Subjoining Equations 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), 49
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.
In fundamental science on general problem solving methods [5], subjoining equations theory (SJET) 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 quantiset [2-5] of n (n > 2; n ∈ N+ = {1, 2, ...}) linear equations
q(i)(ajx + bjy = cj) (j = 1, 2, ... , n) (Ei)
with their own quantities q(i) and 2 unknown variables x and y in a plane with the same coordinates x , y by any given real numbers q(i) > 0, aj , bj , and cj . Suppose that these n straight lines build a polygon P1P2...Pn with n corners (vertices) P1 , P2 , ... , Pn (and, naturally, n sides, or edges) either in the clockwise order or in the anticlockwise order which can be provided via preliminarily renumbering (reindexing) the equations. If necessary and possible, find and use such renumbering (reindexing) of these n corners that polygon P1P2...Pn is convex. If it is impossible, then provide the minimum number of nonconvex corners (vertices) (i.e., with convexity violation) and hence the maximum number of convex corners (vertices) (i.e., without convexity violation). Let Pi be the intersection of straight lines Ei and Ei+1 with their own quantities q(i) and q(i+1) which has its own quantity
p(i) = q(i)q(i+1).
Because of the clear possible periodicity of the indices i = 1, 2, ... , n with period n , consider
Pj+kn = Pj ,
q(i+kn) = q(i) ,
p(i+kn) = p(i)
for any integer k . Then the quantiset [2-5] of n polygon corners (vertices) is
p(1)P1 , p(2)P2 , ... , p(n)Pn .
The quantiset [2-5] of n polygon corners bisectors is
p(1)B1 , p(2)B2 , ... , p(n)Bn .
Nota bene: A bisector of an angle (and of the corresponding angle sides) divides it into two equal parts and is the set of all the points equidistant from the both angle sides [1]. For two intersecting straight lines, there are two mutually perpendicular bisector straight lines. If for a polygon, there is an inscribed circumference, then the desired unique best quasisolution its simply the inscribed circumference center which is the intersection of the bisectors of all the internal corners of this polygon which in this case is always convex. This all holds for any triangle whose case is, therefore, trivial.
Hence further consider case n > 3 only when for a polygon, there is no inscribed circumference.
In a certain sense, subjoining equations theory (SJET) is opposite to equidistance theory (EDT) based on stepwise excluding equations and regarding all the excluded equations at the end. On the contrary, subjoining equations theory (SJET) usually begins with 3 appropriate equations only and then stepwise subjoins the remaining equations.
The main idea, essence, and algorithm of subjoining equations theory (SJET) are as follows:
1) select any three equations determining the straight lines whose intersections are the corners (vertices) of a triangle including the polygon built by the straight lines determined by the set of all the given equations. If necessary renumber (reindex) the equations to provide numbers (indices) 1, 2, and 3 (in any order) for these three equations. Their own quantities are q(1), q(2), and q(3), as well as those straight lines are L1 , L2 , and L3 , respectively.
Nota bene:
A. Such a selection is impossible in the trivial cases of at most two groups of straight lines, all the straight lines in any group being parallel to one another, e.g., by a parallelogram for which the intersection of the diagonals can be considered as the desired quasisolution. Generally, for each of such groups, determine its weighted mean straight line and (by two groups) the intersection of their weighted mean straight line and consider the obtained result as the desired quasisolution.
B. Usually, there are many different possibilities for such a selection. Moreover, the same holds for the further sequence of subjoining the remaining equations. This is very well providing the comparativity of the results obtained by different approaches and methods within this theory itself, as well as estimating and even averaging such results;
2) for this triangle, determine the inscribed circumference (incircle) center C3 (which is the intersection of the bisectors of all the internal corners of this triangle) and the incircle radius (inradius) r3 and give both C3 and r3 the own quantity
q(1) + q(2) + q(3)
which is the sum of the quantities of the three initially selected equations.
Nota bene: The Cartesian coordinates of the incenter (inscribed circumference center) W of a triangle are weighted averages of the corresponding coordinates of the three corners (vertices), the lengths of the triangle sides opposite to these corners (vertices) being the weights of the corresponding corners (vertices) coordinates [1]. Namely, first determine the three corners (vertices) and their coordinates
A(xA , yA), B(xB , yB), C(xC , yC),
further the lengths
a , b , c
of the triangle sides opposite to these corners (vertices), respectively, then the Cartesian coordinates of the incenter (inscribed circumference center) of a triangle
x(C3) = (axA + bxB + cxC)/(a + b + c),
y(C3) = (ayA + byB + cyC)/(a + b + c)
and quanticenter
q(1)+q(2)+q(3)C3[(axA + bxB + cxC)/(a + b + c), (ayA + byB + cyC)/(a + b + c)].
By Heron's formula [1], the triangle area is
S = [s(s - a)(s - b)(s - c)]1/2
where semiperimeter
s = (a + b + c)/2.
Finally, formula
S = r3 s
gives
r3 = [(s - a)(s - b)(s - c)/s]1/2.
Then the quanti-inradius is
q(1)+q(2)+q(3)[(s - a)(s - b)(s - c)/s]1/2;
3) subjoin anyone of the remaining equations and give it number (index) 4. The own quantity of this equation is q(4). Determine straight line L4 defined by this 4th equation and the distance d4 from center C3 to straight line L4 ;
4) determine neutral straight line L04 which is parallel to straight line L4 and provides keeping center C3 , i.e., by replacing straight line L4 with straight line L04 , new center C04 would coincide with C3 .
Nota bene: The distance from center C3 to straight line L04 is r3 . There are two straight lines L04 and L'04 both parallel to straight line L4 with distance r3 from point C3 but by another straight line L'04 , point C3 would be no center;
5) determine straight line L'4 which contains center C3 and is perpendicular to straight line L4 ;
6) determine intersection S04 of straight lines L'4 and L04 ;
7) determine intersection S4 of straight lines L'4 and L4 ;
8) on straight line L'4 , determine displacement vector S04S4 with initial point S04 and endpoint S4 ;
9) determine straight line L''04 which contains center C3 and is parallel to straight line L4 ;
10) determine straight line L''4 via parallel displacement of straight line L''04 by displacement vector S04S4 ;
11) determine intersection D'4 of straight line L''4 with the internal bisector of the both straight lines connected with possibly elongated polygon side lying on straight line L4 ;
12) determine intersection D''4 of straight line L''4 with the internal bisector of the internal bisectors of the both new internal polygon corners by subjoining straight line L4 ;
13) determine intersection D'''4 of straight line L''4 with
13a) either the straight line containing both the intersection of the both straight lines connected with possibly elongated polygon side lying on straight line L4 and the intersection of the internal bisectors of the both new internal polygon corners by subjoining straight line L4 if those both straight lines are intersecting;
13b) or the straight line parallel to both the both straight lines connected with possibly elongated polygon side lying on straight line L4 and containing the intersection of the internal bisectors of the both new internal polygon corners by subjoining straight line L4 if those both straight lines are parallel;
14) determine point D4 to be weighted with center C3 via anyone of the following methods within this subjoining equations theory (SJET):
14a) by external bisector method, take D4 = D'4 ;
14b) by internal bisector method, take D4 = D''4 ;
14c) by pointwise average bisector method, take D4 to be the middle of segment D'4D''4 ;
14d) by anglewise average bisector method, take D4 to be D'''4 ;
15) determine quantipoint
q(4)D4[x(D4), y(D4)];
16) determine center C4 and its quanticenter
q(1)+q(2)+q(3)+q(4)C4[x(C4), y(C4)]
with its coordinates
x(C4) = {[q(1) + q(2) + q(3)]x(C3) + q(4)x(D'4)}/[q(1) + q(2) + q(3) + q(4)],
y(C4) = {[q(1) + q(2) + q(3)]y(C3) + q(4)y(D'4)}/[q(1) + q(2) + q(3) + q(4)];
17) determine quasiradius
r4 = {[q(1) + q(2) + q(3)]r3 + q(4)d4}/[q(1) + q(2) + q(3) + q(4)]
and its quantiquasiradius
q(1)+q(2)+q(3)+q(4)r4 ;
18) continue this process of subjoining the remaining equations with further increasing the polygon sides (edges) number;
19) finish this finite process by obtaining quanticenter
q(1)+q(2)+q(3)+...+q(n)Cn[x(Cn), y(Cn)]
and quantiquasiradius
q(1)+q(2)+q(3)+...+q(n)rn
for the given polygon.
Compare applying subjoining equations theory (SJET), equidistance theory (EDT), opposite sides bisectors theory (OSBT), adjacent corners bisectors theory (ACBT), adjacent sides bisectors theory (ASBT) with one step only, distance quadrat theory (DQT), the least square method (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,
with the unit own quantities of the equations, see Figures 1, 2:
Figure 1
Figure 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).
Subjoining equations theory (SJET) providing simple explicit quasisolutions 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, 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