Solving coupled exponential equations simultaneously with a given condition

Question

I want to solve the following equations simultaneously to obtain x_i 's and T.

 x_i =a_i * exp(-C_i / T)    with the condition  sum(x_i)=1

where i may be up to 30 in the worst case and 3 at best. a_i and c_i are known.

Can any one please suggest an algorithm. I have tried Monte Carlo but could not get the desired result.

Answer 1

Plug the expressions for x_i into the restriction about their sum; you will get an equation for T. You can solve it using Newton's method.