Initial Conditions for Newton Raphson Method for 9 equations 9 unknowns problem set

Hi, I am trying to solve these 9 equations for alpha1, alpha2, alpha3, ...... alpha9

-1+2*(cos(alpha1)-cos(alpha2)+cos(alpha3)-cos(alpha4)+cos(alpha5)-cos(alpha6)+cos(alpha7)-cos(alpha8)+cos(alpha9));
-1+2*(cos(5*alpha1)-cos(5*alpha2)+cos(5*alpha3)-cos(5*alpha4)+cos(5*alpha5)-cos(5*alpha6)+cos(5*alpha7)-cos(5*alpha8)+cos(5*alpha9));
-1+2*(cos(7*alpha1)-cos(7*alpha2)+cos(7*alpha3)-cos(7*alpha4)+cos(7*alpha5)-cos(7*alpha6)+cos(7*alpha7)-cos(7*alpha8)+cos(7*alpha9));
-1+2*(cos(11*alpha1)-cos(11*alpha2)+cos(11*alpha3)-cos(11*alpha4)+cos(11*alpha5)-cos(11*alpha6)+cos(11*alpha7)-cos(11*alpha8)+cos(11*alpha9));
-1+2*(cos(13*alpha1)-cos(13*alpha2)+cos(13*alpha3)-cos(13*alpha4)+cos(13*alpha5)-cos(13*alpha6)+cos(13*alpha7)-cos(13*alpha8)+cos(13*alpha9));
-1+2*(cos(17*alpha1)-cos(17*alpha2)+cos(17*alpha3)-cos(17*alpha4)+cos(17*alpha5)-cos(17*alpha6)+cos(17*alpha7)-cos(17*alpha8)+cos(17*alpha9));
-1+2*(cos(19*alpha1)-cos(19*alpha2)+cos(19*alpha3)-cos(19*alpha4)+cos(19*alpha5)-cos(19*alpha6)+cos(19*alpha7)-cos(19*alpha8)+cos(19*alpha9))
-1+2*(cos(23*alpha1)-cos(23*alpha2)+cos(23*alpha3)-cos(23*alpha4)+cos(23*alpha5)-cos(23*alpha6)+cos(23*alpha7)-cos(23*alpha8)+cos(23*alpha9))
-1+2*(cos(29*alpha1)-cos(29*alpha2)+cos(29*alpha3)-cos(29*alpha4)+cos(29*alpha5)-cos(29*alpha6)+cos(29*alpha7)-cos(29*alpha8)+cos(29*alpha9));

I have used Newton Raphson Method for that, but my results don't converge. I believe the error is with the initial values of alpha1, alpha2, alpha3, ...... alpha9 that I am using. I am using random initial values as of now. All I know is that these angles alpha1 to alpha9 should be in the range of 0 to 90 degrees. I have read about using the correct initial value problem in Newton Raphson method and some articles say that "you should use a rough estimate of the root of the function, if you know it". But the problem is that I do not know the estimate root of the function.

I am attaching a 7-equations in 7 uknowns code that works (I did not find its initial values, they were given to me) and 9-equations in 9 unknowns code that does not work.

I appreciate any help in this regard. Thanks once again.

Answer 1

[Torsten]

It's the same behaviour with "fsolve", MATLAB's nonlinear solver.

The seven equation problem is solved without problems, and the solver fails more or less for the nine equation problem.

Maybe it has no solution - who knows.

m1 = 0:0.01:1.15;
alpha1 = 2*pi/180; 
alpha2 = 16*pi/180;  
alpha3 = 28*pi/180;
alpha4 = 31*pi/180;
alpha5 = 43*pi/180;
alpha6 = 61*pi/180; 
alpha7 = 73*pi/180; 
alpha0 = [alpha1;alpha2;alpha3;alpha4;alpha5;alpha6;alpha7];
sol_alpha = zeros(numel(m1),7);
error = zeros(numel(m1),1);
for i = 1:numel(m1)
  T = [m1(i)*pi/4; 0; 0; 0; 0; 0; 0];
  F = @(alpha1,alpha2,alpha3,alpha4,alpha5,alpha6,alpha7)...
     [-1+2*(cos(alpha1)-cos(alpha2)+cos(alpha3)-cos(alpha4)+cos(alpha5)-cos(alpha6)+cos(alpha7));
      -1+2*(cos(5*alpha1)-cos(5*alpha2)+cos(5*alpha3)-cos(5*alpha4)+cos(5*alpha5)-cos(5*alpha6)+cos(5*alpha7));
      -1+2*(cos(7*alpha1)-cos(7*alpha2)+cos(7*alpha3)-cos(7*alpha4)+cos(7*alpha5)-cos(7*alpha6)+cos(7*alpha7));
      -1+2*(cos(11*alpha1)-cos(11*alpha2)+cos(11*alpha3)-cos(11*alpha4)+cos(11*alpha5)-cos(11*alpha6)+cos(11*alpha7));
      -1+2*(cos(13*alpha1)-cos(13*alpha2)+cos(13*alpha3)-cos(13*alpha4)+cos(13*alpha5)-cos(13*alpha6)+cos(13*alpha7));
      -1+2*(cos(17*alpha1)-cos(17*alpha2)+cos(17*alpha3)-cos(17*alpha4)+cos(17*alpha5)-cos(17*alpha6)+cos(17*alpha7));
      -1+2*(cos(19*alpha1)-cos(19*alpha2)+cos(19*alpha3)-cos(19*alpha4)+cos(19*alpha5)-cos(19*alpha6)+cos(19*alpha7))];
  F = @(alpha1,alpha2,alpha3,alpha4,alpha5,alpha6,alpha7)...
        F(alpha1,alpha2,alpha3,alpha4,alpha5,alpha6,alpha7) - T;
  F = @(alpha) F(alpha(1),alpha(2),alpha(3),alpha(4),alpha(5),alpha(6),alpha(7));
  sol_alpha(i,:) = lsqnonlin(F,alpha0,zeros(7,1),pi/2*ones(7,1),[],[],[],[],[],optimset('Display','none','TolFun',1e-16,'TolX',1e-16));
  alpha0 = sol_alpha(i,:);
  error(i) = norm(F(sol_alpha(i,:)));
end
figure(1)
plot(m1,180/pi*sol_alpha)

alpha1=2*pi/180; 
alpha2=16*pi/180; 
alpha3=28*pi/180; 
alpha4=31*pi/180;
alpha5=43*pi/180;
alpha6=61*pi/180; 
alpha7=73*pi/180; 
alpha8=81*pi/180;
alpha9=88*pi/180; 
alpha0 = [alpha1;alpha2;alpha3;alpha4;alpha5;alpha6;alpha7;alpha8;alpha9];
sol_alpha = zeros(numel(m1),9);
error = zeros(numel(m1),1);
for i = 1:numel(m1)
  T = [m1(i)*pi/4; 0; 0; 0; 0; 0; 0; 0; 0];
  F = @(alpha1,alpha2,alpha3,alpha4,alpha5,alpha6,alpha7,alpha8,alpha9)...
       [-1+2*(cos(alpha1)-cos(alpha2)+cos(alpha3)-cos(alpha4)+cos(alpha5)-cos(alpha6)+cos(alpha7)-cos(alpha8)+cos(alpha9));
        -1+2*(cos(5*alpha1)-cos(5*alpha2)+cos(5*alpha3)-cos(5*alpha4)+cos(5*alpha5)-cos(5*alpha6)+cos(5*alpha7)-cos(5*alpha8)+cos(5*alpha9));
        -1+2*(cos(7*alpha1)-cos(7*alpha2)+cos(7*alpha3)-cos(7*alpha4)+cos(7*alpha5)-cos(7*alpha6)+cos(7*alpha7)-cos(7*alpha8)+cos(7*alpha9));
        -1+2*(cos(11*alpha1)-cos(11*alpha2)+cos(11*alpha3)-cos(11*alpha4)+cos(11*alpha5)-cos(11*alpha6)+cos(11*alpha7)-cos(11*alpha8)+cos(11*alpha9));
        -1+2*(cos(13*alpha1)-cos(13*alpha2)+cos(13*alpha3)-cos(13*alpha4)+cos(13*alpha5)-cos(13*alpha6)+cos(13*alpha7)-cos(13*alpha8)+cos(13*alpha9));
        -1+2*(cos(17*alpha1)-cos(17*alpha2)+cos(17*alpha3)-cos(17*alpha4)+cos(17*alpha5)-cos(17*alpha6)+cos(17*alpha7)-cos(17*alpha8)+cos(17*alpha9));
        -1+2*(cos(19*alpha1)-cos(19*alpha2)+cos(19*alpha3)-cos(19*alpha4)+cos(19*alpha5)-cos(19*alpha6)+cos(19*alpha7)-cos(19*alpha8)+cos(19*alpha9))
        -1+2*(cos(23*alpha1)-cos(23*alpha2)+cos(23*alpha3)-cos(23*alpha4)+cos(23*alpha5)-cos(23*alpha6)+cos(23*alpha7)-cos(23*alpha8)+cos(23*alpha9))
        -1+2*(cos(29*alpha1)-cos(29*alpha2)+cos(29*alpha3)-cos(29*alpha4)+cos(29*alpha5)-cos(29*alpha6)+cos(29*alpha7)-cos(29*alpha8)+cos(29*alpha9))];
  F = @(alpha1,alpha2,alpha3,alpha4,alpha5,alpha6,alpha7,alpha8,alpha9)...
        F(alpha1,alpha2,alpha3,alpha4,alpha5,alpha6,alpha7,alpha8,alpha9) - T;
  F = @(alpha) F(alpha(1),alpha(2),alpha(3),alpha(4),alpha(5),alpha(6),alpha(7),alpha(8),alpha(9));
  sol_alpha(i,:) = lsqnonlin(F,alpha0,zeros(9,1),pi/2*ones(9,1),[],[],[],[],[],optimset('Display','none','TolFun',1e-16,'TolX',1e-16));
  alpha0 = sol_alpha(i,:);
  error(i) = norm(F(sol_alpha(i,:)));
end
figure(2)
plot(m1,180/pi*sol_alpha)

[Haleema Qamar]

Thanks @Torsten for your response. I am bit lost here. The code snippet that you just added shows that the solution does exist for 9-equation system too. But my query was about the initial values. I know that the solution exists for 9-equation system too, but I am stuck at using the correct initial values that will actually give me a solution using Newton Raphson method.

Also, this code snippet does not work for me. Attached is the error I am getting. Thanks.

[Torsten]

If you plot the "error" array of the 9-equations system, you will see that the values "fsolve" returns are not solutions for your equations. So I can neither give you good initial values nor a solution.

@Alex Sha gave you three solutions for your 9-equations system with m1 = 0. He uses a different software (not MATLAB), and my guess is that he obtained them by trial-and-error.

Answer 2

[John D'Errico]

1. Not every system of equations you will write down has a solution.
2. Starting from a random position does not insure you will find a solution. In fact, for complex problems, it may often fail.

Think of an optimizer as putting a blind person down at some place on earth, and asking them to find the location on the surface of the earth with the lowest elevation. All they are given is a cane, to determine a direction to walk that is downhill, and an altimeter (with audio output I hope.)

Would you expect that setting this person down in some random location would let them find the lowest spot on earth? Or, would they likely find themselves stuck in a bad place? That is, unable to move in any direction, because locally, they need to walk uphill?

That is effectively what you are doing when you provide random starting values to the solver.

Is there any magic way to automatically choose good starting values? Of course not. In fact, that is impossible to do for a completely general nonlinear system of equations. This is why there are tools in the Global Optimization toolbox, to try to solve difficult problems where no intelligent starting values are available.

[Sam Chak]

Think of an optimizer as putting a blind person down at some place on earth, and asking them to find the location on the surface of the earth with the lowest elevation.

This analogy may give @Haleema Qamar hope that by deploying numerous scouts across the finite surface of the Earth and allowing sufficient time, one of them will eventually discover the location with the lowest elevation (Challenger Deep). It brings to mind the discovery of the Mariana Trench in 1875 by the Royal Navy's HMS Challenger, which had been specifically commissioned to map the ocean floor.

I might say to her, 'Imagine an optimizer as instructing the most skilled astronomers and astrophysicists equipped with state-of-the-art space telescopes to locate the center of the universe.' This analogy allows her to grasp the incomprehensible vastness of some optimization problems.

[Alex Sha]

For 9-system equations, there are multi-solutions:

1:

alpha1: 1.5707963267949
alpha2: 0.840170695600121
alpha3: 1.0471975511966
alpha4: 1.42686235579836
alpha5: 0.734020226616598
alpha6: 0.667532746594836
alpha7: 0.840170695600122
alpha8: 0.734020226616601
alpha9: 0.379664804601759

2:

alpha1: 1.0471975511966
alpha2: 1.5207762324454
alpha3: 0.301970967263422
alpha4: 0.981148728099019
alpha5: 0.981148728099019
alpha6: 0.301970967263422
alpha7: 1.5207762324454
alpha8: 0.278203493864695
alpha9: 0.278203493864694

3:

alpha1: 0.375776265922601
alpha2: 0.375776265922601
alpha3: 1.0471975511966
alpha4: 0.610961676586485
alpha5: 0.610961676586486
alpha6: 1.46036667264309
alpha7: 1.46036667264309
alpha8: 0.406282347040722
alpha9: 0.406282347040722

...

[Haleema Qamar]

Thanks @Alex Sha.

Could you please let me know how you obtained these three sets of solutions for 9-equation system?

Also, are you sure these values are in correct order? I was expecting alpha1 to be the smallest angle and other angles increasing subsequently, with alpha9 being the largest angle. I do not see that pattern here.

Thanks once again.

[Sam Chak]

I assume you already possess the mathematical knowledge required to test the solutions. Could you please test the third set of solutions?

F = @(alpha1,alpha2,alpha3,alpha4,alpha5,alpha6,alpha7,alpha8,alpha9)...
       [-1+2*(cos(alpha1)-cos(alpha2)+cos(alpha3)-cos(alpha4)+cos(alpha5)-cos(alpha6)+cos(alpha7)-cos(alpha8)+cos(alpha9));
        -1+2*(cos(5*alpha1)-cos(5*alpha2)+cos(5*alpha3)-cos(5*alpha4)+cos(5*alpha5)-cos(5*alpha6)+cos(5*alpha7)-cos(5*alpha8)+cos(5*alpha9));
        -1+2*(cos(7*alpha1)-cos(7*alpha2)+cos(7*alpha3)-cos(7*alpha4)+cos(7*alpha5)-cos(7*alpha6)+cos(7*alpha7)-cos(7*alpha8)+cos(7*alpha9));
        -1+2*(cos(11*alpha1)-cos(11*alpha2)+cos(11*alpha3)-cos(11*alpha4)+cos(11*alpha5)-cos(11*alpha6)+cos(11*alpha7)-cos(11*alpha8)+cos(11*alpha9));
        -1+2*(cos(13*alpha1)-cos(13*alpha2)+cos(13*alpha3)-cos(13*alpha4)+cos(13*alpha5)-cos(13*alpha6)+cos(13*alpha7)-cos(13*alpha8)+cos(13*alpha9));
        -1+2*(cos(17*alpha1)-cos(17*alpha2)+cos(17*alpha3)-cos(17*alpha4)+cos(17*alpha5)-cos(17*alpha6)+cos(17*alpha7)-cos(17*alpha8)+cos(17*alpha9));
        -1+2*(cos(19*alpha1)-cos(19*alpha2)+cos(19*alpha3)-cos(19*alpha4)+cos(19*alpha5)-cos(19*alpha6)+cos(19*alpha7)-cos(19*alpha8)+cos(19*alpha9))
        -1+2*(cos(23*alpha1)-cos(23*alpha2)+cos(23*alpha3)-cos(23*alpha4)+cos(23*alpha5)-cos(23*alpha6)+cos(23*alpha7)-cos(23*alpha8)+cos(23*alpha9))
        -1+2*(cos(29*alpha1)-cos(29*alpha2)+cos(29*alpha3)-cos(29*alpha4)+cos(29*alpha5)-cos(29*alpha6)+cos(29*alpha7)-cos(29*alpha8)+cos(29*alpha9))];

%% Test Solution 1
alpha1  = 1.5707963267949;
alpha2  = 0.840170695600121;
alpha3  = 1.0471975511966;
alpha4  = 1.42686235579836;
alpha5  = 0.734020226616598;
alpha6  = 0.667532746594836;
alpha7  = 0.840170695600122;
alpha8  = 0.734020226616601;
alpha9  = 0.379664804601759;
F1      = F(alpha1, alpha2, alpha3, alpha4, alpha5, alpha6, alpha7, alpha8, alpha9)

F1 = 9x1

1.0e-12 * -0.0056 0.0084 0.0024 0.1334 -0.1975 -0.0837 0.2796 0.2160 -0.1029

%% Test Solution 2
alpha1  = 1.0471975511966;
alpha2  = 1.5207762324454;
alpha3  = 0.301970967263422;
alpha4  = 0.981148728099019;
alpha5  = 0.981148728099019;
alpha6  = 0.301970967263422;
alpha7  = 1.5207762324454;
alpha8  = 0.278203493864695;
alpha9  = 0.278203493864694;
F2      = F(alpha1, alpha2, alpha3, alpha4, alpha5, alpha6, alpha7, alpha8, alpha9)

F2 = 9x1

1.0e-12 * -0.0038 0.0298 -0.0149 0.0466 -0.0639 0.0349 -0.1097 0.0986 0.1725

[Haleema Qamar]

Thanks. For third solution set,

F3 =

1.0e-12 *

-0.0053

0.0191

-0.0158

0.0360

-0.0789

0.0977

-0.0446

0.0469

0.1685

I am new to this method, so I am still trying to figure out what that means. Also, I wanted to know how these three sets of initial values were calculated? And which one is the best one? Thanks

[Sam Chak]

OP: And which one is the best one?

Solutions are regarded as "solutions" in the eyes of pure mathematicians, without any specific preference given. However, the significance lies in understanding how the equation with multiple cosine functions is applicable to your specific research field. Only you possess the knowledge to interpret the solution sets, as the description of your problem does not provide any context.

If I may take a guess, this problem might be related to a 9-degree-of-freedom robotic arm that exhibits three sets of solution angles.

[Haleema Qamar]

@Sam Chak, not really, the problem at hand does not revolve around the robotic arm. Actually, I am trying to eliminiate 9 harmonics from inverter output waveform, for which I need nine angles that would come from the equations above (nine equations in nine unknowns alpha1 to alpha9). There was some data available online from MATLAB community here that I followed for the 7 angle problem, but the same thing is not working for the nine angle problem. I believe I am not using the good initial values for alpha1, alpha2, alpha3,.... alpha9.

[Alex Sha]

@Torsten: no need for "trial-and-error", The code demonstrated below is all the user needs to do

@Haleema Qamar : if you want alpha1 to be the smallest angle and other angles increasing subsequently, with alpha9 being the largest angle, there will be still multi-solutions, even in the range of [0,10]:

Parameter alpha(9)=[0,10];
SubjectTo For(i=1:8)(alpha[i]<=alpha[i+1]);
Function
-1+2*(cos(alpha1)-cos(alpha2)+cos(alpha3)-cos(alpha4)+cos(alpha5)-cos(alpha6)+cos(alpha7)-cos(alpha8)+cos(alpha9));
-1+2*(cos(5*alpha1)-cos(5*alpha2)+cos(5*alpha3)-cos(5*alpha4)+cos(5*alpha5)-cos(5*alpha6)+cos(5*alpha7)-cos(5*alpha8)+cos(5*alpha9));
-1+2*(cos(7*alpha1)-cos(7*alpha2)+cos(7*alpha3)-cos(7*alpha4)+cos(7*alpha5)-cos(7*alpha6)+cos(7*alpha7)-cos(7*alpha8)+cos(7*alpha9));
-1+2*(cos(11*alpha1)-cos(11*alpha2)+cos(11*alpha3)-cos(11*alpha4)+cos(11*alpha5)-cos(11*alpha6)+cos(11*alpha7)-cos(11*alpha8)+cos(11*alpha9));
-1+2*(cos(13*alpha1)-cos(13*alpha2)+cos(13*alpha3)-cos(13*alpha4)+cos(13*alpha5)-cos(13*alpha6)+cos(13*alpha7)-cos(13*alpha8)+cos(13*alpha9));
-1+2*(cos(17*alpha1)-cos(17*alpha2)+cos(17*alpha3)-cos(17*alpha4)+cos(17*alpha5)-cos(17*alpha6)+cos(17*alpha7)-cos(17*alpha8)+cos(17*alpha9));
-1+2*(cos(19*alpha1)-cos(19*alpha2)+cos(19*alpha3)-cos(19*alpha4)+cos(19*alpha5)-cos(19*alpha6)+cos(19*alpha7)-cos(19*alpha8)+cos(19*alpha9));
-1+2*(cos(23*alpha1)-cos(23*alpha2)+cos(23*alpha3)-cos(23*alpha4)+cos(23*alpha5)-cos(23*alpha6)+cos(23*alpha7)-cos(23*alpha8)+cos(23*alpha9));
-1+2*(cos(29*alpha1)-cos(29*alpha2)+cos(29*alpha3)-cos(29*alpha4)+cos(29*alpha5)-cos(29*alpha6)+cos(29*alpha7)-cos(29*alpha8)+cos(29*alpha9));

Result-1:

alpha1: 0.904864015998845
alpha2: 0.904864015998846
alpha3: 1.0471975511966
alpha4: 1.28107290992584
alpha5: 1.28107290992584
alpha6: 2.57604127845461
alpha7: 3.70714402872498
alpha8: 3.78084219090445
alpha9: 3.78084219090445

Result-2:

alpha1: 1.28942939221469
alpha2: 2.44302192019987
alpha3: 2.79296583578312
alpha4: 3.01134262414758
alpha5: 3.38382449460789
alpha6: 4.18879020478639
alpha7: 7.08815101735809
alpha8: 8.02895359176611
alpha9: 9.29452793132717

Result-3:

alpha1: 3.51243597107882
alpha2: 3.59682914661921
alpha3: 3.89638229542025
alpha4: 4.18879020478639
alpha5: 5.52839566534913
alpha6: 5.77201268279289
alpha7: 5.91234198969055
alpha8: 6.73842180020901
alpha9: 6.79435793156629

Result-4:

alpha1: 1.0471975511966
alpha2: 1.7417664625754
alpha3: 4.54141884460419
alpha4: 6.12717894980294
alpha5: 6.32942168913572
alpha6: 6.32942168913572
alpha7: 6.43919166455624
alpha8: 6.72447132594417
alpha9: 6.72447132594417

....

[Torsten]

@Alex Sha

Another condition was 0 <= alpha <= pi/2.

Seems there is no solution satisfying this condition, is it ?

[Alex Sha]

There are still multi-solutions, even in the range of [0, pi/2], and satisfy the conditions of alpha1<alpha2...<alpha9:

1:

alpha1: 3.91624697106773E-8
alpha2: 0.180888491664553
alpha3: 0.180888491664577
alpha4: 0.346063517935811
alpha5: 0.160265702458496
alpha6: 0.416265628990047
alpha7: 0.630931922206561
alpha8: 1.0471975511966
alpha9: 1.46346318018664

2:

alpha1: 0.203193493585978
alpha2: 0.369672232654069
alpha3: 0.677525318542554
alpha4: 0.844004057610605
alpha5: 1.04719755119661
alpha6: 1.25039104470494
alpha7: 1.25039551959259
alpha8: 1.25039551967021
alpha9: 1.41686978385068

3:

alpha1: 1.19157084434692E-8
alpha2: 0.442501414173056
alpha3: 0.604696137023543
alpha4: 1.0471975511966
alpha5: 1.18529705757435
alpha6: 1.18529705757436
alpha7: 1.34499213444553
alpha8: 1.34499213444553
alpha9: 1.48969896536965

4:

alpha1: 3.47228539110217E-8
alpha2: 0.32285578727148
alpha3: 0.563928125697787
alpha4: 0.563928125697789
alpha5: 0.724341763925112
alpha6: 0.780892412945105
alpha7: 0.780892412945107
alpha8: 1.04719755119659
alpha9: 1.37005333846807

...

further try the range of [0,pi/3], still multi-solutions:

1:

alpha1: 2.45551080796824E-8
alpha2: 2.54044609352637E-8
alpha3: 0.311985704857291
alpha4: 0.311985704857292
alpha5: 0.540750687957892
alpha6: 0.540750687957893
alpha7: 0.786520503128533
alpha8: 0.786520503128533
alpha9: 1.0471975511966

2:

alpha1: 1.95732379177253E-8
alpha2: 2.1556791215656E-8
alpha3: 0.226048467057157
alpha4: 0.226048467057158
alpha5: 0.691609241228221
alpha6: 0.691609241228221
alpha7: 0.85800647331531
alpha8: 0.858006473315311
alpha9: 1.0471975511966

3:

alpha1: 4.12738820583618E-9
alpha2: 6.99668668257787E-9
alpha3: 0.305707153981926
alpha4: 0.305707153981926
alpha5: 0.545459810429233
alpha6: 0.545459810429233
alpha7: 0.841022766210289
alpha8: 0.841022766210289
alpha9: 1.0471975511966

...