On 2014-05-14, Ailurus <ailuru...@gmail.com> wrote:
> Hi all,
>
> I'm trying to solve the following system of two equations,
>
> Eq1 = 2*a^2*cos(pi/n)^2 - 2*a - b - (a*sin((2*pi)/n)*(-2*(cos((2*pi)/n) -
> sin((2*pi)/n)*i)*(a^2*cos((2*pi)/n) - 4*a - 2*b + a^2 + 2))^(1/2))/2 +
> 2^(1/2)*a*cos(pi/n)^2*(-(cos((2*pi)/n) -
> sin((2*pi)/n)*i)*(a^2*cos((2*pi)/n) - 4*a - 2*b + a^2 + 2))^(1/2)*i + 1 ==
> 1/2
> Eq2 = a^2*(cos((4*pi)/n) + 1) - 2*a - b +
> (a*sin((4*pi)/n)*(-2*(cos((4*pi)/n) - sin((4*pi)/n)*i)*(a^2*cos((4*pi)/n) -
> 4*a - 2*b + a^2 + 2))^(1/2))/2 - 2^(1/2)*a*(cos((4*pi)/n)/2 +
> 1/2)*(-(cos((4*pi)/n) - sin((4*pi)/n)*i)*(a^2*cos((4*pi)/n) - 4*a - 2*b +
> a^2 + 2))^(1/2)*i + 1 == 1/4
>
> where the three variables are defined using a,b,n = var('a,b,n'). Note that
> both expressions are quadratic in a and linear in b. Solving for a and b, I
> expect expressions in terms of (some function, e.g. cos, of) n. The command
> I tried for solving is solve([Eq1, Eq2], a, b). The input, so Eq1 and Eq2,
> is generated by MATLAB and Sage seems to be quite happy with it. However, I
> don't get any solutions — Sage prints the following output,
>
> [I*sqrt(2)*sqrt(-(a^2*cos(2*pi/n) + a^2 - 4*a - 2*b + 2)*(cos(2*pi/n) -
> I*sin(2*pi/n)))*a*cos(pi/n)^2 + 2*a^2*cos(pi/n)^2 -
> 1/2*sqrt(-(a^2*cos(2*pi/n) + a^2 - 4*a - 2*b + 2)*(2*cos(2*pi/n) -
> 2*I*sin(2*pi/n)))*a*sin(2*pi/n) - 2*a - b + 1 == (1/2),
> -1/2*I*sqrt(2)*sqrt(-(a^2*cos(4*pi/n) + a^2 - 4*a - 2*b + 2)*(cos(4*pi/n) -
> I*sin(4*pi/n)))*a*(cos(4*pi/n) + 1) + a^2*(cos(4*pi/n) + 1) +
> 1/2*sqrt(-(a^2*cos(4*pi/n) + a^2 - 4*a - 2*b + 2)*(2*cos(4*pi/n) -
> 2*I*sin(4*pi/n)))*a*sin(4*pi/n) - 2*a - b + 1 == (1/4)]
>
> What's happening and how can I fix it? Thanks.
I guess Sage has no idea how to deal with equations depending in a
nontrivial way on the parameter n.
On the practical side, certainly you can eliminate b and get one equation,
quadratic in a. Then you can solve it directly.
>
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