function [SOLS, err] = solve32( u, PP  , select_polysolve) 
%function [SOLS, err] = solve32( u, PP , select_polysolve) 
%
%   WARNING!
% This code depends heavily on the maple link being functional. Without 
% maple there will be failure. The code is also very far from being optimal. 
%
% u is the matrix containing the observations.  
% row 1 is camera-id 
% row 2 is time-id 
% row 3 is also camera id. 
% row 4-6 is the observed image point. 
%
% SOLS contains a1,b1,a2,b2 so that [ai,bi;-bi ai] gives rotation 
%  matrix i. 
%
% By setting the select_polysolve variable the strategy for solving the 
%  polynomial is changed. 
%
% It should not be very hard to build a better solver than this one. 
%
%
%
%
%%%%  EXAMPLE CODE
%clear all
%
%Create a random configuration of vehicle stations, space points and vehicle relative cameras.
%[T,X,P] = planar_mov_randconf( 3,2 );
%a1 = T{2}(1,1);
%b1 = T{2}(1,2);
%a2 = T{3}(1,1);
%b2 = T{3}(1,2);
%
%[l,u]  = observe_plucker_as_cell( T,X,P );
%
%for kk=0:1
%  if not(kk )
%    [SOLSold, err] = solve32(  u , P );
%  else
%    [SOLSold, err] = solve32(  u , P ,1);
%  end
%  % Tar bort de l%G�%@ningar som %G�%@ f%G�%@ m%G�%@ga!
%  %
%  [dummy, I] = sortrows( abs( SOLSold')  );
%  %%
%  %% Must rotate the solutions so that we can check if they are correct.
%  %%  [ a1;b1;c1;d1; a2;b2;c2;d2; a3;b3;c3;d3; z1;y2;z2];
%
%  for i=1:size(SOLSold,2)
%     R0=[SOLSold(1,i) SOLSold(2,i); -SOLSold(2,i) SOLSold(1,i)  ] ;
%     R1=[SOLSold(5,i) SOLSold(6,i); -SOLSold(6,i) SOLSold(5,i)  ] ;
%     R2=[SOLSold(9,i) SOLSold(10,i); -SOLSold(10,i) SOLSold(9,i)  ] ;
%     R1 = R1*inv(R0);
%     R2 = R2*inv(R0);
%     S2(:,i) = [R1(1,1); R2(1,1); R1(1,2);R2(1,2)];
%  end
%  SOLSold = S2;
%  terr = SOLSold - ([a1;a2;b1;b2]*ones(1,size(S2,2)));
%
%  if not(kk )
%    err_old = min(sum(abs(terr)));
%  else
%    err_new = min(sum(abs(terr)));
%  end
%
%end
%disp( [err_old err_new])
%
%If using this code, please refer to any of the two publications below. 
%
%@PHDTHESIS{stewenius-pdh-2005,
%  AUTHOR = {H. Stew\'enius},
%  TITLE = {Gr{\"o}bner Basis Methods for Minimal Problems in Computer Vision},
%  SCHOOL = {Lund University},
%  YEAR = 2005,
%  MONTH = APR,
%  PDF = {http://www.maths.lth.se/matematiklth/personal/stewe/THESIS/stewenius_phd_2005.pdf}
%}
%
%
%@INPROCEEDINGS{stewenius-astrom-eccv-2004,
%  AUTHOR = {H. Stew\'enius and K. {\AA}str{\"o}m},
%  TITLE = {Structure and Motion Problems for Multiple Rigidly Moving Cameras},
%  BOOKTITLE = {European Conference on Computer Vision (ECCV)},
%  YEAR = 2004,
%  MONTH = MAY,
%  ADDRESS = {Prague , Czech Republic},
%  PAGES = {238ff},
%  NOTE = {An improved version of this in Chapter 9 of my thesis}
%}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



for i=1:2
   for j=1:3
      for k=1:4
	 eval( sprintf( 'p%i%i%i = PP{%i}(%i,%i);\n', i,j,k,i,j,k) ) ;
      end
   end
end


for uu =u
   trackid = uu(1);
   timeid = uu(2);
   camid = uu(3);
   uu = uu(4:end); 
   eval( sprintf( 'u%i%i1 = uu(1);\n',trackid, timeid ));  
   eval( sprintf( 'u%i%i2 = uu(2);\n',trackid, timeid ));  
end

% Här kommer block som kommer från 
%   create_buildAB_code.m
% Maskingenererat! 

A=[[0,0,0,0,0,0];[0,0,0,0,0,0];[0,0,0,0,0,0];[p212,0,0,p211,0,0];[p222,0,0,p221,0,0];[p232,0,0,p231,0,0];[0,0,0,0,0,0];[0,0,0,0,0,0];[0,0,0,0,0,0];[0,p212,0,0,p211,0];[0,p222,0,0,p221,0];[0,p232,0,0,p231,0];[0,0,0,0,0,0];[0,0,0,0,0,0];[0,0,0,0,0,0];[0,0,p212,0,0,p211];[0,0,p222,0,0,p221];[0,0,p232,0,0,p231]];
B=[[p113,0,p111,0,0,p112,0,0,-1.*u111,0,0,0,0,0,p114];[p123,0,p121,0,0,p122,0,0,-1.*u112,0,0,0,0,0,p124];[p133,0,p131,0,0,p132,0,0,-1.,0,0,0,0,0,p134];[0,p213,p211,0,0,p212,0,0,0,0,0,-1.*u211,0,0,p214];[0,p223,p221,0,0,p222,0,0,0,0,0,-1.*u212,0,0,p224];[0,p233,p231,0,0,p232,0,0,0,0,0,-1.,0,0,p234];[p113,0,0,p111,0,0,p112,0,0,-1.*u121,0,0,0,0,p114];[p123,0,0,p121,0,0,p122,0,0,-1.*u122,0,0,0,0,p124];[p133,0,0,p131,0,0,p132,0,0,-1.,0,0,0,0,p134];[0,p213,0,p211,0,0,p212,0,0,0,0,0,-1.*u221,0,p214];[0,p223,0,p221,0,0,p222,0,0,0,0,0,-1.*u222,0,p224];[0,p233,0,p231,0,0,p232,0,0,0,0,0,-1.,0,p234];[p113,0,0,0,p111,0,0,p112,0,0,-1.*u131,0,0,0,p114];[p123,0,0,0,p121,0,0,p122,0,0,-1.*u132,0,0,0,p124];[p133,0,0,0,p131,0,0,p132,0,0,-1.,0,0,0,p134];[0,p213,0,0,p211,0,0,p212,0,0,0,0,0,-1.*u231,p214];[0,p223,0,0,p221,0,0,p222,0,0,0,0,0,-1.*u232,p224];[0,p233,0,0,p231,0,0,p232,0,0,0,0,0,-1.,p234]];

% slut block




N = [];
h1 = [1 2 1];
h2 = [2 7 7];
for var = 1:6
   for i=1:3
      hole1 = h1(i);
      hole2 = h2(i);
      I=[1:(hole1-1) (hole1+1):(hole2-1) (hole2+1):18];
      N(i,var) = det( [B(I,1) A(I,var) B(I,2:end)]);
   end
end

vec = null(N);


N = inv(N(1:3,1:3))*N;
N(1:3,1:3) = eye(3);

syms b1 b2 b3 
bvec = [b1;b2;b3];

a1 = -N(1,4:end)*bvec;
a2 = -N(2,4:end)*bvec;
a3 = -N(3,4:end)*bvec;


eq = [maple('expand', a1^2+b1^2-1  );
      maple('expand', a2^2+b2^2-1  );
      maple('expand', a3^2+b3^2-1  );];

MONS2 = [2  1  0  1  0  0  1  0  0  0
	 0  1  2  0  1  0  0  1  0  0
	 0  0  0  1  1  2  0  0  1  0];
	 

mcoeff=inline(['maple(''coeff'', '...
               'maple(''coeff'', '...
               'maple(''coeff'', '...
               'f,''b1'',n(1)),''b2'',n(2)),''b3'',n(3))'] ...
              ,'f','n');

M = zeros( 3,10);
for i=1:3
   for j=1:10
      M(i,j) = eval( mcoeff( eq(i), MONS2(:,j)));
   end
end



M = inv(M(:,1:3))*M;
M(:,1:3) = eye(3);



MONS3 = [3 2 1 0 2 1 0 1 0 0 2 1 0 1 0 0 1 0 0 0
0 1 2 3 0 1 2 0 1 0 0 1 2 0 1 0 0 1 0 0
	   0 0 0 0 1 1 1 2 2 3 0 0 0 1 1 2 0 0 1 0]; 

MONS4 = [4 3 2 1 0 3 2 1 0 2 1 0 1 0 0 3 2 1 0 2 1 0 1 0 0 2 1 0 1 0 0 1 0 0 0
0 1 2 3 4 0 1 2 3 0 1 2 0 1 0 0 1 2 3 0 1 2 0 1 0 0 1 2 0 1 0 0 1 0 0 
  0 0 0 0 0 1 1 1 1 2 2 2 3 3 4 0 0 0 0 1 1 1 2 2 3 0 0 0 1 1 2 0 0 1 0 ]; 


M23ett = build_multiplication_matrix( MONS2,MONS3, [0 0 0] );
M23b1 = build_multiplication_matrix( MONS2,MONS3, [1 0 0] );
M23b2 = build_multiplication_matrix( MONS2,MONS3, [0 1 0] );
M23b3 = build_multiplication_matrix( MONS2,MONS3, [0 0 1] );

M34ett = build_multiplication_matrix( MONS3,MONS4, [0 0 0] );
M34b1 = build_multiplication_matrix( MONS3,MONS4, [1 0 0] );
M34b2 = build_multiplication_matrix( MONS3,MONS4, [0 1 0] );
M34b3 = build_multiplication_matrix( MONS3,MONS4, [0 0 1] );


M2 = [M*M23ett
M*M23b1
M*M23b2
M*M23b3 ];


%   M2([4 5 6 9:end 1:3 7 8 ],: )

M2 = inv( M2(:,[1:9 11 12 13] ))*M2;
M2(:,[1:9 11 12 13] )= eye(12);



M4 = [M2*M34ett
M2*M34b1
M2*M34b2
M2*M34b3 ];

M4 = (M4(:,[1:24 26:28] ))\M4;
M4(:,[1:24 26:28] ) = eye(27);


outher_points = MONS4(:,[1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 26 27 28]);
inner_points = MONS4(:,[25 29 30 31 32 33 34 35]);
M = [];
for p=inner_points
   p = p+[0;0;1];
   t=  ismember( inner_points' , p', 'rows' )' ;
   if any( t )
%      disp( t ) ;
         M= [M; t];
   else
      t=  ismember(outher_points' , p', 'rows' )' ;
      I = find( t );
%      disp( I );
      M = [M; -M4(I,[25 29 30 31 32 33 34 35])];
   end
end






syms alpha1 alpha2 alpha3
r =  vec*[alpha1;alpha2;alpha3];
r= maple('evalf', r );
p = [r(1)^2+r(4)^2-1;
     r(2)^2+r(5)^2-1;
     r(3)^2+r(6)^2-1;];
if exist( 'select_polysolve' )
   alpha = polysolve222_2( p );
else
   alpha = polysolve222( p );
end


%maple( 'solve', ['{' char( p(1)) ','  char(p(2)) ',' char(p(3))  '}' ] )

ab_sols = vec*alpha';


% ab_sols = ab_sols(:, find( not(imag(ab_sols(1,:))) ));


err=[];
ab_sols;
for  i=1:size( ab_sols,2)   
   AA = A*ab_sols(:,i);
   MMM = [B(:,1) AA B(:,2:end)];


   rest_sols = null(MMM );
   rest_sols = rest_sols/rest_sols(end);

   a1= ab_sols(1,i);
   a2= ab_sols(2,i);
   a3= ab_sols(3,i);
   b1= ab_sols(4,i);
   b2= ab_sols(5,i);
   b3= ab_sols(6,i);
   z1 = rest_sols(1 );
   y2 = rest_sols(2 );
   z2 = rest_sols(3 );
   c1 = rest_sols(4 );
   c2 = rest_sols(5 );
   c3 = rest_sols(6 );
   d1 = rest_sols(7 );
   d2 = rest_sols(8 );
   d3 = rest_sols(9 );
   lambda11 = rest_sols(10 );
   lambda12 = rest_sols(11 );
   lambda13 = rest_sols(12 );
   lambda21 = rest_sols(13 );
   lambda22 = rest_sols(14);
   lambda23 = rest_sols(15 );

   T1 = [a1 b1 0 c1;-b1 a1 0 d1 ; 0 0 1 0 ; 0 0 0 1];
   T2 = [a2 b2 0 c2;-b2 a2 0 d2 ; 0 0 1 0 ; 0 0 0 1];
   T3 = [a3 b3 0 c3;-b3 a3 0 d3 ; 0 0 1 0 ; 0 0 0 1];
   T= {T1, T2,T3};


   X1 = [0;0;z1;1];
   X2 = [0;y2;z2;1];

   lambda{1} = {lambda11, lambda12, lambda13 };
   lambda{2} = {lambda21, lambda22, lambda23 };

   X= {X1, X2};
   M= []*b1;
   for uu =u
      trackid = uu(1);
      timeid = uu(2);
      camid = uu(3);
      uu = uu(4:end);
      t =  PP{camid}*T{timeid}*X{trackid}- lambda{trackid}{timeid}*uu ;
      M= [M;t];
   end
   
   err = [err  norm(M ) ];
   SOLS(:,i) = [ a1;b1;c1;d1; a2;b2;c2;d2; a3;b3;c3;d3; z1;y2;z2];
end

if norm( err)  > 1e-10
   err
   warning('Very large residuals in solve32.m' ) 
end



function alpha = polysolve222( p ) 

coeff = inline('maple(''coeff'',f,x)','f','x');
if 0 
   load vec.mat 
   syms alpha1 alpha2 alpha3
   r =  vec*[alpha1;alpha2;alpha3];
   r= maple('evalf', r );
   p = [r(1)^2+r(4)^2-1;
	r(2)^2+r(5)^2-1;
	r(3)^2+r(6)^2-1;];
end

for i=1:3
   p(i) = maple('expand', p(i) );
end


q11 = eval(coeff( p(1) , 'alpha1^2'));
q12 = eval(coeff( p(1) , 'alpha2^2'));
q13 = eval(coeff( p(1) , 'alpha3^2'));
q14 = eval(coeff(coeff(p(1), 'alpha1'), 'alpha2'));
q15 = eval(coeff(coeff(p(1), 'alpha1'), 'alpha3'));
q16 = eval(coeff(coeff(p(1), 'alpha2'), 'alpha3'));

q21 = eval(coeff( p(2) , 'alpha1^2'));
q22 = eval(coeff( p(2) , 'alpha2^2'));
q23 = eval(coeff( p(2) , 'alpha3^2'));
q24 = eval(coeff(coeff(p(2), 'alpha1'), 'alpha2'));
q25 = eval(coeff(coeff(p(2), 'alpha1'), 'alpha3'));
q26 = eval(coeff(coeff(p(2), 'alpha2'), 'alpha3'));

q31 = eval(coeff( p(3) , 'alpha1^2'));
q32 = eval(coeff( p(3) , 'alpha2^2'));
q33 = eval(coeff( p(3) , 'alpha3^2'));
q34 = eval(coeff(coeff(p(3), 'alpha1'), 'alpha2'));
q35 = eval(coeff(coeff(p(3), 'alpha1'), 'alpha3'));
q36 = eval(coeff(coeff(p(3), 'alpha2'), 'alpha3'));




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23,q26,q22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,q25,q24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,q21,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0];[q23,q26,q22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,q25,q24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,q21,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0];[0,q23,q26,q22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,q25,q24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,q21,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0];[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,q23,q26,q22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,q25,q24,0,0,0,0,0,0,0,0,0,0,0,0,0,q21,0,-1,0,0,0,0,0];[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,q23,q26,q22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,q25,q24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,q21,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0];[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,q23,q26,q22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,q25,q24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,q21,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0];[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,q23,q26,q22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,q25,q24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,q21,0,-1,0,0,0,0,0,0];[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,q33,q36,q32,0,0,0,0,0,0,0,0,0,0,0,0,q35,q34,0,0,0,0,0,0,0,0,q31,0,-1];[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,q33,q36,q32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,q35,q34,0,0,0,0,0,0,0,0,0,0,0,0,0,0,q31,0,0,0,-1,0,0,0,0,0,0,0,0,0,0];[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,q33,q36,q32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,q35,q34,0,0,0,0,0,0,0,0,0,0,0,0,0,0,q31,0,0,0,-1,0,0,0,0,0,0,0,0,0];[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,q33,q36,q32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,q35,q34,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,q31,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0];[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,q33,q36,q32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,q35,q34,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,q31,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0];[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,q33,q36,q32,0,0,0,0,0,0,0,0,0,0,0,0,0,q35,q34,0,0,0,0,0,0,0,0,0,q31,0,-1,0];[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,q33,q36,q32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,q35,q34,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,q31,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0];[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,q33,q36,q32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,q35,q34,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,q31,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0];[0,0,0,0,0,0,0,0,0,0,0,0,q33,q36,q32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,q35,q34,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,q31,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0];[0,0,0,0,0,0,0,0,0,0,0,0,0,q33,q36,q32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,q35,q34,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,q31,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0];[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,q33,q36,q32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,q35,q34,0,0,0,0,0,0,0,0,0,0,q31,0,-1,0,0];[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,q33,q36,q32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,q35,q34,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,q31,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0];[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,q33,q36,q32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,q35,q34,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,q31,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0];[0,0,0,0,0,0,0,0,q33,q36,q32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,q35,q34,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,q31,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0];[0,0,0,0,0,0,0,0,0,q33,q36,q32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,q35,q34,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,q31,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0];[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,q33,q36,q32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,q35,q34,0,0,0,0,0,0,0,0,0,0,0,q31,0,-1,0,0,0];[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,q33,q36,q32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,q35,q34,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,q31,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0];[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,q33,q36,q32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,q35,q34,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,q31,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0];[0,0,0,0,q33,q36,q32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,q35,q34,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,q31,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0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[l,u] = lu(M ) ;
p1 = ( u(end-5, end-8:end) );
alpha1 = roots(p1);
p2 = ( u(end-6, end-9:end) );
alpha2 = -polyval( p2(2:end), alpha1)/p2(1);
p3 = ( u(end-7, end-10:end) );
alpha3 = -(p3(2)*alpha2 + polyval( p3(3:end), alpha1))/p3(1);
alpha = [alpha1 alpha2 alpha3];


if 0 
   for i=1:8
      alpha1 = alpha(i,1);
      alpha2 = alpha(i,2);
      alpha3 = alpha(i,3);
      disp( norm(eval(p)) ) 
   end
end




function alpha = polysolve222_2( p ) 


MONS2 =  [2 1 1 0 0 0 0 ;
	 0 1 0 2 1 0 0 ; 
	 0 0 1 0 1 2 0 ; ];


mcoeff = inline( 'maple(''coeff'',maple(''coeff'',maple(''coeff'',f,''alpha1'',v(1)),''alpha2'',v(2)),''alpha3'',v(3))','f','v');


M = zeros(3,7);
for i=1:3
   for j=1:7
      M(i,j) = eval( mcoeff( p(i), MONS2(:,j) ));
   end
end

MONS3 = [3 2 1 0 2 1 0 1 0 0 2 1 0 1 0 0 1 0 0 0
0 1 2 3 0 1 2 0 1 0 0 1 2 0 1 0 0 1 0 0
	   0 0 0 0 1 1 1 2 2 3 0 0 0 1 1 2 0 0 1 0]; 


MONS4 = [4 3 2 1 0 3 2 1 0 2 1 0 1 0 0 3 2 1 0 2 1 0 1 0 0 2 1 0 1 0 0 1 0 0 0
0 1 2 3 4 0 1 2 3 0 1 2 0 1 0 0 1 2 3 0 1 2 0 1 0 0 1 2 0 1 0 0 1 0 0 
  0 0 0 0 0 1 1 1 1 2 2 2 3 3 4 0 0 0 0 1 1 1 2 2 3 0 0 0 1 1 2 0 0 1 0 ]; 


m24_ett = build_multiplication_matrix( MONS2, MONS4, [0;0;0] );
m24_11 = build_multiplication_matrix( MONS2, MONS4, [2;0;0] );
m24_12 = build_multiplication_matrix( MONS2, MONS4, [1;1;0] );
m24_13 = build_multiplication_matrix( MONS2, MONS4, [1;0;1] );
m24_22 = build_multiplication_matrix( MONS2, MONS4, [0;2;0] );
m24_23 = build_multiplication_matrix( MONS2, MONS4, [0;1;1] );
m24_33 = build_multiplication_matrix( MONS2, MONS4, [0;0;2] );


M4 = [M*m24_ett ;
      M*m24_11 ;
      M*m24_12 ;
      M*m24_13 ;
      M*m24_22 ;
      M*m24_23 ;
      M*m24_33 ;];
I = find( sum( abs( M4 ) ));

M4 = M4(:,I(1:18))\M4 ;
M4(:,I(1:18)) = eye(18);


M4 = M4(:,I);
MONS4 = MONS4(:,I);


%%Nu vill jag betrakta action matrisen för alpha1^2




inner_points = MONS4(:,19:22);
outher_points = MONS4(:,1:18);
M = [];
for pp=inner_points
   pp = pp+[0;0;2];
   t=  ismember( inner_points' , pp', 'rows' )' ;
   if any( t )
      M= [M; t];
   else
      t=  ismember( outher_points' , pp', 'rows' )' ;
      I = find( t ) ;
%      disp( I )
      M = [M; -M4(I,19:22)];
   end
end

[V,D] = eig( M);
alpha3 = sqrt( transpose(diag( D)));

V = V./(ones(4,1)*V(end,:));

alpha1 = V(1,:)./alpha3;
alpha2 = V(2,:)./alpha3;

alpha = transpose( [alpha1;alpha2;alpha3]  ) ;





function M = build_multiplication_matrix( MONS_from, MONS_to, mult )
mult = mult(:)';
M =[];
for i=1:size(MONS_from,2)
   p = MONS_from(:,i)';
   M=[M ismember(  MONS_to', p+mult , 'rows' ) ];
end
M = M';