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

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

MaPal93 170 Maple
eval rootof simplification allvalues duplicate-question
May 08 2024
1

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

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restart;
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local gamma:

Equations:

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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:

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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]:

RootOf((8*rho^3+24*rho^2+24*rho+8)*_Z^4+(-12*gamma*rho^3*sigma__d*sigma__v-12*gamma*rho^2*sigma__d*sigma__v+12*gamma*rho*sigma__d*sigma__v+12*gamma*sigma__d*sigma__v)*_Z^3+(5*gamma^2*rho^3*sigma__d^2*sigma__v^2-5*gamma^2*rho^2*sigma__d^2*sigma__v^2-5*gamma^2*rho*sigma__d^2*sigma__v^2+5*gamma^2*sigma__d^2*sigma__v^2-4*rho^2-8*rho-4)*_Z^2+(4*gamma*rho^2*sigma__d*sigma__v-4*gamma*sigma__d*sigma__v)*_Z-gamma^2*sigma__v^2*sigma__d^2*rho^2+2*rho*gamma^2*sigma__v^2*sigma__d^2-gamma^2*sigma__v^2*sigma__d^2)

 (1)

Check: TOO SLOW

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


 

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