I need a method for solving a linear system (x1..xn) of type

x1*(k11+k12/(x1+x2+..xn)) x2*k13 x3*k14 ....

... x2*(k22+k23/(x1+x2+..xn)) x3*k24 ....

... ..... x3*(k34+k35/(x1+x2+..xn)) x4*k36

and so on, size about 400 x 400 elements,

I am unable to find a solution with Thomas or similar efficient methods

cause diagonal elements contains 1/(x1+x2+...+xn) which require separate processing,

what am I missing here ?

Is there a simple solution for these systems ?

Thanks for help.