Unierror Quadrat Theories in Fundamental Science on General Problem Reserve

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), 51

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 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.

Show the essence of unierror quadrat theories (EQT) in fundamental science on general problem reserve [5] 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) (L_j)

with m unknowns x_k (k = 1, 2, ... , m) and any given real numbers a_kj and c_j .

For each inexact object I, its unierror [2-5]

A(I) > 0

and reserve [2-5]

R(I) = - A(I).

In particular, this holds for any pseudosolution [2-5] to a contradictory problem, e.g., to the above overdetermined set of linear equations (L_j). Maximizing such a negative reserve is equivalent to minimizing such a positive unierror, which corresponds to the principle of tolerable simplicity [2-5] and seems to be more suitable.

Linear unierror quadrat theory (LEQT) is based on linear unierror [2-5]

¹A_j = |Σ_k=1^m a_kjx_k - c_j|/(Σ_k=1^m |a_kjx_k| + |c_j|)

of linear equation (L_j) by any pseudosolution

[_k=1^m x_k] = (x₁ , x₂ , ... , x_m)

and the following iteration algorithm:

1) take any initial, e.g., 1st iteration (which is a pseudosolution)

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

to the desired quasisolution to the above overdetermined set of linear equations (L_j), e.g., via the least square method (LSM) [1];

2) for any already known ith iteration

[_k=1^{m i}x_k] = (ⁱx₁ , ⁱx₂ , ... , ⁱx_m)

(i ∈ N⁺ = {1, 2, ...}) to the desired quasisolution to the above overdetermined set of linear equations (L_j), provide the elementary step of iteration due to:

2.1) equivalently transforming linear equation (L_j) via dividing all its factors a_kj and c_j namely by the denominator of linear unierror ¹A_j of this linear equation (L_j) by this ith iteration and replacing x_k with its desired i+1st iteration:

Σ_k=1^m a_kj / (Σ_k=1^m |a_kj ⁱx_k| + |c_j|) ⁱ⁺¹x_k = c_j / (Σ_k=1^m |a_kj ⁱx_k| + |c_j|) (j = 1, 2, ... , n) (ⁱL_j);

2.2) solving this overdetermined set of linear equations (ⁱL_j) via the least square method (LSM) [1] to obtain the desired next, i+1st iteration

[_k=1^{m i+1}x_k] = (ⁱ⁺¹x₁ , ⁱ⁺¹x₂ , ... , ⁱ⁺¹x_m);

3) if sequence

[_k=1^{m i}x_k] = (ⁱx₁ , ⁱx₂ , ... , ⁱx_m) (i ∈ N⁺ = {1, 2, ...})

has a limit

[_k=1^m X_k] = (X₁ , X₂ , ... , X_m),

then it is considered to be the desired quasisolution to the above overdetermined set of linear equations (L_j).

Square unierror quadrat theory (SEQT) is based on square unierror [2-5]

²A_j = |Σ_k=1^m a_kjx_k - c_j|/[(m + 1)(Σ_k=1^m a_kj²x_k² + c_j²)]^1/2

of linear equation (L_j) by any pseudosolution

[_k=1^m x_k] = (x₁ , x₂ , ... , x_m)

and the following iteration algorithm:

1) take any initial, e.g., 1st iteration (which is a pseudosolution)

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

to the desired quasisolution to the above overdetermined set of linear equations (L_j), e.g., via the least square method (LSM) [1];

2) for any already known ith iteration

[_k=1^{m i}x_k] = (ⁱx₁ , ⁱx₂ , ... , ⁱx_m)

(i ∈ N⁺ = {1, 2, ...}) to the desired quasisolution to the above overdetermined set of linear equations (L_j), provide the elementary step of iteration due to:

2.1) equivalently transforming linear equation (L_j) via dividing all its factors a_kj and c_j namely by the denominator of quadratic unierror ²A_j of this linear equation (L_j) by this ith iteration and replacing x_k with its desired i+1st iteration:

Σ_k=1^m a_kj / [(m + 1)(Σ_k=1^m a_kj²x_k² + c_j²)]^{1/2 i+1}x_k = c_j / [(m + 1)(Σ_k=1^m a_kj²x_k² + c_j²)]^1/2 (j = 1, 2, ... , n) (ⁱL_j);

2.2) solving this overdetermined set of linear equations (ⁱL_j) via the least square method (LSM) [1] to obtain the desired next, i+1st iteration

[_k=1^{m i+1}x_k] = (ⁱ⁺¹x₁ , ⁱ⁺¹x₂ , ... , ⁱ⁺¹x_m);

3) if sequence

[_k=1^{m i}x_k] = (ⁱx₁ , ⁱx₂ , ... , ⁱx_m) (i ∈ N⁺ = {1, 2, ...})

has a limit

[_k=1^m X_k] = (X₁ , X₂ , ... , X_m),

then it is considered to be the desired quasisolution to the above overdetermined set of linear equations (L_j).

For example, by m = 2, replace x₁ with x , x₂ with y , a_1j with a_j , and a_2j with b_j . Then obtain

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

Linear unierror quadrat theory (LEQT) gives

¹A_j = |a_jx + b_jy - c_j|/(|a_jx| + |b_jy| + |c_j|);

²S = Σ_j=1ⁿ (a_jx + b_jy - c_j)² ;

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

²S'_y = Σ_j=1ⁿ 2b_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 ,

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

¹x = x = (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],

¹y = y = (_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];

a_j / (|a_j ⁱx| + |b_j ⁱy| + |c_j|) ⁱ⁺¹x + b_j / (|a_j ⁱx| + |b_j ⁱy| + |c_j|) ⁱ⁺¹y = c_j / (|a_j ⁱx| + |b_j ⁱy| + |c_j|) (j = 1, 2, ... , n) (ⁱL_j).

Square unierror quadrat theory (SUEQT) gives

²A_j = |a_jx + b_jy - c_j|/[3(a_j²x² + b_j²y² + c_j²)]^1/2 ;

²S = Σ_j=1ⁿ (a_jx + b_jy - c_j)² ;

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

²S'_y = Σ_j=1ⁿ 2b_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 ,

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

¹x = x = (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],

¹y = y = (_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];

a_j / [3(a_j^{2 i}x² + b_j^{2 i}y² + c_j²)]^{1/2 i+1}x + b_j / [3(a_j^{2 i}x² + b_j^{2 i}y² + c_j²)]^{1/2 i+1}y = c_j / [3(a_j^{2 i}x² + b_j^{2 i}y² + c_j²)]^1/2 (j = 1, 2, ... , n) (ⁱL_j).

Compare applying both linear unierror quadrat theory (LEQT) and square unierror quadrat theory (SEQT) with distance quadrat theory (DQT) [5] and the least square method (LSM) [1] 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

Nota bene:

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

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

Unierror quadrat theories (EQT) providing simple explicit quasisolutions (further improvable, e.g., via using unierror biquadrat theories) to even contradictory problems are 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, 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