# Question:Verify a solution of a 2-equations system using eval()

## Question:Verify a solution of a 2-equations system using eval()

Maple

I am trying to verify a solution by checking whether eval() returns 0 but it's taking me forever. If I recall correctly it once returned [0,0] so I am quite confident that the first and only positive root of that polynomial solves my system. I am now running again the same calculation but somehow eval() is stuck in "Evaluating...". I am not sure if it matters here, but parameters gamma, sigma_v, and sigma_d are strictly positive while -1<rho<+1 (rho is a correlation coefficient).

How else can I verify such solution?

EDIT: THIS IS NOT A DUPLICATE QUESTION AND SHOULDN'T BE TAGGED AS SUCH

 > restart;
 > local gamma:

Equations:

 > eq1 := (gamma*sigma__v^2*(-1 + rho__v) - 2*lambda__2)*(rho__v*sigma__v^2 + sigma__v^2)/((2*gamma*(lambda__1 + lambda__2)*(-1 + rho__v)*sigma__v^2 - 8*lambda__1*lambda__2)*((gamma*sigma__v^2*(-1 + rho__v) - 2*lambda__2)^2*(2*rho__v*sigma__v^2 + 2*sigma__v^2)/(2*gamma*(lambda__1 + lambda__2)*(-1 + rho__v)*sigma__v^2 - 8*lambda__1*lambda__2)^2 + (-lambda__1*(gamma*sigma__v^2*(-1 + rho__v) - 4*lambda__2)/(2*gamma*(lambda__1 + lambda__2)*(-1 + rho__v)*sigma__v^2 - 8*lambda__1*lambda__2) + 1)^2*sigma__d^2 + gamma^2*lambda__2^2*sigma__v^4*(-1 + rho__v)^2*sigma__d^2/(2*gamma*(lambda__1 + lambda__2)*(-1 + rho__v)*sigma__v^2 - 8*lambda__1*lambda__2)^2)): eq2 := (gamma*sigma__v^2*(-1 + rho__v) - 2*lambda__1)*(rho__v*sigma__v^2 + sigma__v^2)/((2*gamma*(lambda__1 + lambda__2)*(-1 + rho__v)*sigma__v^2 - 8*lambda__1*lambda__2)*((gamma*sigma__v^2*(-1 + rho__v) - 2*lambda__1)^2*(2*rho__v*sigma__v^2 + 2*sigma__v^2)/(2*gamma*(lambda__1 + lambda__2)*(-1 + rho__v)*sigma__v^2 - 8*lambda__1*lambda__2)^2 + (-(gamma*sigma__v^2*(-1 + rho__v) - 4*lambda__1)*lambda__2/(2*gamma*(lambda__1 + lambda__2)*(-1 + rho__v)*sigma__v^2 - 8*lambda__1*lambda__2) + 1)^2*sigma__d^2 + gamma^2*lambda__1^2*sigma__v^4*(-1 + rho__v)^2*sigma__d^2/(2*gamma*(lambda__1 + lambda__2)*(-1 + rho__v)*sigma__v^2 - 8*lambda__1*lambda__2)^2)): Eq1, Eq2 := eq1 - lambda__1, eq2 - lambda__2:

Solution:

 > Gamma := gamma*sigma__v*sigma__d: L__2 := RootOf(-8*(rho + 1)^4*_Z^4 + 12*(rho + 1)^3*Gamma*(rho - 1)*_Z^3 - 5*(rho + 1)^2*(-4/5 + Gamma^2*rho^2 + 2*(-2/5 - Gamma^2)*rho + Gamma^2)*_Z^2 - 4*(rho + 1)*Gamma*(rho^2 - 1)*_Z + Gamma^2*(rho + 1)*(rho - 1)^2); l__2 := L__2*sigma__v*(rho__v + 1)/sigma__d: quartic_solution := lambda__2 = simplify(allvalues~([l__2]))[1]:
 (1)

Check: TOO SLOW

 > simplify(eval([eval(Eq1, lambda__1 = lambda__2), eval(Eq2, lambda__1 = lambda__2)], quartic_solution));