Unierror Biquadrat 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), 53

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 biquadrat theory (BQT) in fundamental science of solving general problems.

Show the essence of unierror biquadrat theories (EBQT) 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 in the Cartesian m-dimensional "space" [1].

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 biquadrat theory (LEBQT) 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) naturally introducing new equation factors (k = 1, 2, ... , m; j = 1, 2, ... , n)

a_kj = a'_kj / (Σ_k=1^m |a'_kj ⁱx_k| + |c'_j|),

c_kj = c'_kj / (Σ_k=1^m |a'_kj ⁱx_k| + |c'_j|);

2.3) representing the obtained set of normalized linear equations (ⁱL_j) as

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

Nota bene: This normalization is depends on specific pseudosolutions [2-5] to this set of equations and also holds for the absolute errors [1] of such equations;

2.4) solving this overdetermined set of linear equations (ⁱL_j) via applying biquadrat theory (BQT) (see below) in fundamental science of solving general problems [5] to this set of normalized linear equations (ⁱL_j) 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 biquadrat theory (SEBQT) is based on quadratic 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) naturally introducing new equation factors (k = 1, 2, ... , m; j = 1, 2, ... , n)

a_kj = a'_kj / [(m + 1)(Σ_k=1^m a'_kj²x_k² + c'_j²)]^1/2 ,

c_kj = c'_kj / [(m + 1)(Σ_k=1^m a'_kj²x_k² + c'_j²)]^1/2 ;

2.3) representing the obtained set of normalized linear equations (ⁱL_j) as

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

Nota bene: This normalization is depends on specific pseudosolutions [2-5] to this set of equations and also holds for the absolute errors [1] of such equations;

2.4) solving this overdetermined set of linear equations (ⁱL_j) via applying the least biquadratic method (LBQM) (see below) in fundamental science of solving general problems [5] to this set of normalized linear equations (ⁱL_j) 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 biquadrat theory (LEBQT) gives

¹A_j = |a'_jx + b'_jy - c'_j|/(|a'_jx| + |b'_jy| + |c'_j|);

a'_j / (|a'_jx| + |b'_jy| + |c'_j|) ⁱ⁺¹x + b'_j / (|a'_jx| + |b'_jy| + |c'_j|) ⁱ⁺¹y = c'_j / (|a'_jx| + |b'_jy| + |c'_j|) (j = 1, 2, ... , n) (ⁱL_j);

a_j = a'_j / (|a'_jx| + |b'_jy| + |c'_j|),

b_j = b'_j / (|a'_jx| + |b'_jy| + |c'_j|),

c_j = c'_j / (|a'_jx| + |b'_jy| + |c'_j|);

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

It can be suitable to further replace ⁱ⁺¹x with x and ⁱ⁺¹y with y . Then obtain

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

Square unierror biquadrat theory (SEBQT) gives

²A_j = |a'_jx + b'_jy - c'_j|/[3(a'_j²x² + b'_j²y² + c'_j²)]^1/2 ;

a'_j / [3(a'_j²x² + b'_j²y² + c'_j²)]^{1/2 i+1}x + b'_j / [3(a'_j²x² + b'_j²y² + c'_j²)]^{1/2 i+1}y = c'_j / [3(a'_j²x² + b'_j²y² + c'_j²)]^1/2 (j = 1, 2, ... , n) (ⁱL_j);

a_j = a'_j / [3(a'_j²x² + b'_j²y² + c'_j²)]^1/2 ,

b_j = b'_j / [3(a'_j²x² + b'_j²y² + c'_j²)]^1/2 ,

c_j = c'_j / [3(a'_j²x² + b'_j²y² + c'_j²)]^1/2 ;

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

It can be suitable to further replace ⁱ⁺¹x with x and ⁱ⁺¹y with y . Then obtain

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

Now explicitly show applying the biquadrat theory (BQT) in fundamental science of solving general problems [5] to the above set of normalized linear equations

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

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] already normalized above

e_j = |Σ_k=1^m a_kjx_k - c_j|

of equations ⁱL_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 unierror biquadrat theories (EBQT) in fundamental science on general problem reserve [5] vs. distance biquadrat theory (DBQT) [5], biquadrat theory (BQT) [5], unierror quadrat theories (EQT) [5], distance quadrat theory (DQT) [2-5], 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 via linear unierror quadrat theory only.

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 it 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 sensitivity and 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 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.

7. Unierror biquadrat theories (EBQT) replace the absolute errors [1] with unierrors due to preliminary equations set normalization.

8. Both linear unierror biquadrat theory (LEBQT) and square unierror biquadrat theory (SEBQT) give relatively near results. Therefore, in Figure 1, we have shown the results obtained via linear unierror biquadrat theory (LEBQT) only.

9. Comparing the results of applying unierror biquadrat theories (EBQT) vs. unierror quadrat theories (EQT) also to other test equation sets shows that increasing the power from 2 to 4 provides very substantially improving theory sensitivity.

Unierror biquadrat theories (EBQT) providing simple explicit quasisolutions 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