Algebraic Geodesy and Geoinformatics - 2009- PART I METHODS 6 Algebraic LEast Square Solution (ALESS) 6-1 Introduction In geodesy, very frequently one has to handle overdetermined systems of nonlinear equations. In that case we have more equations than unknowns, therefore "the solution " of the system can be interpreted in least square sense. It means that the original problem will be transformed into a minimization problem. It goes without saying that this technique works in case of determined system, too. See for example the nonlinear equation solver of MATLAB. Let us suppose, that our nonlinear system is a system of multivariate polynomial equations, then the following theorem can be considered: Given m algebraic (polynomial) observational equations, where m is the dimension of the observation space Y of order l in n unknown variables, and n is the dimension of the parameter space X. There exists n normal equations of the polynomial order (2 l - 1) to be solved with algebraic methods (ALESS). 6.2 Overdetermined vs. determined problems Clear@"Global‘*"D Let us consider the following system, f1@x_, y_D := x^2 + y - 3 f2@x_, y_D := x + 1 8y^2 - 1 f3@x_, y_D := x - y here n = 2, m = 2 and l = 2. The objective function to be minimized is, obj = f1@x, yD 2 + f2@x, yD 2 + f3@x, yD 2 Expand 10 - 2x - 4x 2 + x 4 - 6y - 2xy + 2x 2 y + 7y 2 4 + xy 2 4 + y 4 64 The order of the objective function is 2 l = 4. This overdetermined system has one solution from the point of view of least square sense. NMinimize @obj, 8x, y<D 80.232361, 8x fi 1.24747, y fi 1.27393<< We are looking for the solution of the original system as the solution of a determined problem. Considering the necessary condition for the minimum,
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Algebraic Geodesy and Geoinformatics - 2009- PART I METHODS
6 Algebraic LEast Square Solution (ALESS)
6-1 Introduction
In geodesy, very frequently one has to handle overdetermined systems of nonlinear equations. In that case we have moreequations than unknowns, therefore "the solution " of the system can be interpreted in least square sense. It means that theoriginal problem will be transformed into a minimization problem. It goes without saying that this technique works in case of determined system, too. See for example the nonlinear equationsolver of MATLAB.Let us suppose, that our nonlinear system is a system of multivariate polynomial equations, then the following theorem canbe considered:Given m algebraic (polynomial) observational equations, where m is the dimension of the observation space Y of order l in nunknown variables, and n is the dimension of the parameter space X. There exists n normal equations of the polynomial order(2 l - 1) to be solved with algebraic methods (ALESS).
6.2 Overdetermined vs. determined problems
Clear@"Global‘*"DLet us consider the following system,
f1@x_, y_D := x^2 + y - 3
f2@x_, y_D := x + 1 � 8 y^2 - 1
f3@x_, y_D := x - y
here n = 2, m = 2 and l = 2. The objective function to be minimized is,
obj = f1@x, yD2 + f2@x, yD2 + f3@x, yD2 �� Expand
10 - 2 x - 4 x2 + x4 - 6 y - 2 x y + 2 x2 y +7 y2
4+x y2
4+y4
64
The order of the objective function is 2 l = 4. This overdetermined system has one solution from the point of view of leastsquare sense.
NMinimize@obj, 8x, y<D80.232361, 8x ® 1.24747, y ® 1.27393<<
We are looking for the solution of the original system as the solution of a determined problem. Considering the necessarycondition for the minimum,
eq1 = D@obj, xD-2 - 8 x + 4 x3 - 2 y + 4 x y +
y2
4
eq2 = D@obj, yD-6 - 2 x + 2 x2 +
7 y
2+x y
2+y3
16
The order of this determined system is 2 l - 1 = 4 -1 = 3. Now, employing Global Numerical Solver, the solution of thissystem,
This means, we should find out all of real solutions of the determined system representing the original overdetermined one!You can hardly do it with local numerical methods!
6- 3 Finding all of the roots of polynomial systems
6- 3- 1 Symbolic computer algebra solution
We have seen in the previous chapters that two basic types of the algebraic methods are at our disposal:
- symbolic solution using computer algebra like resultants or Groebner basis,- global numerical methods like linear homotopy
As an illustration, considering our problem, we solve the polynomial system via reduced Groebner basis,
Let us consider the following 3D Ranging by Local Positioning Systems (LPS). The coordinates of the reference points areknown, {xi,yi,zi} as well as their distances, si, i = 1, 2, 3. Then the following 3 equations can be considered for computing
the unknown coordinates, {x0, y0, z0},
e1 = Hx1 - x0L2 + Hy1 - y0L2 + Hz1 - z0L2 - s12;
e2 = Hx2 - x0L2 + Hy2 - y0L2 + Hz2 - z0L2 - s22;
e3 = Hx3 - x0L2 + Hy3 - y0L2 + Hz3 - z0L2 - s32;
However, having more than 3 reference points, our system becomes overdetermined. Let us suppose we have 7 referencestations,
d = 83, 3, 3<;we should track 33= 27 paths, if we use the automatic start system generator. In order to reduce the computation task, wehave already demonstrated, how one can generate start system in a different way. For example in Section 5- 10- 2, weconsidered the corresponding univariate parts of the equations.Now, we show an other general way to create a start system, which results a reduced initial value set. Let us consider now the