Resolution of a system with sine and cosine terms

I write a code like this, but the run continues indefinitely; it seems that Matlab is not able to solve this system. Is that really the case or am I wrong somewhere?

clc
clear
close all

syms phi1
syms theta1
syms phi2
syms theta2
syms phi3
syms theta3
syms phi4
syms theta4

ro = sqrt(3);

csi1 = ro*sin(phi1)*cos(theta1);
eta1 = ro*sin(phi1)*sin(theta1);
zeta1 = ro*cos(phi1);
csi2 = ro*sin(phi2)*cos(theta2);
eta2 = ro*sin(phi2)*sin(theta2);
zeta2 = ro*cos(phi2);
csi3 = ro*sin(phi3)*cos(theta3);
eta3 = ro*sin(phi3)*sin(theta3);
zeta3 = ro*cos(phi3);
csi4 = ro*sin(phi4)*cos(theta4);
eta4 = ro*sin(phi4)*sin(theta4);
zeta4 = ro*cos(phi4);

eqns = [csi1^2 + csi2^2 + csi3^2 + csi4^2 == 4, eta1^2 + eta2^2 + eta3^2 + eta4^2 == 4, zeta1^2 + zeta2^2 + zeta3^2 + zeta4^2 == 4, csi1*eta1 + csi2*eta2 + csi3*eta3 + csi4*eta4 == 0, eta1*zeta1 + eta2*zeta2 + eta3*zeta3 + eta4*zeta4 == 0, zeta1*csi1 + zeta2*csi2 + zeta3*csi3 + zeta4*csi4 == 0, csi1 + csi2 + csi3 + csi4 == 0, eta1 + eta2 + eta3 + eta4 == 0, zeta1 + zeta2 + zeta3 + zeta4 == 0,...
    theta1>0, theta1<2*pi, phi1>0, phi1<pi, theta2>0, theta2<2*pi, phi2>0, phi2<pi, theta3>0, theta3<2*pi, phi3>0, phi3<pi, theta4>0, theta4<2*pi, phi4>0, phi4<pi];

[phi1, theta1, phi2, theta2, phi3, theta3, phi4, theta4] = solve(eqns, [phi1, theta1, phi2, theta2, phi3, theta3, phi4, theta4]);

[Torsten]

I count 9 equations in 8 unknowns - thus usually impossible to solve.

[Sam Chak]

Me too... 9 equations and 8 unknowns. At first glance, the system is overdetermined because there are more equations than unknowns.

eqns = [csi1^2 + csi2^2 + csi3^2 + csi4^2 == 4, ...                 % Eq.(1)
        eta1^2 + eta2^2 + eta3^2 + eta4^2 == 4, ...                 % Eq.(2)
        zeta1^2 + zeta2^2 + zeta3^2 + zeta4^2 == 4, ...             % Eq.(3)
        csi1*eta1 + csi2*eta2 + csi3*eta3 + csi4*eta4 == 0, ...     % Eq.(4)
        eta1*zeta1 + eta2*zeta2 + eta3*zeta3 + eta4*zeta4 == 0, ... % Eq.(5)
        zeta1*csi1 + zeta2*csi2 + zeta3*csi3 + zeta4*csi4 == 0, ... % Eq.(6)
        csi1 + csi2 + csi3 + csi4 == 0, ...                         % Eq.(7)
        eta1 + eta2 + eta3 + eta4 == 0, ...                         % Eq.(8)
        zeta1 + zeta2 + zeta3 + zeta4 == 0, ...                     % Eq.(9)
        theta1>0, theta1<2*pi, phi1>0, phi1<pi, theta2>0, theta2<2*pi, phi2>0, phi2<pi, theta3>0, theta3<2*pi, phi3>0, phi3<pi, theta4>0, theta4<2*pi, phi4>0, phi4<pi];

[phi1, theta1, phi2, theta2, phi3, theta3, phi4, theta4] = solve(eqns, [phi1, theta1, phi2, theta2, phi3, theta3, phi4, theta4]);

[Andrea Cesaro]

Ok thank you, but although I add ro like an other unknown Matlab is not able to solve it

[Andrea Cesaro]

Like this:

clc

clear

close all

syms phi1

syms theta1

syms phi2

syms theta2

syms phi3

syms theta3

syms phi4

syms theta4

syms ro

csi1 = ro*sin(phi1)*cos(theta1);

eta1 = ro*sin(phi1)*sin(theta1);

zeta1 = ro*cos(phi1);

csi2 = ro*sin(phi2)*cos(theta2);

eta2 = ro*sin(phi2)*sin(theta2);

zeta2 = ro*cos(phi2);

csi3 = ro*sin(phi3)*cos(theta3);

eta3 = ro*sin(phi3)*sin(theta3);

zeta3 = ro*cos(phi3);

csi4 = ro*sin(phi4)*cos(theta4);

eta4 = ro*sin(phi4)*sin(theta4);

zeta4 = ro*cos(phi4);

eqns = [csi1^2 + csi2^2 + csi3^2 + csi4^2 == 4, eta1^2 + eta2^2 + eta3^2 + eta4^2 == 4, zeta1^2 + zeta2^2 + zeta3^2 + zeta4^2 == 4, csi1*eta1 + csi2*eta2 + csi3*eta3 + csi4*eta4 == 0, eta1*zeta1 + eta2*zeta2 + eta3*zeta3 + eta4*zeta4 == 0, zeta1*csi1 + zeta2*csi2 + zeta3*csi3 + zeta4*csi4 == 0, csi1 + csi2 + csi3 + csi4 == 0, eta1 + eta2 + eta3 + eta4 == 0, zeta1 + zeta2 + zeta3 + zeta4 == 0,...

theta1>0, theta1<2*pi, phi1>0, phi1<pi, theta2>0, theta2<2*pi, phi2>0, phi2<pi, theta3>0, theta3<2*pi, phi3>0, phi3<pi, theta4>0, theta4<2*pi, phi4>0, phi4<pi];

[phi1, theta1, phi2, theta2, phi3, theta3, phi4, theta4, ro] = solve(eqns, [phi1, theta1, phi2, theta2, phi3, theta3, phi4, theta4, ro]);

Answer 1

[Dheeraj]

Hi,

If the script is running for a long time without producing results, it's possible that the symbolic solver is struggling to find a solution due to the complexity of the equations or the constraints. Few ways you could work around to solve the problem are,

1. Simplifying the equations or by breaking down the problem to subproblems and solve them separately and try to combine the results at the end.
2. Using MATLAB parallel computing toolbox to speed up calculations. You may refer to this document on parallel computing for better understanding. https://in.mathworks.com/help/parallel-computing/getting-started-with-parallel-computing-toolbox.html

[John D'Errico]

Yes. Note also that inequalities are difficult things to handle, and there are MANY of them here.

[Andrea Cesaro]

Thank you very much