Power Theories in Fundamental Science of Solving General Problems

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), 55

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 of solving general problems.

Show the essence of power theories (PT) in fundamental science of solving general problems [5] 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].

Minimize the sum

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

of the 2pth powers (p ∈ N⁺ = {1, 2, ...} is a power exponent) of the absolute errors [1] (or of the 2pth powers of the distances between the above point and everyone of the n "planes" (L_j))

e_j = |Σ_i=1^m a_ijx_i + c_j|

of 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.

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

Σ_j=1ⁿ a_i'j Σ_i=1^m a_ijc_j^2p-2 x_i = Σ_j=1ⁿ a_i'j Σ_i=1^m a_ijc_j^2p-2 x_i - Σ_j=1ⁿ 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 a_i'j Σ_i=1^m a_ijc_j^{2p-2 k+1}x_i = Σ_j=1ⁿ a_i'j Σ_i=1^m a_ijc_j^{2p-2 k}x_i - Σ_j=1ⁿ 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:

a_jx + b_jy + c_j = 0 (j = 1, 2, ... , n) (L_j),

e_j = |a_jx + b_jy + c_j|,

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

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

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

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

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

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

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

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

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

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

Σ_j=1ⁿ a_jb_jc_j^{2p-2 k+1}x + Σ_j=1ⁿ b_j²c_j^{2p-2 k+1}y = Σ_j=1ⁿ a_jb_jc_j^{2p-2 k}x + Σ_j=1ⁿ b_j²c_j^{2p-2 k}y - Σ_j=1ⁿ 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 the least square method (LSM) [1] giving here

Σ_j=1ⁿ a_j(a_jx + b_jy + c_j) = 0,

Σ_j=1ⁿ b_j(a_jx + b_jy + c_j) = 0;

Σ_j=1ⁿ a_j² x + Σ_j=1ⁿ a_jb_j y + Σ_j=1ⁿ a_jc_j = 0,

Σ_j=1ⁿ a_jb_j x + Σ_j=1ⁿ b_j² y+ Σ_j=1ⁿ b_jc_j = 0;

Σ_j=1ⁿ a_j² x + Σ_j=1ⁿ a_jb_j y = - Σ_j=1ⁿ a_jc_j ,

Σ_j=1ⁿ a_jb_j x + Σ_j=1ⁿ b_j² y= - Σ_j=1ⁿ b_jc_j ;

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

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

Notata bene:

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

2. To further improve power theories (PT) with using their ideas, there are at least two ways:

2.1) 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 theory sensitivity and its results;

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

Power theories (PT) [5] are applicable to even contradictory problems and very efficient by solving some of them, have advantages as compared to the least square method (LSM) [1] and biquadrat theory (BQT) [5], and open ways for creating much more universal theories in fundamental science of solving general problems [5].

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

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, 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