Distance Quadrat Theory 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)

10 (2010), 1

The least square method (LSM) [1] by Legendre and Gauss only usually applies to contradictory (e.g., overdetermined) problems. Overmathematics [2-4] and the system of fundamental sciences on general problems [5] have discovered many principal shortcomings [2-6] 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 fundamental science on general problem distance [5], distance quadrat theory (DQT) is valid by coordinate system rotation invariance. Show the essence of this theory in the simplest but most typical case providing linear solving with giving the unique best quasisolution [2-5] to a finite overdetermined set of n (n > m; m , n ∈ N⁺ = {1, 2, ...}) linear equations

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

with m unknowns x_k (k = 1, 2, ... , m) and any given real numbers a_kj and c_j . The distance between the jth m-1-dimensional "plane" (1) in an m-dimensional space of points

[_k=1^m x_k] = (x₁ , x₂ , ... , x_m) (2)

and any point (2) is

d_j = |Σ_k=1^m a_kjx_k - c_j|/(Σ_k=1^m a_kj²)^1/2 .

The sum of the squares of the distances between this point (2) and everyone of the n "planes" (1) is

²S = Σ_j=1ⁿ d_j² = Σ_j=1ⁿ (Σ_k=1^m a_kjx_k - c_j)²/Σ_k=1^m a_kj² .

This nonnegative differentiable function has and takes its minimum at a point with vanishing all the first order derivatives of this function by every x_i (i = 1, 2, ... , m):

²S'_xi = Σ_j=1ⁿ 2a_ij(Σ_k=1^m a_kjx_k - c_j)/Σ_k=1^m a_kj² = 0 (i = 1, 2, ... , m),

which gives the following determined set of m equations with m unknowns x_k :

Σ_k=1^m [Σ_j=1ⁿ a_ija_kj/Σ_k=1^m a_kj²] x_k = Σ_j=1ⁿ a_ijc_j / Σ_k=1^m a_kj² (i = 1, 2, ... , m).

For example, by m = 2, replacing x₁ with x , x₂ with y , a_1j with a_j , and a_2j with b_j , we finally obtain:

d_j = |a_jx + b_jy - c_j|/(a_j² + b_j²)^1/2 ,

²S = Σ_j=1ⁿ d_j² = Σ_j=1ⁿ (a_jx + b_jy - c_j)²/(a_j² + b_j²),

²S'_x = Σ_j=1ⁿ 2a_j/(a_j² + b_j²) (a_jx + b_jy - c_j) = 0,

²S'_y = Σ_j=1ⁿ 2b_j/(a_j² + b_j²) (a_jx + b_jy - c_j) = 0,

Σ_j=1ⁿ a_j²/(a_j² + b_j²) x + Σ_j=1ⁿ a_jb_j/(a_j² + b_j²) y = Σ_j=1ⁿ a_jc_j/(a_j² + b_j²),

Σ_j=1ⁿ a_jb_j/(a_j² + b_j²) x + Σ_j=1ⁿ b_j²/(a_j² + b_j²) y = Σ_j=1ⁿ b_jc_j/(a_j² + b_j²),

x = [Σ_j=1ⁿ a_jb_j/(a_j² + b_j²) Σ_j=1ⁿ b_jc_j/(a_j² + b_j²) - Σ_j=1ⁿ a_jc_j/(a_j² + b_j²) Σ_j=1ⁿ b_j²/(a_j² + b_j²)]/

{Σ_j=1ⁿ a_j²/(a_j² + b_j²) Σ_j=1ⁿ b_j²/(a_j² + b_j²) - [Σ_j=1ⁿ a_jb_j/(a_j² + b_j²)]²},

y = [Σ_j=1ⁿ a_jb_j/(a_j² + b_j²) Σ_j=1ⁿ a_jc_j/(a_j² + b_j²) - Σ_j=1ⁿ a_j²/(a_j² + b_j²) Σ_j=1ⁿ b_jc_j/(a_j² + b_j²)]/

{Σ_j=1ⁿ a_j²/(a_j² + b_j²) Σ_j=1ⁿ b_j²/(a_j² + b_j²) - [Σ_j=1ⁿ a_jb_j/(a_j² + b_j²)]²}.

Compare applying DQT, the LSM, least normed square method (LNSM), unierror equalizing method (EEM), and direct solution method (DSM) [2-4] to test equation set

29x + 21y = 50,

50x - 17y = 33,

x + 2y = 7,

2x - 3y = 0,

see Figures 1, 2. The LSM gives x ≈ 1.0023, y ≈ 1.0075 practically ignoring the last two equations with smaller factors (unlike DQT, the EEM, DSM, and even LNSM):

Figure 1

Figure 2

Distance quadrat theory (DQT) providing simple explicit quasisolutions (further improvable, e.g., via using biquadratic distances) 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

[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

[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