ;

[_ω∈Ω _s(ω)z_ω]

is a quantiset of indexed quantielements _s(ω)z_ω .

_q(λ){L_λ{_φ∈Φ _r(φ)f_φ[_ω∈Ω _s(ω)z_ω]} R_λ 0} (λ∈Λ).

When replacing all the unknowns (unknown functions) with their possible ”values” (some known functions), the above quantiset of inequations is transformed into the corresponding quantiset of formal functional inequalities. To conserve the quantiset form, let us use for these known functions the same designations f_φ .

By using unstrict inequality relations such as ≈ , ∼ , ≤ , ≥ , etc. only, a general pure inequations problem clearly further generalizes a general pure equations problem.

General Problem Pseudosolution

General problem pseudosolution theory (GPPST) in fundamental science of general problem pseudosolution defines both a pseudosolution to a general problem and arts (particular cases) of a pseudosolution which are conditional pseudosolutions.

Further we need some useful definitions and agreements [2-5].

A pseudosolution to a general problem is an arbitrary quantisystem of such values of all the corresponding variables that, after replacing each variable with its value, the general problem becomes a determinable (e.g., true or false) known quantisystem. In the above particular case, this is a known quantiset of relations containing known elements only, and each of its relations becomes determinable (e.g., true or false).

A (precise) solution to a general problem is an arbitrary quantisystem of such values of all the corresponding variables that, after replacing each variable with its value, the general problem becomes a true known quantisystem. In the above particular case, this is a known quantiset of relations containing known elements only, and each of its relations becomes true.

A quasisolution to a general problem by a specific realization of a certain method or theory is a pseudosolution (to this general problem) which has the least unierror and/or the greatest reserve (by this realization of this method or theory) among all the pseudosolutions to this general problem.

Nota bene: A quasisolution is not necessarily a solution, which is especially important in contradictory general problems that have no solutions in principle but can possess quasisolutions.

A supersolution to a general problem by a specific realization of a certain method or theory is a solution (to this general problem) which has the greatest reserve (by this realization of this method or theory) among all the solutions to this general problem.

Nota bene: A supersolution a general problem not necessarily coincides with its quasisolution because the set of the solutions is a subset of the set of the pseudosolutions. If the both exist, then the quasisolution (which is not necessarily a solution) has a not less reserve in comparison with the supersolution. If in the last comparison, namely the strict inequality holds, then the quasisolution is certainly no solution.

An antisolution to a general problem by a specific realization of a certain method or theory is a pseudosolution (to this general problem) which has the greatest unierror and/or the least reserve (by this realization of this method or theory) among all the pseudosolutions to this general problem.

Notata bene:

1. Quasisolutions and supersolutions, as well as antisolutions, not necessarily exist because a set of unierrors or reserves not necessarily contains its greatest lower bound and its least upper one, respectively.

2. The concepts of conditional pseudosolutions (in particular, quasisolutions, supersolutions, and antisolutions) are relative depending not only on the corresponding condition, criterion, method, or theory, but also on the precise setting of a general problem. For example, a quasisolution to a contradictory general problem is namely a quasisolution if the precise setting of a general problem is precisely satisfying all the contradictory conditions of a given general problem. But the same quasisolution becomes a precise solution to the same contradictory general problem by its other setting when general problem contradictoriness measure minimization (instead of precisely satisfying all the contradictory conditions of a given general problem) is required (desired). All the more, an antisolution to a contradictory general problem is namely an antisolution if the precise setting of a general problem is precisely satisfying all the contradictory conditions of a given general problem. But the same antisolution becomes a precise solution to the same contradictory general problem by its other setting when general problem contradictoriness measure maximization (instead of precisely satisfying all the contradictory conditions of a given general problem) is required (desired).

General Approximation Problem

Let us now consider a general approximation problem.

Let

Z ⊆ X × Y

be any given subset of the direct product of two sets X and Y and have a projection Z/X on X consisting of all x ∈ X really represented in Z , i.e., of all such x that for each of them there is a y ∈ Y such that

(x, y) ∈ Z .

Let further

{ y = F(x) }

where

x ∈ X

y ∈ Y

be a certain class of functions defined on X with range in Y .

Then the graph of such a function is a curve in X × Y .

The problem consists in finding (in class { y = F(x) }) functions with graphs nearest to Z in a certain reasonable sense.

To exactly fit this with a specific function

y = F(x),

the set Z ⊆ X × Y has to be included in the graph of this function:

Z ⊆ { (x, F(x)) | x ∈ X },

or, equivalently,

F(x) = y

for each

x ∈ Z/X .

But this inclusion (or equality) does not necessarily hold in the general case. Then it seems to be reasonable to estimate the error

E( F(x) =? y | x ∈ Z/X )

of the formal equality (true or false)

F(x) =? y

on this set Z/X via a certain error function E defined at least on Z/X .

To suitably construct such an error function, it seems to be reasonable to first consider two stages of its building:

1) defining local error functions to estimate errors at separate points x ;

2) defining global error functions using the values of local error functions to estimate errors on the whole set Z/X .

Possibly the simplest and most straightforward approach includes the following steps:

1) defining on Y × Y certain nonnegative functions r_yy’(y, y’) generally individual for different y , y’ and, e.g., similar to a distance [1] between any two elements y, y’ of Y (but not necessarily with holding the distance axioms [1]),

2) defining certain nonnegative functions R_x(r(F(x), y)) generally individual for different x ,

3) summing (possibly including integrating) their values on Z/X , and

4) using this sum (possibly including integrals) as a nearness measure.

General Problem Iteration

Now suppose that for each quantisystem of solutions to this general problem, it is possible to explicitly represent each unknown function f_φ as an operator L_φ of all the unknown (desired) functions (and, naturally, possibly, of any known functions):

f_φ[_ω∈Ω z_ω] = L_φ[_φ∈Φ f_φ[_ω∈Ω z_ω]].

Expressions in the right-hand parts of such representations provide themselves no solutions to this general problem because those expressions contain unknown (desired) functions. But such representations can make it possible to approximately determine solutions to this general problem via usual iteration approach to solving a general problem.

To begin with, consider any source (initial pseudosolution)

[_φ∈Φ ¹f_φ[_ω∈Ω z_ω]]

and rationally transform the above representations to forms

ⁱ⁺¹f_φ[_ω∈Ω z_ω] = L_φ[_φ∈Φ ⁱf_φ[_ω∈Ω z_ω]]

expressing any i+1st (i ∈ N⁺ = {1, 2, ...}) iteration approximation (which is also a pseudosolution)

[_φ∈Φ ⁱ⁺¹f_φ[_ω∈Ω z_ω]]

to a solution

[_φ∈Φ f_φ[_ω∈Ω z_ω]]

to a general problem via the preceding, ith iteration approximation (which is also a pseudosolution)

[_φ∈Φ ⁱf_φ[_ω∈Ω z_ω]].

Another usual iteration approach to solving a general problem is still more general. Suppose that we have a representation

ⁱ⁺¹f_φ[_ω∈Ω z_ω] = ⁱ⁺¹M_φ{^i+1,
1L_φ[_φ∈Φ ¹f_φ[_ω∈Ω z_ω]], ^i+1,
2L_φ[_φ∈Φ ²f_φ[_ω∈Ω z_ω]], ... , ^i+1,
iL_φ[_φ∈Φ ⁱf_φ[_ω∈Ω z_ω]]}

of any i+1st iteration

[_φ∈Φ ⁱ⁺¹f_φ[_ω∈Ω z_ω]]

to a solution

[_φ∈Φ f_φ[_ω∈Ω z_ω]]

to a general problem via all the preceding iteration approximations

[_φ∈Φ ¹f_φ[_ω∈Ω z_ω]], [_φ∈Φ ²f_φ[_ω∈Ω z_ω]], ... , [_φ∈Φ ⁱf_φ[_ω∈Ω z_ω]]

with using any known operators

ⁱ⁺¹M_φ ; ^{i+1, 1}L_φ , ^{i+1, 2}L_φ , ... , ^{i+1, i}L_φ

whose upper index i+1 is the present iteration number whereas the second upper index 1, 2, ... , i of each operator L_φ is the corresponding number of one of the previous iterations.

Nota bene: The first upper index i+1 of each operator L_φ is necessary because each of these operators can vary by representing different iterations.

If sequence

[_φ∈Φ ¹f_φ[_ω∈Ω z_ω]], [_φ∈Φ ²f_φ[_ω∈Ω z_ω]], ... , [_φ∈Φ ⁱf_φ[_ω∈Ω z_ω]], ...

converges, then its limit

[_φ∈Φ f_φ[_ω∈Ω z_ω]]

is a solution to a general problem whereas iteration approximations with sufficiently great numbers providing sufficient nearness of these approximations to the precise solution could be suitable enough. To estimate the difference between them, use its unierrors, reserves, reliabilities, and risks [2-5] rather than absolute and relative errors [1] having [2-5] many principal shortcomings. Also use theories and methods of the system of fundamental sciences on general problems [2-5] rather than the least square method (LSM) [1] by Legendre and Gauss which is the unique well-known method applicable to contradictory (e.g. overdetermined) problems. Overmathematics and the system of fundamental sciences on general problems [2-5] have discovered many principal shortcomings of this method, by methods of finite elements, points, etc. Additionally consider its simplest approach which is typical. Minimizing the sum of the squared differences of the alone preselected coordinates (e.g., ordinates in a two-dimensional problem) of the graph of the desired approximation function and of everyone among the given functions depends on this preselection, ignores the remaining coordinates, and provides no coordinate system rotation invariance and hence no objective sense of the result.

In some cases, the above convergence can be slow enough and requires computer aided calculation to provide its practical real-time feasibility. Pure programming languages without checking each operation and estimating its result could be inadequate and not very suitable whereas stopping calculation after each operation strongly reduces any advantages of computer aided calculation by this approach. Then, by a finite number of unknowns, table calculation software such as Excel in Microsoft Office or OpenOffice.org Calc seems to be much more suitable.

Single-source iteration theory (SSIT) in fundamental science on general problem essence provides a lot of methods and know-how with rationally compactly placing all the calculations of each iteration approximation via the previous iteration approximation in a certain rectangle which should be as small as possible but can be relatively great by extensive calculation. The intention is to obtain each next iteration approximation at once via copying such a rectangle and pasting this copy into the adjacent (from the right) rectangle of the same width (number of columns) and height (number of rows) with automatically performing all the necessary calculations. Name such a rectangle a calculation rectangle. Selecting namely the right direction provides unrestrictedly calculating any number of iteration approximations. Such placing seems to be always possible but its practical real-time finding needs great creativity, phantasy, imaginativeness, inventiveness, intuition, and ingenuity. This is not only science but also art. This needs creativity not only of a mathematician but also of an artist or architect. In the successful case, such computer aided calculation provides not only practical real-time feasibility, acceleration, and performance but also much inspiration, enjoyment, enthusiasm, pleasure, and happiness. Human emotions are especially important for such extensive calculation because they are performed by a human and for a human.

Single-source iteration theory (SSIT) includes further improvements, too. It is very suitable to place (if possible) all the parameters of each iteration approximation in a finite column (by one-parameter unknowns) or, more generally, a finite number of adjacent finite columns building a rectangle (by multiple-parameter unknowns). Name such a rectangle a representation rectangle. Naturally, its numbers of columns and rows can differ from those of a calculation rectangle. For example, by solving a finite overdetermined quantiset [2-5] of n (n > m; m , n ∈ N⁺ = {1, 2, ...}) linear equations

_q(j)(Σ_k=1^m a_kjx_k = c_j) (j = 1, 2, ... , n)

with their own quantities q(j) and m unknowns x_k (k = 1, 2, ... , m) by any given real numbers q(j) > 0, a_j , b_j , and c_j, a calculation rectangle could have n (the number of different equations) rows whereas a representation rectangle could have m (the number of unknowns) rows. But it is always possible to mentally add some blank rows if necessary to attach a representation rectangle to a calculation rectangle to obtain a unified calculation and representation rectangle. It is also possible to add to it the representation rectangles for previous iteration approximations to provide their comparison, e.g., estimating the difference

[_φ∈Φ ⁱ⁺¹f_φ[_ω∈Ω z_ω] - ⁱf_φ[_ω∈Ω z_ω]]

of the i+1st and ith iteration approximations. For further comparability, extra create a sufficient number of blank rows to attach all the representation rectangles to one another.

Single-source iteration theory (SSIT) in fundamental science on general problem iteration is very efficient by solving many urgent general (including contradictory) problems.

References

[1] Encyclopaedia of Mathematics. Ed. M. Hazewinkel. Volumes 1 to 10. Kluwer Academic Publ., Dordrecht, 1988-1994

[2] Lev Gelimson. Elastic Mathematics. General Strength Theory. The ”Collegium” International Academy of Sciences Publishers, Munich, 2004

[3] Lev Gelimson. Providing Helicopter Fatigue Strength: Flight Conditions. In: Structural Integrity of Advanced Aircraft and Life Extension for Current Fleets – Lessons Learned in 50 Years After the Comet Accidents, Proceedings of the 23rd ICAF Symposium, Dalle Donne, C. (Ed.), 2005, Hamburg, Vol. II, 405-416

[4] Lev Gelimson. Overmathematics: Fundamental Principles, Theories, Methods, and Laws of Science. The ”Collegium” All World Academy of Sciences Publishers, Munich, 2010

[5] Lev Gelimson. General Problem Fundamental Sciences System. The ”Collegium” All World Academy of Sciences Publishers, Munich, 2011