Distance Power Theories in Fundamental Science on General Problem Distance

by

© 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), 56

The least square method (LSM) [1] by Legendre and Gauss only usually applies to contradictory (e.g. overdetermined) problems. Overmathematics [2-4] and fundamental science of solving general problems [5] have discovered many principal shortcomings [2-6] of this method, by methods of finite elements, points, etc. Additionally, by more than 4 data points, the second power can paradoxically give smaller errors of better approximations and can be increased due to the least biquadratic method (LBQM) in fundamental science on general problem solving methods [5].

In fundamental science on general problem distance [5], distance power theories (DPT) (with any power exponent t > 1) are valid by coordinate system rotation invariance. Show the essence of these theories in the simplest but most typical case providing linear solving with giving the unique best quasisolution [2-5] to a finite overdetermined quantiset [2-5] of n (n > m; m , n ∈ N⁺ = {1, 2, ...}) linear equations

_q(j)(Σ_i=1^m a'_ijx_i + c'_j = 0) (j = 1, 2, ... , n) (L'_j)

with their own positive number quantities q(i), m pure number unknown variables x_i (i = 1, 2, ... , m), and any given real numbers a'_ij and c'_j in the Cartesian m-dimensional "space" [1].

To begin with, apply distance transformation theory (DTT) in fundamental science of general problem transformation [5] to this quantiset of linear equations.

The distance between the jth m-1-dimensional "plane" (1) in an m-dimensional space of points

[_i=1^m x_i] = (x₁ , x₂ , ... , x_m)

and namely this point is

d_j = |Σ_i=1^m a'_ijx_i + c'_j|/(Σ_i=1^m a'_ij²)^1/2 .

Hence preliminarily equivalently transform each linear equation (L'_j) via dividing all its factors a'_ij and c'_j namely by the denominator of the last formula

_q(j)[Σ_i=1^m a'_ij / (Σ_i=1^m a'_ij²)^1/2 x_i + c'_j / (Σ_i=1^m a'_ij²)^1/2 = 0] (j = 1, 2, ... , n) (L_j),

naturally introduce new equation factors (i = 1, 2, ... , m; j = 1, 2, ... , n)

a_ij = a'_ij / (Σ_i=1^m a'_ij²)^1/2 ,

c_ij = c'_ij / (Σ_i=1^m a'_ij²)^1/2 ,

and represent the obtained quantiset of normalized linear equations (L_j) as

_q(j)(Σ_i=1^m a_ijx_i + c_j = 0) (j = 1, 2, ... , n) (L_j).

Nota bene: This normalization is independent of specific pseudosolutions [2-5] to this set of equations and provides its universalization, i.e. quasisolutions [2-5] independency of multiplying equations by any different real nonzero factors.

Now apply power theories (PT) in fundamental science of solving general problems [5] to this quantiset of normalized linear equations (L_j).

Nota bene: This equations set universalization also holds for the absolute errors [1] of these already normalized linear equations (L_j) (j = 1, 2, ... , n).

To begin with, consider power exponent t > 1 to be an even natural number 2p (p ∈ N⁺ = {1, 2, ...}.

Minimize the sum weighted with their own positive number quantities q(i)

^2pS(x₁ , x₂ , ... , x_m) = Σ_j=1ⁿ q(j)e_j^2p = Σ_j=1ⁿ q(j)(Σ_i=1^m a_ijx_i + c_j)^2p

of the 2pth powers of the absolute errors [1]

e_j = |Σ_i=1^m a_ijx_i + c_j|

of these already normalized linear equations (L_j) of n m-1-dimensional "planes" by substituting the coordinates of any point

[_i=1^m x_i] = (x₁ , x₂ , ... , x_m).

of the m-dimensional space.

Nota bene: This sum equals the weighted sum of the 2pth powers of the distances between this point and everyone of the n "planes" defined and determined by the given linear equations (L'_j).

This nonnegative function ^2pS(x₁ , x₂ , ... , x_m) everywhere differentiable has and takes its minimum at a point with vanishing all the first order derivatives

^2pS'_x(i') = Σ_j=1ⁿ 2pq(j)a_i'j(Σ_i=1^m a_ijx_i + c_j)^2p-1 = 0 (i' = 1, 2, ... , m)

of this function by every

x(i') = x_i' (i' = 1, 2, ... , m),

which gives the following determined set

Σ_j=1ⁿ q(j)a_i'j(Σ_i=1^m a_ijx_i + c_j)^2p-1 = 0 (i' = 1, 2, ... , m)

of m equations with m unknowns x_i' to determine all the possibly extremum points and, finally, the desired minimum point.

This set of m equations (whose power is 2p - 1) is suitable for testing pseudosolutions but in any nontrivial case (even by p = 2) NOT for finding quasisolutions [2-5]. Therefore, transform this set via iteration preparation to obtain a set of m namely linear equations using power expansion. We may rationally represent:

(2p - 1)Σ_j=1ⁿ q(j)a_i'j Σ_i=1^m a_ijc_j^2p-2 x_i = (2p - 1)Σ_j=1ⁿ a_i'j Σ_i=1^m q(j)a_ijc_j^2p-2 x_i - Σ_j=1ⁿ q(j)a_i'j(Σ_i=1^m a_ijx_i + c_j)^2p-1 (i' = 1, 2, ... , m),

Σ_j=1ⁿ q(j)a_i'j Σ_i=1^m a_ijc_j^2p-2 x_i = Σ_j=1ⁿ q(j)a_i'j Σ_i=1^m a_ijc_j^2p-2 x_i - Σ_j=1ⁿ q(j)a_i'j(Σ_i=1^m a_ijx_i + c_j)^2p-1/ (2p - 1) (i' = 1, 2, ... , m),

and finally desired set

Σ_j=1ⁿ q(j)a_i'j Σ_i=1^m a_ijc_j^{2p-2 k+1}x_i = Σ_j=1ⁿ q(j)a_i'j Σ_i=1^m a_ijc_j^{2p-2 k}x_i - Σ_j=1ⁿ q(j)a_i'j(Σ_i=1^m a_ij ^kx_i + c_j)^2p-1/ (2p - 1) (i' = 1, 2, ... , m)

of m namely linear equations expressing any k+1st approximation iteration

[_i=1^{m k+1}x_i] = (^k+1x₁ , ^k+1x₂ , ... , ^k+1x_m) (k ∈ N⁺ = {1, 2, ...})

to a quasisolution via kth approximation iteration

[_i=1^{m k}x_i] = (^kx₁ , ^kx₂ , ... , ^kx_m).

In particular, by m = 2, replacing x₁ with x , x₂ with y , a_1j with a_j , and a_2j with b_j , we obtain:

e_j = |a_jx + b_jy + c_j| ,

^2pS = Σ_j=1ⁿ q(j)e_j^2p = Σ_j=1ⁿ q(j)(a_jx + b_jy + c_j)^2p,

^2pS'_x = Σ_j=1ⁿ 2pq(j)a_j(a_jx + b_jy + c_j)^2p-1 = 0,

^2pS'_y = Σ_j=1ⁿ 2pq(j)b_j(a_jx + b_jy + c_j)^2p-1 = 0;

Σ_j=1ⁿ q(j)a_j(a_jx + b_jy + c_j)^2p-1 = 0,

Σ_j=1ⁿ q(j)b_j(a_jx + b_jy + c_j)^2p-1 = 0;

(2p - 1)[Σ_j=1ⁿ q(j)a_j²c_j^2p-2x + Σ_j=1ⁿ q(j)a_jb_jc_j^2p-2y] = (2p - 1)[Σ_j=1ⁿ q(j)a_j²c_j^2p-2x + Σ_j=1ⁿ q(j)a_jb_jc_j^2p-2y] - Σ_j=1ⁿ q(j)a_j(a_jx + b_jy + c_j)^2p-1,

(2p - 1)[Σ_j=1ⁿ q(j)a_jb_jc_j^2p-2x + Σ_j=1ⁿ q(j)b_j²c_j^2p-2y] = (2p - 1)[Σ_j=1ⁿ q(j)a_jb_jc_j^2p-2x + Σ_j=1ⁿ q(j)b_j²c_j^2p-2y] - Σ_j=1ⁿ q(j)b_j(a_jx + b_jy + c_j)^2p-1;

Σ_j=1ⁿ q(j)a_j²c_j^2p-2x + Σ_j=1ⁿ q(j)a_jb_jc_j^2p-2y = Σ_j=1ⁿ q(j)a_j²c_j^2p-2x + Σ_j=1ⁿ q(j)a_jb_jc_j^2p-2y - Σ_j=1ⁿ q(j)a_j(a_jx + b_jy + c_j)^2p-1/ (2p - 1),

Σ_j=1ⁿ q(j)a_jb_jc_j^2p-2x + Σ_j=1ⁿ q(j)b_j²c_j^2p-2y = Σ_j=1ⁿ q(j)a_jb_jc_j^2p-2x + Σ_j=1ⁿ q(j)b_j²c_j^2p-2y - Σ_j=1ⁿ q(j)b_j(a_jx + b_jy + c_j)^2p-1/ (2p - 1);

Σ_j=1ⁿ q(j)a_j²c_j^{2p-2 k+1}x + Σ_j=1ⁿ q(j)a_jb_jc_j^{2p-2 k+1}y = Σ_j=1ⁿ q(j)a_j²c_j^{2p-2 k}x + Σ_j=1ⁿ q(j)a_jb_jc_j^{2p-2 k}y - Σ_j=1ⁿ q(j)a_j(a_j^kx + b_j^ky + c_j)^2p-1/ (2p - 1),

Σ_j=1ⁿ q(j)a_jb_jc_j^{2p-2 k+1}x + Σ_j=1ⁿ q(j)b_j²c_j^{2p-2 k+1}y = Σ_j=1ⁿ q(j)a_jb_jc_j^{2p-2 k}x + Σ_j=1ⁿ q(j)b_j²c_j^{2p-2 k}y - Σ_j=1ⁿ q(j)b_j(a_j^kx + b_j^ky + c_j)^2p-1/ (2p - 1).

The obtained desired set of m = 2 namely linear equations expresses any k+1st approximation iteration

(^k+1x , ^k+1y) (k ∈ N⁺ = {1, 2, ...})

to a quasisolution via kth approximation iteration

(^kx , ^ky).

One of many reasonable possibilities to take first iteration (x₁ , y₁) is using distance quadrat theory (DQT) [2-5] (p = 1) giving here

Σ_j=1ⁿ q(j)a_j(a_jx + b_jy + c_j) = 0,

Σ_j=1ⁿ q(j)b_j(a_jx + b_jy + c_j) = 0;

Σ_j=1ⁿ q(j)a_j² x + Σ_j=1ⁿ q(j)a_jb_j y + Σ_j=1ⁿ q(j)a_jc_j = 0,

Σ_j=1ⁿ q(j)a_jb_j x + Σ_j=1ⁿ q(j)b_j² y+ Σ_j=1ⁿ q(j)b_jc_j = 0;

Σ_j=1ⁿ q(j)a_j² x + Σ_j=1ⁿ q(j)a_jb_j y = - Σ_j=1ⁿ q(j)a_jc_j ,

Σ_j=1ⁿ q(j)a_jb_j x + Σ_j=1ⁿ q(j)b_j² y= - Σ_j=1ⁿ q(j)b_jc_j ;

x₁ = [Σ_j=1ⁿ q(j)a_jb_j Σ_j=1ⁿ q(j)b_jc_j - Σ_j=1ⁿ q(j)a_jc_j Σ_j=1ⁿ q(j)b_j²]/{Σ_j=1ⁿ q(j)a_j² Σ_j=1ⁿ q(j)b_j² - [Σ_j=1ⁿ q(j)a_jb_j]²},

y₁ = [Σ_j=1ⁿ q(j)a_jb_j Σ_j=1ⁿ q(j)a_jc_j - Σ_j=1ⁿ q(j)a_j² Σ_j=1ⁿ q(j)b_jc_j]/{Σ_j=1ⁿ q(j)a_j² Σ_j=1ⁿ q(j)b_j² - [Σ_j=1ⁿ q(j)a_jb_j]²}.

Compare applying distance biquadrat theory (DBQT) vs. the least square method (LSM) [1], the least biquadratic method (LBQM) [5], distance quadrat theory (DQT) [2-5], linear unierror quadrat theory (LEQT), and square root unierror quadrat theory (SREQT) to test equation set

29x + 21y = 50,

50x - 17y = 33,

x + 2y = 7,

2x - 3y = 0,

see Figure 1 and Table 1:

Figure 1

Table 1

Notata bene:

1. The least square method (LSM) [1] practically ignores the last two equations with smaller factors (unlike least squared distance theory and both the linear unierror method and the quadratic unierroro method in least squared reserve theory).

2. Both linear unierror quadrat theory (LEQT) and square root unierror quadrat theory (SREQT) give relatively near results. Therefore, in Figure 1, we have shown the results obtained by linear unierror quadrat theory (LEQT) only.

3. Comparing the results of applying the least biquadratic method (LBQM) [5] vs. the least square method (LSM) [1] also to other test equation sets shows that increasing the power from 2 to 4 provides very substantially improving the method sensitivity. But it is not sufficient because, like the least square method (LSM), the least biquadratic method (LBQM) is also based on the absolute error [1] which is not invariant by equivalent transformations of a problem and hence has no objective sense.

4. To further improve the least biquadratic method (LBQM) with using its ideas, there are at least two ways:

4.1) further increasing the power from 4 to 6, 8, etc. (excluding odd integer powers provides avoiding absolute values and hence simplifying analytic expressions) which alone leads to much more complicated formulae and relatively slowly improving the method sensitivity and its results;

4.2) replacing the absolute errors [1] with distances and unierrors which both are invariant by equivalent transformations of a problem and hence have objective sense.

5. Distance quadrat theory (DQT) [2-5] and distance biquadrat theory (DBQT) replaces the absolute errors [1] with distances due to preliminary equations set universalization via its normalization.

6. Comparing the results of applying distance biquadrat theory (DBQT) vs. distance quadrat theory (DQT) also to other test equation sets shows that increasing the power from 2 to 4 provides improving theory sensitivity.

Distance biquadrat theory (DBQT) providing simple explicit quasisolutions to even contradictory problems is very efficient by solving many urgent problems.

Acknowledgements to Anatolij Gelimson for our constructive discussions on coordinate system transformation invariances and his very useful remarks.

References

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[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, 2010

[6] Lev Gelimson. Corrections and Generalizations of the Least Square Method. In: Review of Aeronautical Fatigue Investigations in Germany during the Period May 2007 to April 2009, Ed. Dr. Claudio Dalle Donne, Pascal Vermeer, CTO/IW/MS-2009-076 Technical Report, International Committee on Aeronautical Fatigue, ICAF 2009, EADS Innovation Works Germany, 2009, 59-60