Biquadrat Theory 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), 50

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 biquadrat theory (BQT) 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 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 in the Cartesian m-dimensional "space" [1].

Minimize the sum

⁴S(x₁ , x₂ , ... , x_m) = Σ_j=1ⁿ (Σ_k=1^m a_kjx_k - c_j)⁴

of the 4th powers of the absolute errors [1]

e_j = |Σ_k=1^m a_kjx_k - c_j|

of equations L_j of n m-1-dimensional "planes" by substituting the coordinates of any point

[_k=1^m x_k] = (x₁ , x₂ , ... , x_m)

of the m-dimensional space.

This nonnegative function ⁴S(x₁ , x₂ , ... , x_m) everywhere differentiable has and takes its minimum at a point with vanishing all the first order derivatives

⁴S'_xk = Σ_j=1ⁿ 4a_kj(Σ_k=1^m a_kjx_k - c_j)³ = 0 (k = 1, 2, ... , m)

of this function by every x_k (k = 1, 2, ... , m), which gives the following determined set

Σ_j=1ⁿ a_kj(Σ_k=1^m a_kjx_k - c_j)³ = 0 (k = 1, 2, ... , m)

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

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:

e_j = |a_jx + b_jy - c_j| ,

⁴S = Σ_j=1ⁿ e_j⁴ = Σ_j=1ⁿ (a_jx + b_jy - c_j)⁴,

⁴S'_x = Σ_j=1ⁿ 4a_j(a_jx + b_jy - c_j)³ = 0,

⁴S'_y = Σ_j=1ⁿ 4b_j(a_jx + b_jy - c_j)³ = 0;

Σ_j=1ⁿ a_j(a_jx + b_jy - c_j)³ = 0,

Σ_j=1ⁿ b_j(a_jx + b_jy - c_j)³ = 0;

Σ_j=1ⁿ a_j⁴ x³ + 3Σ_j=1ⁿ a_j³b_j x²y + 3Σ_j=1ⁿ a_j²b_j² xy² + Σ_j=1ⁿ a_jb_j³ y³ - 3Σ_j=1ⁿ a_j³c_j x² - 6Σ_j=1ⁿ a_j²b_jc_j xy - 3Σ_j=1ⁿ a_jb_j²c_j y² +

3Σ_j=1ⁿ a_j²c_j² x + 3Σ_j=1ⁿ a_jb_jc_j² y - Σ_j=1ⁿ a_jc_j³ = 0,

Σ_j=1ⁿ a_j³b_j x³ + 3Σ_j=1ⁿ a_j²b_j² x²y + 3Σ_j=1ⁿ a_jb_j³ xy² + Σ_j=1ⁿ b_j⁴ y³ - 3Σ_j=1ⁿ a_j²b_jc_j x² - 6Σ_j=1ⁿ a_jb_j²c_j xy - 3Σ_j=1ⁿ b_j³c_j y² +

3Σ_j=1ⁿ a_jb_jc_j² x + 3Σ_j=1ⁿ b_j²c_j² y - Σ_j=1ⁿ b_jc_j³ = 0;

3Σ_j=1ⁿ a_j²c_j² x + 3Σ_j=1ⁿ a_jb_jc_j² y = Σ_j=1ⁿ a_jc_j³ + 3Σ_j=1ⁿ a_j³c_j x² + 6Σ_j=1ⁿ a_j²b_jc_j xy + 3Σ_j=1ⁿ a_jb_j²c_j y² - Σ_j=1ⁿ a_j⁴ x³ - 3Σ_j=1ⁿ a_j³b_j x²y - 3Σ_j=1ⁿ a_j²b_j² xy² - Σ_j=1ⁿ a_jb_j³ y³ ,

3Σ_j=1ⁿ a_jb_jc_j² x + 3Σ_j=1ⁿ b_j²c_j² y = Σ_j=1ⁿ b_jc_j³ + 3Σ_j=1ⁿ a_j²b_jc_j x² + 6Σ_j=1ⁿ a_jb_j²c_j xy + 3Σ_j=1ⁿ b_j³c_j y² - Σ_j=1ⁿ a_j³b_j x³ - 3Σ_j=1ⁿ a_j²b_j² x²y - 3Σ_j=1ⁿ a_jb_j³ xy² - Σ_j=1ⁿ b_j⁴ y³ .

Solve this set of linearized cubic equations iteratively using formulae

3Σ_j=1ⁿ a_j²c_j² x_i+1 + 3Σ_j=1ⁿ a_jb_jc_j² y_i+1 = Σ_j=1ⁿ a_jc_j³ + 3Σ_j=1ⁿ a_j³c_j x_i² + 6Σ_j=1ⁿ a_j²b_jc_j x_iy_i + 3Σ_j=1ⁿ a_jb_j²c_j y_i² - Σ_j=1ⁿ a_j⁴ x_i³ - 3Σ_j=1ⁿ a_j³b_j x_i²y_i - 3Σ_j=1ⁿ a_j²b_j² x_iy_i² - Σ_j=1ⁿ a_jb_j³ y_i³ ,

3Σ_j=1ⁿ a_jb_jc_j² x_i+1 + 3Σ_j=1ⁿ b_j²c_j² y_i+1 = Σ_j=1ⁿ b_jc_j³ + 3Σ_j=1ⁿ a_j²b_jc_j x_i² + 6Σ_j=1ⁿ a_jb_j²c_j x_iy_i + 3Σ_j=1ⁿ b_j³c_j y_i² - Σ_j=1ⁿ a_j³b_j x_i³ - 3Σ_j=1ⁿ a_j²b_j² x_i²y_i - 3Σ_j=1ⁿ a_jb_j³ x_iy_i² - Σ_j=1ⁿ b_j⁴ y_i³

to obtain i+1st iteration (x_i+1 , y_i+1) via ith iteration (x_i , y_i) for any i ∈ N⁺ = {1, 2, ...}. One of many reasonable possibilities to take first iteration (x₁ , y₁) is using the least square method (LSM) [1] giving here

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 - 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].

Compare applying biquadrat theory (BQT) in fundamental science of solving general problems vs. 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:

Figure 1

Notata bene:

1. The least square method (LSM) [1] gives

x ≈ 1.0023,

y ≈ 1.0075

practically ignoring the last two equations with smaller factors.

2. Biquadrat theory (BQT) [5] gives

x ≈ 1.0500,

y ≈ 1.0500

better considering the last two equations with smaller factors.

3. Comparing the results of applying biquadrat theory (BQT) 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 sensitivity. But this is not sufficient because, like the least square method (LSM), biquadrat theory (BQT) 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 biquadrat theory (BQT) 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 theory 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.

Biquadrat theory (BQT) [5] is applicable to even contradictory problems and very efficient by solving some of them, has advantages as compared to the least square method (LSM) [1], and opens ways for creating much more universal theories for 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