Problem when trying to solve a rotation matrix with symbolic elements: (Master thesis)

Hi, I'm a master student in Robotics and I've been stuck with this problem for some weeks and I really need some help to continue my work:

So, after multiplying multiple rotation matrices, I get a final rotation matrix which is:

Final rotation matrix:
[                                                                                            sin(a)*(sin(a)^2/2 + (2^(1/2)*cos(a))/2 + (cos(a)*sin(a))/2) - cos(a)*((2^(1/2)*sin(a))/4 - (2^(1/2)*cos(a))/4 - sin(a)/2 + (2^(1/2)*cos(a)^2)/4 + (2^(1/2)*cos(a)*sin(a))/4),                                                              sin(a)/2 - (2^(1/2)*cos(a))/4 + (2^(1/2)*sin(a))/4 - (2^(1/2)*cos(a)^2)/4 - (2^(1/2)*cos(a)*sin(a))/4,                                                                                            sin(a)*((2^(1/2)*sin(a))/4 - (2^(1/2)*cos(a))/4 - sin(a)/2 + (2^(1/2)*cos(a)^2)/4 + (2^(1/2)*cos(a)*sin(a))/4) + cos(a)*(sin(a)^2/2 + (2^(1/2)*cos(a))/2 + (cos(a)*sin(a))/2)]
[                       (sin(a)^2*(cos(2*a + pi/4)*(2^(1/2) + 2) + 2^(1/2) + 2*2^(1/2)*cos(a) - 1))/4 - cos(a)*((2^(1/2)*(2^(1/2)/4 + sin(2*a + pi/4)*(2^(1/2)/4 + 1/2) - 1/4))/2 - sin(a)^2/2 + (2^(1/2)*cos(a)*(2^(1/2)/4 + cos(2*a + pi/4)*(2^(1/2)/4 + 1/2) - 1/4))/2),          sin(a)^2/2 + (2^(1/2)*(2^(1/2)/4 + sin(2*a + pi/4)*(2^(1/2)/4 + 1/2) - 1/4))/2 - (2^(1/2)*cos(a)*(2^(1/2)/4 + cos(2*a + pi/4)*(2^(1/2)/4 + 1/2) - 1/4))/2,                  sin(a)*((2^(1/2)*(2^(1/2)/4 + sin(2*a + pi/4)*(2^(1/2)/4 + 1/2) - 1/4))/2 - sin(a)^2/2 + (2^(1/2)*cos(a)*(2^(1/2)/4 + cos(2*a + pi/4)*(2^(1/2)/4 + 1/2) - 1/4))/2) + (cos(a)*sin(a)*(cos(2*a + pi/4)*(2^(1/2) + 2) + 2^(1/2) + 2*2^(1/2)*cos(a) - 1))/4]
[ sin(a)*(sin(a)*(sin(2*a + pi/4)*(2^(1/2)/4 + 1/2) - 2^(1/2)/4 + 1/4) - (2^(1/2)*cos(a)^2)/2) - cos(a)*((cos(a)*sin(a))/2 - (2^(1/2)*(cos(2*a + pi/4)*(2^(1/2)/4 + 1/2) - 2^(1/2)/4 + 1/4))/2 + (2^(1/2)*cos(a)*(sin(2*a + pi/4)*(2^(1/2)/4 + 1/2) - 2^(1/2)/4 + 1/4))/2), - (2^(1/2)*(cos(2*a + pi/4)*(2^(1/2)/4 + 1/2) - 2^(1/2)/4 + 1/4))/2 - (cos(a)*sin(a))/2 - (2^(1/2)*cos(a)*(sin(2*a + pi/4)*(2^(1/2)/4 + 1/2) - 2^(1/2)/4 + 1/4))/2, cos(a)*(sin(a)*(sin(2*a + pi/4)*(2^(1/2)/4 + 1/2) - 2^(1/2)/4 + 1/4) - (2^(1/2)*cos(a)^2)/2) + sin(a)*((cos(a)*sin(a))/2 - (2^(1/2)*(cos(2*a + pi/4)*(2^(1/2)/4 + 1/2) - 2^(1/2)/4 + 1/4))/2 + (2^(1/2)*cos(a)*(sin(2*a + pi/4)*(2^(1/2)/4 + 1/2) - 2^(1/2)/4 + 1/4))/2)]

(with (a) being a syms variable defined previously).

It may look complex but it's just a 3x3 matrix with huge elements.

I DO know that this final matrix shall be equal to the identity matrix, so I've tried to solve with things such as:

 eq_1 = X_total(1,1) == 1; %first element of the matrix == 1, as X_total should be equal to the identity matrix
  1.  [sola] = solve(eq_1);
  2.  [sola] = solve(eq_1,'IgnoreAnalyticConstraints',true);

but all ways give me solutions which look like:

    -log(root(2^(1/2)*z^6*(1 - 1i) + z^6*(2 - 2i) - 2^(1/2)*z^5*(2 - 2i) + z^5*4i + 2^(1/2)*z^4*(3 - 1i) - z^4*(2 - 6i) - 4*2^(1/2)*z^3 + 32*z^3 + 2^(1/2)*z^2*(3 + 1i) - z^2*(2 + 6i) - 2^(1/2)*z*(2 + 2i) - z*4i + 2^(1/2)*(1 + 1i) + (2 + 2i), z, 1))*1i

I also tried solving for cos(a) and sin(a) using 2 variables: syms sina and syms cosa and then using two equations to solve the system like this:

eq_1 = X_tot(1,1) - 1 == 0;
eq_2 = X_tot(2,2) - 1 == 0; %equations defined as previously
  [solacosa,solasina] = solve(eq_1,eq_2,sina,cosa);

the result is a 9x1 vector with all elements have the same value

    (1356945683269916498*2^(1/2)*root(z^9 + (z^8*(40646370540593230529675376111439*2^(1/2) + 57482648479749177740913514589635))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) - (z^7*(48950395506894713531281417549686*2^(1/2) + 69226313209377518582549939536733))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z^6*(59135621925949926270484844611882*2^(1/2) + 83630398547046153213000249050231))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z^5*(255070896824454141176498238274983*2^(1/2) + 360724721655811468397476671471853))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z^4*(69682769222483480998602050740137*2^(1/2) + 98546317298150630332891610577466))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) - (z^3*(3384262362996977766641442698518*2^(1/2) + 4786069732379144454076875275372))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z^2*(156240974119420649562191631285482*2^(1/2) + 220958104598068424057609399598508))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z*(18544628785686575814198409310776*2^(1/2) + 26226065537892456179585517114620))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) - (71753040946492451590476100638776*2^(1/2) + 101474123648041645683480942072580)/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988), z, 1)^2 + 304812207722910491*2^(1/2)*root(z^9 + (z^8*(40646370540593230529675376111439*2^(1/2) + 57482648479749177740913514589635))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) - (z^7*(48950395506894713531281417549686*2^(1/2) + 69226313209377518582549939536733))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z^6*(59135621925949926270484844611882*2^(1/2) + 83630398547046153213000249050231))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z^5*(255070896824454141176498238274983*2^(1/2) + 360724721655811468397476671471853))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z^4*(69682769222483480998602050740137*2^(1/2) + 98546317298150630332891610577466))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) - (z^3*(3384262362996977766641442698518*2^(1/2) + 4786069732379144454076875275372))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z^2*(156240974119420649562191631285482*2^(1/2) + 220958104598068424057609399598508))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z*(18544628785686575814198409310776*2^(1/2) + 26226065537892456179585517114620))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) - (71753040946492451590476100638776*2^(1/2) + 101474123648041645683480942072580)/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988), z, 1)^3 + 2717933758233735543*2^(1/2)*root(z^9 + (z^8*(40646370540593230529675376111439*2^(1/2) + 57482648479749177740913514589635))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) - (z^7*(48950395506894713531281417549686*2^(1/2) + 69226313209377518582549939536733))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z^6*(59135621925949926270484844611882*2^(1/2) + 83630398547046153213000249050231))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z^5*(255070896824454141176498238274983*2^(1/2) + 360724721655811468397476671471853))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z^4*(69682769222483480998602050740137*2^(1/2) + 98546317298150630332891610577466))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) - (z^3*(3384262362996977766641442698518*2^(1/2) + 4786069732379144454076875275372))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z^2*(156240974119420649562191631285482*2^(1/2) + 220958104598068424057609399598508))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z*(18544628785686575814198409310776*2^(1/2) + 26226065537892456179585517114620))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) - (71753040946492451590476100638776*2^(1/2) + 101474123648041645683480942072580)/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988), z, 1)^4 + 2585209551038856213*2^(1/2)*root(z^9 + (z^8*(40646370540593230529675376111439*2^(1/2) + 57482648479749177740913514589635))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) - (z^7*(48950395506894713531281417549686*2^(1/2) + 69226313209377518582549939536733))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z^6*(59135621925949926270484844611882*2^(1/2) + 83630398547046153213000249050231))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z^5*(255070896824454141176498238274983*2^(1/2) + 360724721655811468397476671471853))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z^4*(69682769222483480998602050740137*2^(1/2) + 98546317298150630332891610577466))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) - (z^3*(3384262362996977766641442698518*2^(1/2) + 4786069732379144454076875275372))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z^2*(156240974119420649562191631285482*2^(1/2) + 220958104598068424057609399598508))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z*(18544628785686575814198409310776*2^(1/2) + 26226065537892456179585517114620))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) - (71753040946492451590476100638776*2^(1/2) + 101474123648041645683480942072580)/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988), z, 1)^5 - 849637155152696990*2^(1/2)*root(z^9 + (z^8*(40646370540593230529675376111439*2^(1/2) + 57482648479749177740913514589635))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) - (z^7*(48950395506894713531281417549686*2^(1/2) + 69226313209377518582549939536733))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z^6*(59135621925949926270484844611882*2^(1/2) + 83630398547046153213000249050231))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z^5*(255070896824454141176498238274983*2^(1/2) + 360724721655811468397476671471853))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z^4*(69682769222483480998602050740137*2^(1/2) + 98546317298150630332891610577466))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) - (z^3*(3384262362996977766641442698518*2^(1/2) + 4786069732379144454076875275372))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z^2*(156240974119420649562191631285482*2^(1/2) + 220958104598068424057609399598508))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z*(18544628785686575814198409310776*2^(1/2) + 26226065537892456179585517114620))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) - (71753040946492451590476100638776*2^(1/2) + 101474123648041645683480942072580)/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988), z, 1)^6 + 795184158069056*2^(1/2)*root(z^9 + (z^8*(40646370540593230529675376111439*2^(1/2) + 57482648479749177740913514589635))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) - (z^7*(48950395506894713531281417549686*2^(1/2) + 69226313209377518582549939536733))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z^6*(59135621925949926270484844611882*2^(1/2) + 83630398547046153213000249050231))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z^5*(255070896824454141176498238274983*2^(1/2) + 360724721655811468397476671471853))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z^4*(69682769222483480998602050740137*2^(1/2) + 98546317298150630332891610577466))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) - (z^3*(3384262362996977766641442698518*2^(1/2) + 4786069732379144454076875275372))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z^2*(156240974119420649562191631285482*2^(1/2) + 220958104598068424057609399598508))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z*(18544628785686575814198409310776*2^(1/2) + 26226065537892456179585517114620))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) - (71753040946492451590476100638776*2^(1/2) + 101474123648041645683480942072580)/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988), z, 1)^7 + 881305591100011572*2^(1/2)*root(z^9 + (z^8*(40646370540593230529675376111439*2^(1/2) + 57482648479749177740913514589635))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) - (z^7*(48950395506894713531281417549686*2^(1/2) + 69226313209377518582549939536733))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z^6*(59135621925949926270484844611882*2^(1/2) + 83630398547046153213000249050231))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z^5*(255070896824454141176498238274983*2^(1/2) + 360724721655811468397476671471853))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z^4*(69682769222483480998602050740137*2^(1/2) + 98546317298150630332891610577466))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) - (z^3*(3384262362996977766641442698518*2^(1/2) + 4786069732379144454076875275372))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z^2*(156240974119420649562191631285482*2^(1/2) + 220958104598068424057609399598508))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z*(18544628785686575814198409310776*2^(1/2) + 26226065537892456179585517114620))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) - (71753040946492451590476100638776*2^(1/2) + 101474123648041645683480942072580)/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988), z, 1)^8 + 1919010988683942134*root(z^9 + (z^8*(40646370540593230529675376111439*2^(1/2) + 57482648479749177740913514589635))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) - (z^7*(48950395506894713531281417549686*2^(1/2) + 69226313209377518582549939536733))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z^6*(59135621925949926270484844611882*2^(1/2) + 83630398547046153213000249050231))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z^5*(255070896824454141176498238274983*2^(1/2) + 360724721655811468397476671471853))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z^4*(69682769222483480998602050740137*2^(1/2) + 98546317298150630332891610577466))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) - (z^3*(3384262362996977766641442698518*2^(1/2) + 4786069732379144454076875275372))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z^2*(156240974119420649562191631285482*2^(1/2) + 220958104598068424057609399598508))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z*(18544628785686575814198409310776*2^(1/2) + 26226065537892456179585517114620))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) - (71753040946492451590476100638776*2^(1/2) + 101474123648041645683480942072580)/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988), z, 1)^2 + 431069558138625080*root(z^9 + (z^8*(40646370540593230529675376111439*2^(1/2) + 57482648479749177740913514589635))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) - (z^7*(48950395506894713531281417549686*2^(1/2) + 69226313209377518582549939536733))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z^6*(59135621925949926270484844611882*2^(1/2) + 83630398547046153213000249050231))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z^5*(255070896824454141176498238274983*2^(1/2) + 360724721655811468397476671471853))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z^4*(69682769222483480998602050740137*2^(1/2) + 98546317298150630332891610577466))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) - (z^3*(3384262362996977766641442698518*2^(1/2) + 4786069732379144454076875275372))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z^2*(156240974119420649562191631285482*2^(1/2) + 220958104598068424057609399598508))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z*(18544628785686575814198409310776*2^(1/2) + 26226065537892456179585517114620))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) - (71753040946492451590476100638776*2^(1/2) + 101474123648041645683480942072580)/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988), z, 1)^3 + 3843738782525825599*root(z^9 + (z^8*(40646370540593230529675376111439*2^(1/2) + 57482648479749177740913514589635))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) - (z^7*(48950395506894713531281417549686*2^(1/2) + 69226313209377518582549939536733))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z^6*(59135621925949926270484844611882*2^(1/2) + 83630398547046153213000249050231))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z^5*(255070896824454141176498238274983*2^(1/2) + 360724721655811468397476671471853))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z^4*(69682769222483480998602050740137*2^(1/2) + 98546317298150630332891610577466))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) - (z^3*(3384262362996977766641442698518*2^(1/2) + 4786069732379144454076875275372))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z^2*(156240974119420649562191631285482*2^(1/2) + 220958104598068424057609399598508))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z*(18544628785686575814198409310776*2^(1/2) + 26226065537892456179585517114620))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) - (71753040946492451590476100638776*2^(1/2) + 101474123648041645683480942072580)/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988), z, 1)^4 + 3656038408655610529*root(z^9 + (z^8*(40646370540593230529675376111439*2^(1/2) + 57482648479749177740913514589635))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) - (z^7*(48950395506894713531281417549686*2^(1/2) + 69226313209377518582549939536733))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z^6*(59135621925949926270484844611882*2^(1/2) + 83630398547046153213000249050231))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z^5*(255070896824454141176498238274983*2^(1/2) + 360724721655811468397476671471853))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z^4*(69682769222483480998602050740137*2^(1/2) + 98546317298150630332891610577466))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) - (z^3*(3384262362996977766641442698518*2^(1/2) + 4786069732379144454076875275372))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z^2*(156240974119420649562191631285482*2^(1/2) + 220958104598068424057609399598508))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z*(18544628785686575814198409310776*2^(1/2) + 26226065537892456179585517114620))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) - (71753040946492451590476100638776*2^(1/2) + 101474123648041645683480942072580)/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988), z, 1)^5 - 1201568387913037680*root(z^9 + (z^8*(40646370540593230529675376111439*2^(1/2) + 57482648479749177740913514589635))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) - (z^7*(48950395506894713531281417549686*2^(1/2) + 69226313209377518582549939536733))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z^6*(59135621925949926270484844611882*2^(1/2) + 83630398547046153213000249050231))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z^5*(255070896824454141176498238274983*2^(1/2) + 360724721655811468397476671471853))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z^4*(69682769222483480998602050740137*2^(1/2) + 98546317298150630332891610577466))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) - (z^3*(3384262362996977766641442698518*2^(1/2) + 4786069732379144454076875275372))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z^2*(156240974119420649562191631285482*2^(1/2) + 220958104598068424057609399598508))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z*(18544628785686575814198409310776*2^(1/2) + 26226065537892456179585517114620))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) - (71753040946492451590476100638776*2^(1/2) + 101474123648041645683480942072580)/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988), z, 1)^6 + 1124560220925490*root(z^9 + (z^8*(40646370540593230529675376111439*2^(1/2) + 57482648479749177740913514589635))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) - (z^7*(48950395506894713531281417549686*2^(1/2) + 69226313209377518582549939536733))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z^6*(59135621925949926270484844611882*2^(1/2) + 83630398547046153213000249050231))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z^5*(255070896824454141176498238274983*2^(1/2) + 360724721655811468397476671471853))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z^4*(69682769222483480998602050740137*2^(1/2) + 98546317298150630332891610577466))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) - (z^3*(3384262362996977766641442698518*2^(1/2) + 4786069732379144454076875275372))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z^2*(156240974119420649562191631285482*2^(1/2) + 220958104598068424057609399598508))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z*(18544628785686575814198409310776*2^(1/2) + 26226065537892456179585517114620))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) - (71753040946492451590476100638776*2^(1/2) + 101474123648041645683480942072580)/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988), z, 1)^7 + 1246354319528873616*root(z^9 + (z^8*(40646370540593230529675376111439*2^(1/2) + 57482648479749177740913514589635))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) - (z^7*(48950395506894713531281417549686*2^(1/2) + 69226313209377518582549939536733))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z^6*(59135621925949926270484844611882*2^(1/2) + 83630398547046153213000249050231))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z^5*(255070896824454141176498238274983*2^(1/2) + 360724721655811468397476671471853))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z^4*(69682769222483480998602050740137*2^(1/2) + 98546317298150630332891610577466))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) - (z^3*(3384262362996977766641442698518*2^(1/2) + 4786069732379144454076875275372))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z^2*(156240974119420649562191631285482*2^(1/2) + 220958104598068424057609399598508))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z*(18544628785686575814198409310776*2^(1/2) + 26226065537892456179585517114620))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) - (71753040946492451590476100638776*2^(1/2) + 101474123648041645683480942072580)/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988), z, 1)^8 - 501710985731959272*2^(1/2) + 937020290048174705*2^(1/2)*root(z^9 + (z^8*(40646370540593230529675376111439*2^(1/2) + 57482648479749177740913514589635))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) - (z^7*(48950395506894713531281417549686*2^(1/2) + 69226313209377518582549939536733))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z^6*(59135621925949926270484844611882*2^(1/2) + 83630398547046153213000249050231))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z^5*(255070896824454141176498238274983*2^(1/2) + 360724721655811468397476671471853))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z^4*(69682769222483480998602050740137*2^(1/2) + 98546317298150630332891610577466))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) - (z^3*(3384262362996977766641442698518*2^(1/2) + 4786069732379144454076875275372))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z^2*(156240974119420649562191631285482*2^(1/2) + 220958104598068424057609399598508))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z*(18544628785686575814198409310776*2^(1/2) + 26226065537892456179585517114620))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) - (71753040946492451590476100638776*2^(1/2) + 101474123648041645683480942072580)/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988), z, 1) + 1325146802404899932*root(z^9 + (z^8*(40646370540593230529675376111439*2^(1/2) + 57482648479749177740913514589635))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) - (z^7*(48950395506894713531281417549686*2^(1/2) + 69226313209377518582549939536733))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z^6*(59135621925949926270484844611882*2^(1/2) + 83630398547046153213000249050231))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z^5*(255070896824454141176498238274983*2^(1/2) + 360724721655811468397476671471853))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z^4*(69682769222483480998602050740137*2^(1/2) + 98546317298150630332891610577466))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) - (z^3*(3384262362996977766641442698518*2^(1/2) + 4786069732379144454076875275372))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z^2*(156240974119420649562191631285482*2^(1/2) + 220958104598068424057609399598508))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) + (z*(18544628785686575814198409310776*2^(1/2) + 26226065537892456179585517114620))/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988) - (71753040946492451590476100638776*2^(1/2) + 101474123648041645683480942072580)/(32893336723025330371259778013941*2^(1/2) + 46518202905407400950493588584988), z, 1) - 709526480413711184)/(810738853893036773*2^(1/2) + 1146557882718351754)

I've also tried to use vpasolve with vpasolve(eq_1,a) and I get as a result empty: 0 by 1.

Does anyone know why I'm getting this kind of solution? Can anyone suggest another software or way of solving this symbolic non-linear equation? I really need to fix this as it's been blocking my master thesis for almost a month.

Thank you very much,

Sincerely, R.A

Answer 1

[Walter Roberson]

Those equations are not consistent with being equal to an identity matrix.

If you solve at position (2,2) then the fundamental solutions are [Pi, (1/4)*Pi, arctan(-sqrt(2), I), arctan(-sqrt(2), -I)]. If you substitute each of these in turn into your equations, you will find that you cannot get (1,1) to match: the corner elements at (1,1), (1,3), (3,1), (3,3) all come out as +/- sqrt(2) for example.

Answer 2

[Roc Arandes]

Wow, I was so sure the method was correct. But you are completely right, now I corrected it and I finally have the good final matrix.

[ cos(a) + cos(a)^2/2 - cos(a)^3 + cos(a)^4/4 + (2^(1/2)*sin(a))/2 - (2^(1/2)*cos(a)*sin(a))/2 - (2^(1/2)*cos(a)^2*sin(a))/2 + (2^(1/2)*cos(a)^3*sin(a))/2 + 1/4,                                                   -((cos(a) - 1)*(3*cos(a) + cos(a)^2 - cos(a)^3 - 4*2^(1/2)*cos(a)*sin(a) - 2*2^(1/2)*cos(a)^2*sin(a) + 1))/4,                 cos(a)^4 - (3*cos(a)^2)/2 - (2^(1/2)*sin(a))/4 - (2^(1/2)*cos(a)*sin(a))/4 + (3*2^(1/2)*cos(a)^2*sin(a))/4 - (2^(1/2)*cos(a)^3*sin(a))/4 + 1/2]
[                 (3*cos(a)^2)/2 - cos(a)^4 + (2^(1/2)*sin(a))/4 + (2^(1/2)*cos(a)*sin(a))/4 - (3*2^(1/2)*cos(a)^2*sin(a))/4 + (2^(1/2)*cos(a)^3*sin(a))/4 - 1/2, cos(a) + cos(a)^2/2 - 2*cos(a)^3 - cos(a)^4 + (2^(1/2)*sin(a))/4 - (2^(1/2)*cos(a)*sin(a))/4 - (2^(1/2)*cos(a)^2*sin(a))/4 + (2^(1/2)*cos(a)^3*sin(a))/4 + 1/2,                                                                            ((cos(a) + 1)*(3*cos(a) - 3*cos(a)^2 + cos(a)^3 + 2*2^(1/2)*cos(a)^2*sin(a) - 1))/2]
[                                          -(sin(a)*(sin(a) - cos(a)^2*sin(a) - 4*2^(1/2)*cos(a) + 2*2^(1/2)*cos(a)^2 + 2*2^(1/2)*cos(a)^3 + 2*cos(a)*sin(a)))/4,                                  -((cos(a) + 1)*(cos(a)^3 - cos(a)^2 - cos(a) - 2*2^(1/2)*sin(a) + 4*2^(1/2)*cos(a)*sin(a) + 2*2^(1/2)*cos(a)^2*sin(a) + 1))/4, cos(a) + cos(a)^2/2 - 2*cos(a)^3 - cos(a)^4 + (2^(1/2)*sin(a))/4 - (2^(1/2)*cos(a)*sin(a))/4 - (2^(1/2)*cos(a)^2*sin(a))/4 + (2^(1/2)*cos(a)^3*sin(a))/4 + 1/2]

I checked the numerical values of this one for a certain "a" and they look like a "identity" matrix. But instead of ones and zeros , I have the diagonal elements equal to 0.89, 0.90, etc. And the other elements that should be 0 are actually 0.004,0.003,etc.

I think to find a solution for (a) I should try to minimize the least square error between two elements of the matrix.

Any clue on how to correctly implement this?

THanks

[Walter Roberson]

That has solutions a = pi and a = arctan(2*sqrt(2)).

The upper right corner also has solution a = 0, but putting that in leads to [1 0 0; 0 -1 0; 0 0 -1]

[Roc Arandes]

Could you tell me how are you solving it please?

The thing is I will have to calculate this matrix 180 times with different initial values to find a relation between angles. (between (a) and another angle that is already numerically introduced on the matrix i sent you).

Really thanks for your time

[Roc Arandes]

How are you solving for (a) in Matlab? I can't find the results using vpasolve neither solve.

[Walter Roberson]

I used Maple.

In MATLAB you can solve() the first entry. That will get you 0 and -log((2^(1/2)*2i)/3 + 1/3)*1i . You can rewrite(ans,'atan') to get pi/2 - atan(2^(1/2)/4) which is the same value as arctan(2*sqrt(2)).

If you solve() with 'ignoreanalytic', true then you get back pi as well.

[Roc Arandes]

Thank you very much, I will ask someone on my lab if they have Maple to solve it! Really I didn't expect someone to help me as much as you did

Answer 3

[Roc Arandes]

I know I can get the result with WolframAlpha, but as I need to solve this 180 times, I'm looking for a function/command that allows me to obtain the same results (ex: a = pi and a = arctan(2*sqrt(2))) in Matlab. Thx