algebraic least square solution...

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,

sol = NSolve@8eq1 � 0, eq2 � 0<, 8x, y<D88x ® 2.0573 + 1.28006 ä, y ® -0.0282082 - 7.43985 ä<,

8x ® 2.0573 - 1.28006 ä, y ® -0.0282082 + 7.43985 ä<,8x ® 0.33941 + 0.679091 ä, y ® -0.606933 - 7.81887 ä<,8x ® 0.33941 - 0.679091 ä, y ® -0.606933 + 7.81887 ä<,8x ® -0.987112 + 1.25442 ä, y ® 2.21036 + 1.61614 ä<,8x ® -0.987112 - 1.25442 ä, y ® 2.21036 - 1.61614 ä<, 8x ® -1.371, y ® -0.177989<,8x ® 1.24747, y ® 1.27393<, 8x ® -2.69566, y ® -4.24638<<

The real solutions are

solR = Select@sol, Im@ð@@1, 2DDD � 0 &D88x ® -1.371, y ® -0.177989<, 8x ® 1.24747, y ® 1.27393<, 8x ® -2.69566, y ® -4.24638<<

However, the only real solution can be considered, which provides the least value of the residual,

res = Map@obj �. ð &, solRD88.71186, 0.232361, 4.48362<

Select@solR, Min@resD == obj �. ð &D88x ® 1.24747, y ® 1.27393<<

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,

solpx = GroebnerBasis@8eq1, eq2<, 8x, y<, yD9-318 + 256 x - 231 x2 - 292 x3 + 166 x4 + 186 x5 + 44 x6 - 56 x7 + 8 x9=

solpy = GroebnerBasis@8eq1, eq2<, 8x, y<, xD9-24 576 - 110 592 y + 147 456 y2 - 39 168 y3 + 2016 y5 - 48 y6 + 104 y7 + y9=

Selecting real solutions of the two univariate polynomials,

2 ALESS_06.nb

Selecting real solutions of the two univariate polynomials,

solxR = Select@NSolve@solpx � 0, xD, Im@ð@@1, 2DDD � 0 &D �� Flatten

8x ® -2.69566, x ® -1.371, x ® 1.24747<

and

solyR = Select@NSolve@solpy � 0, yD, Im@ð@@1, 2DDD � 0 &D �� Flatten

8y ® -4.24638, y ® -0.177989, y ® 1.27393<

The pairs of solutions,

solxyR = Transpose@8solxR, solyR<D88x ® -2.69566, y ® -4.24638<, 8x ® -1.371, y ® -0.177989<, 8x ® 1.24747, y ® 1.27393<<

6- 3- 2 Global numerical solution via homotopy

Demonstrating the global numerical method, let us employ linear homotopy. The target system,

F = 8eq1, eq2< �� Expand

:-2 - 8 x + 4 x3 - 2 y + 4 x y +y2

4, -6 - 2 x + 2 x2 +

7 y

2+x y

2+y3

16>

The variables,

X = 8x, y<;The degrees of the equations,

d = 83, 3<;Remark: Homotopy functions of Mathematica have been implemented in the GeoAlgebra package. Let us load it,

<< GeoAlgebra‘LinearHomotopy‘

Generation of the start system,

? StartingSystem

Computes the start system for polynomial system.

Input parameters:

F - list of functions of the target system,

X - list of variables,

d - list of orders of the equations.

Output parameters:

G - list of functions of the start system,

X0 - solution of the start system, the initial values for homotopy

ss = StartingSystem@F, X, dD;G = ss@@1DD9H-0.626564 - 0.77937 äL IH0.653448 + 0.756971 äL + x3M,

H-0.228897 - 0.973451 äL IH0.998698 + 0.0510185 äL + y3M=

its initial values, the solutions of the start system are,

ALESS_06.nb 3

X0 = ss@@2DD88-0.959317 - 0.28233 ä, -0.999855 - 0.0170127 ä<,

8-0.959317 - 0.28233 ä, 0.514661 - 0.857394 ä<,8-0.959317 - 0.28233 ä, 0.485194 + 0.874406 ä<,80.724163 - 0.689628 ä, -0.999855 - 0.0170127 ä<,80.724163 - 0.689628 ä, 0.514661 - 0.857394 ä<,80.724163 - 0.689628 ä, 0.485194 + 0.874406 ä<,80.235154 + 0.971958 ä, -0.999855 - 0.0170127 ä<,80.235154 + 0.971958 ä, 0.514661 - 0.857394 ä<,80.235154 + 0.971958 ä, 0.485194 + 0.874406 ä<<

The number of paths,

Length@X0D9

Γ = 81, 1<;Let us employ the direct path tracing technique,

? LinearHomotopyFR

Computes the homotopy paths with direct path tracing.

Input parameters:

F - list of functions of the target system,

G - list of functions of the start system,


X0 - list of initial values,

Γ - list of complex weigths,

n - number of subintervals,

Λ - dummy variable.

Output variables:

sol@@1DD - list of the solutions,

sol@@2DD - list of homotopy paths

sol = LinearHomotopyFR@F, G, X, X0, Γ, 100, ΛD;sol@@1DD88-0.987112 + 1.25442 ä, 2.21036 + 1.61614 ä<, 80.33941 + 0.679091 ä, -0.606933 - 7.81887 ä<,

8-1.371, -0.177989<, 80.33941 - 0.679091 ä, -0.606933 + 7.81887 ä<, 8-2.69566, -4.24638<,8-0.987112 - 1.25442 ä, 2.21036 - 1.61614 ä<, 82.0573 - 1.28006 ä, -0.0282082 + 7.43985 ä<,82.0573 + 1.28006 ä, -0.0282082 - 7.43985 ä<, 81.24747, 1.27393<<

The different paths can be visualized,

? Paths

Display homotopy paths.

Input parameters:


sol - list of homotopy paths,

X0 - list of initial values

Paths@X, sol@@2DD, X0, ΛD

4 ALESS_06.nb

-1.8 -1.6 -1.4 -1.2 -1.0

-0.5

0.0

0.5

1.0

Re

Im

xHΛL

-1.0 -0.5 0.0 0.5 1.0 1.5 2.00.0

0.5

1.0

1.5

2.0

Re

Im

yHΛL

-0.8 -0.6 -0.4 -0.2 0.0 0.2

-0.2

0.0

0.2

0.4

0.6

Re

Im

xHΛL

-0.5 0.0 0.5 1.0 1.5 2.0 2.5-8

-7

-6

-5

-4

-3

-2

-1

ReIm

yHΛL

-1.4 -1.3 -1.2 -1.1 -1.0

-0.35

-0.30

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

Re

Im

xHΛL

-0.1 0.0 0.1 0.2 0.3 0.4 0.5-0.2

0.0

0.2

0.4

0.6

0.8

Re

Im

yHΛL

ALESS_06.nb 5

0.2 0.3 0.4 0.5 0.6 0.7-0.7

-0.6

-0.5

-0.4

-0.3

Re

Im

xHΛL

-4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.00

2

4

6

8

Re

Im

yHΛL

-2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

Re

Im

xHΛL

-4 -3 -2 -1 0-2.0

-1.5

-1.0

-0.5

0.0

ReIm

yHΛL

-1.0 -0.5 0.0 0.5-1.3

-1.2

-1.1

-1.0

-0.9

-0.8

-0.7

Re

Im

xHΛL

0.5 1.0 1.5 2.0 2.5

-1.5

-1.0

-0.5

0.0

0.5

Re

Im

yHΛL

6 ALESS_06.nb

0.5 1.0 1.5 2.0

-1.0

-0.5

0.0

0.5

1.0

Re

Im

xHΛL

-3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.00

2

4

6

Re

Im

yHΛL

0.5 1.0 1.5 2.0

0.6

0.8

1.0

1.2

1.4

Re

Im

xHΛL

0.0 0.5 1.0 1.5 2.0 2.5

-7

-6

-5

-4

-3

-2

-1

ReIm

yHΛL

0.4 0.6 0.8 1.0 1.2 1.40.0

0.2

0.4

0.6

0.8

1.0

Re

Im

xHΛL

0.6 0.8 1.0 1.20.0

0.2

0.4

0.6

0.8

Re

Im

yHΛL

The real solutions are,

Select@sol@@1DD, HIm@ð@@1DDD � 0L ì HIm@ð@@2DDD � 0L &D88-1.371, -0.177989<, 8-2.69566, -4.24638<, 81.24747, 1.27393<<

6- 4 Examples

6- 4- 1 2D Ranging

Let us illustrate this technique with a 2D positioning problem. Considering 4 points with known coordinates, Pi(xi,yi) and

their distances (ti) from a point P0 with unknown coordinates, these coordinates (x0,y0) should be computed.

The values are in the following table,

Table 6.1 Data for the 2D Ranging problem

ALESS_06.nb 7

Pi x@mD y@mD ti@mD1 48 177.62 6531.28 611.023

2 49 600.15 7185.19 1529.482

3 49 830.93 5670.69 1323.884

4 47 863.91 5077.24 1206.524

In order to increase the precision of the computation the values of the coordinates and the distances are rationalized,

x1 = Rationalize@48 177.62D; y1 = [email protected]; x2 = Rationalize@49 600.15D;y2 = [email protected]; x3 = Rationalize@49 830.93D; y3 = [email protected];x4 = Rationalize@47 863.91D; y4 = [email protected];t1 = [email protected]; t2 = [email protected];t3 = [email protected]; t4 = [email protected];

The observation equations,

f1 = Hx1 - x0L^2 + Hy1 - y0L^2 - t12;

f2 = Hx2 - x0L^2 + Hy2 - y0L^2 - t22;

f3 = Hx3 - x0L^2 + Hy3 - y0L^2 - t32;

f4 = Hx4 - x0L^2 + Hy4 - y0L^2 - t42;

The objective function in least square sense,

f = f1^2 + f2^2 + f3^2 + f4^2;

The necessary conditions,

eq1 = D@f, x0D-4

4 786 391

100- x0 -

90 981 260 161

62 500+

4 786 391

100- x0

2

+126 931

25- y0

2

-

44 983 093

100- x0 -

109 541 802 841

62 500+

4 983 093

100- x0

2

+567 069

100- y0

2

-

42 408 881

50- x0 -

373 349 106 529

1 000 000+

2 408 881

50- x0

2

+163 282

25- y0

2

-

4992 003

20- x0 -

584 828 797 081

250 000+

992 003

20- x0

2

+718 519

100- y0

2

eq2 = D@f, y0D-4 -

90 981 260 161

62 500+

4 786 391

100- x0

2

+126 931

25- y0

2 126 931

25- y0 -

4 -109 541 802 841

62 500+

4 983 093

100- x0

2

+567 069

100- y0

2 567 069

100- y0 -

4 -373 349 106 529

1 000 000+

2 408 881

50- x0

2

+163 282

25- y0

2 163 282

25- y0 -

4 -584 828 797 081

250 000+

992 003

20- x0

2

+718 519

100- y0

2 718 519

100- y0

Applying Groebner basis,

8Eq1, Eq2< = GroebnerBasis@8eq1, eq2<, 8x0, y0<D;We have two univariate polynomials,

8 ALESS_06.nb

We have two univariate polynomials,

Eq1

-39 369 019 248 577 328 985 507 405 416 860 886 066 553 926 555 833 743 199 +

32 585 023 481 961 061 411 254 948 630 177 541 361 813 554 433 324 625 y0 -

10 823 598 730 743 148 937 633 137 512 777 294 201 932 729 843 750 y02 +

1 803 503 952 825 841 577 317 972 532 850 605 686 914 062 500 y03 -

150 751 683 232 320 916 417 491 276 600 000 000 000 000 y04 +

5 057 308 137 681 786 539 515 625 000 000 000 000 y05

Eq2

42 210 533 352 947 442 557 577 252 977 233 774 082 157 324 219 852 680 329 194 625 882 868 228 131 410 �

358 286 389 +

13 167 566 633 260 944 810 742 152 454 148 405 962 406 219 645 348 856 741 260 699 955 307 255 252 �

600 200 x0 -

29 608 709 236 850 114 150 995 164 567 767 970 036 108 393 999 220 194 530 608 130 003 561 639 745 �

364 041 525 y0 +

7 676 001 438 966 718 396 390 975 989 019 652 409 337 479 332 520 917 254 056 946 221 863 467 942 187 �

500 y02 -

885 305 242 413 149 182 018 499 982 799 118 281 692 505 513 077 450 092 377 250 021 550 000 000 000

y03 +

38 341 812 392 316 437 600 753 830 179 791 266 425 664 293 477 941 644 471 675 000 000 000 000 y04

Let us compute the roots,

sol = N@Solve@Eq1 � 0, y0D, 20D88y0 ® 6058.9781636967393724<, 8y0 ® 5747.6051307602141861 - 710.6398373518648874 ä<,

8y0 ® 5747.6051307602141861 + 710.6398373518648874 ä<,8y0 ® 6127.2461131347275826 - 873.7323165685666507 ä<,8y0 ® 6127.2461131347275826 + 873.7323165685666507 ä<<

The real solution is,

y0 = Select@y0 �. sol, Im@ðD � 0 &D@@1DD6058.9781636967393724

Then the second equation,

Eq2

-6.394864280001290 ´ 1083 +

13 167 566 633 260 944 810 742 152 454 148 405 962 406 219 645 348 856 741 260 699 955 307 255 252 �

600 200 x0

and its solution,

N@Solve@Eq2 � 0, x0D, 20D88x0 ® 48 565.26994021828<<

In order to check the result, we solve the problem via global minimization employing the built function, NMinimize,

Clear@y0, x0DNMinimize@f, 8x0, y0<, WorkingPrecision ® 20D8543.59239525607154765, 8x0 ® 48 565.269940218279911, y0 ® 6058.9781636967393724<<

ALESS_06.nb 9

6- 4- 2 3D Ranging

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,

datan = 8x1 -> 4 157 246.5346, y1 -> 671 877.0281, z1 -> 4 774 581.6314, s1 -> 566.8635,

x2 -> 4 156 749.5977, y2 -> 672 711.4554, z2 -> 4 774 981.5459, s2 -> 1324.2380,

x3 -> 4 156 748.6829, y3 -> 671 171.9385, z3 -> 4 775 235.5483, s3 -> 542.2609,

x4 -> 4 157 066.8851, y4 -> 671 064.9381, z4 -> 4 774 865.8238, s4 -> 364.9797,

x5 -> 4 157 266.6181, y5 -> 671 099.1577, z5 -> 4 774 689.8536, s5 -> 430.5286,

x6 -> 4 157 307.5147, y6 -> 671 171.7006, z6 -> 4 774 690.5691, s6 -> 400.5837,

x7 -> 4 157 244.9515, y7 -> 671 338.5915, z7 -> 4 774 699.9070, s7 -> 269.2309

<;Then the prototype equation,

e = Hxi - x0L2 + Hyi - y0L2 + Hzi - z0L2 - si2;

The objective to be minimized in least square sense,

f = ApplyAPlus, TableAe2, 8i, 1, 7<EEI-s1

2 + H-x0 + x1L2+ H-y0 + y1L2

+ H-z0 + z1L2M2+

I-s22 + H-x0 + x2L2

+ H-y0 + y2L2+ H-z0 + z2L2M2

+ I-s32 + H-x0 + x3L2

+ H-y0 + y3L2+ H-z0 + z3L2M2

+

I-s42 + H-x0 + x4L2

+ H-y0 + y4L2+ H-z0 + z4L2M2

+ I-s52 + H-x0 + x5L2

+ H-y0 + y5L2+ H-z0 + z5L2M2

+

I-s62 + H-x0 + x6L2

+ H-y0 + y6L2+ H-z0 + z6L2M2

+ I-s72 + H-x0 + x7L2

+ H-y0 + y7L2+ H-z0 + z7L2M2

From the necessary condition, we get

g1 = D@f, x0D;g2 = D@f, y0D;g3 = D@f, z0D;

Now we solve this square system (g1, g2, g3) = 0.

6- 4- 3 Groebner Basis Solution

Let us use high precision data with numerical reduced Groenber basis (see Section 4- 8) for variable x0,

gbx0 = GroebnerBasis@SetPrecision@8g1, g2, g3< �. datan, 300D,8x0, y0, z0<, 8y0, z0<, CoefficientDomain ® InexactNumbersD;

This is an univariate polynomial of order seven,

Exponent@gbx0, x0D87<

Its roots are,

10 ALESS_06.nb

Its roots are,

Solx0 = SetPrecision@NSolve@gbx0 � 0, x0D, 20D99x0 ® 4.1570478280643144805 ´ 106 - 17.7824555537030 ä=,

9x0 ® 4.1570478280643144805 ´ 106 + 17.7824555537030 ä=,9x0 ® 4.1570661115296523158 ´ 106=,9x0 ® 4.1571001585939538923 ´ 106 - 169.0166013474395 ä=,9x0 ® 4.1571001585939538923 ´ 106 + 169.0166013474395 ä=,9x0 ® 4.1571279473843755008 ´ 106 - 270.2601421331170 ä=,9x0 ® 4.1571279473843755008 ´ 106 + 270.2601421331170 ä==

Selecting real solution,

x0s = Select@Map@x0 �. ð &, Flatten@Solx0DD, Im@ðD � 0 &D94.1570661115296523158 ´ 106=

Similarly for variables y0 and z0, we get,

gby0 = GroebnerBasis@SetPrecision@8g1, g2, g3< �. datan, 300D,8x0, y0, z0<, 8x0, z0<, CoefficientDomain ® InexactNumbersD;

Soly0 = NSolve@gby0 � 0, y0D;y0s = SetPrecision@Select@Map@y0 �. ð &, Flatten@Soly0DD, Im@ðD � 0 &D, 20D8671 429.66547894619317<

and

gbz0 = GroebnerBasis@SetPrecision@8g1, g2, g3< �. datan, 300D,8x0, y0, z0<, 8x0, y0<, CoefficientDomain ® InexactNumbersD;

Solz0 = NSolve@gbz0 � 0, z0D;z0s = SetPrecision@Select@Map@z0 �. ð &, Flatten@Solz0DD, Im@ðD � 0 &D, 20D94.7748793703096208740 ´ 106=

6- 4- 4 Linear Homotopy Solution

The system,

F = 8g1, g2, g3< �. datan �� Expand

9-4.71777 ´ 1021 + 2.10263 ´ 1015 x0 - 3.49196 ´ 108 x02 + 28 x03 +

1.56321 ´ 1014 y0 - 3.76035 ´ 107 x0 y0 - 1.16399 ´ 108 y02 + 28 x0 y02 +

1.11156 ´ 1015 z0 - 2.6739 ´ 108 x0 z0 - 1.16399 ´ 108 z02 + 28 x0 z02,

-7.62057 ´ 1020 + 1.56321 ´ 1014 x0 - 1.88017 ´ 107 x02 + 1.16012 ´ 1015 y0 -

2.32797 ´ 108 x0 y0 + 28 x02 y0 - 5.64052 ´ 107 y02 + 28 y03 +

1.7955 ´ 1014 z0 - 2.6739 ´ 108 y0 z0 - 1.88017 ´ 107 z02 + 28 y0 z02,

-5.41882 ´ 1021 + 1.11156 ´ 1015 x0 - 1.33695 ´ 108 x02 + 1.7955 ´ 1014 y0 -

1.33695 ´ 108 y02 + 2.41161 ´ 1015 z0 - 2.32797 ´ 108 x0 z0 +

28 x02 z0 - 3.76035 ´ 107 y0 z0 + 28 y02 z0 - 4.01085 ´ 108 z02 + 28 z03=

The variables,

X = 8x0, y0, z0<;Since

ALESS_06.nb 11

Since

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

linear parts of the equations, g1, g2 and g3,

G1 = Map@Select@ð, NumberQD &, 8F@@1DD, Coefficient@F@@1DD, x0D,Coefficient@F@@1DD, y0D, Coefficient@F@@1DD, z0D<D.81, x0, y0, z0<

-4.71777 ´ 1021 + 2.10263 ´ 1015 x0 + 1.56321 ´ 1014 y0 + 1.11156 ´ 1015 z0


-7.62057 ´ 1020 + 1.56321 ´ 1014 x0 + 1.16012 ´ 1015 y0 + 1.7955 ´ 1014 z0


-5.41882 ´ 1021 + 1.11156 ´ 1015 x0 + 1.7955 ´ 1014 y0 + 2.41161 ´ 1015 z0

Then the normalized start system,

G = 8G1, G2, G3< 10-14;

The initial value is the solution of the start system. In equation form,

eqs = Map@ð � 0 &, GD;The matrix form of the linear equation system,

8b, m< = CoefficientArrays@eqs, 8x0, y0, z0<D;The solution of the system,

X0 = 8LinearSolve@m, -bD<991.3857 ´ 106, 223 830., 1.59161 ´ 106==

Now, we use real Γ in order to keep the path remaining in real region,

Γ = 81, 1, 1<;Applying direct path tracing,

sol = LinearHomotopyFR@F, G, X, X0, Γ, 1000, ΛD;sol@@1DD �� NumberForm@ð, 16D &

994.157067373572651 ´ 106, 671429.8556776534, 4.774880675934527 ´ 106==

12 ALESS_06.nb

algebraic least square solution...

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