Solution of a system of non-linear equations

Question

I have this system of $n$ non-linear equations in $n$ unknowns, that I'm unable to solve. Given that $x_0=1$, I have to solve for $(x_1,x_2,\ldots,x_n).$ $$\sum_{i=0}^n x_i^2+2\sum_{j=1}^n\sum_{i=0}^{n-j}x_ix_{i+j}=1$$ $$\sum_{j=1}^n\sum_{i=0}^{n-j}~j^2~x_ix_{i+j}=0$$ $$\sum_{j=1}^n\sum_{i=0}^{n-j}~j^4~x_ix_{i+j}=0$$ $$\cdots ~\cdots ~\cdots$$ $$\sum_{j=1}^n\sum_{i=0}^{n-j}~j^{2(n-1)}~x_ix_{i+j}=0$$ Is there any way to find exact/approximate solutions? Is there any algorithm available for approximate solutions of such systems? Any help would be greatly appreciated.

Answer 1

Hint: Reformulate the problem as an optimization problem. Let us call the left-hand side of the first equation $F_1(x)$, the left-hand side of the second equation $F_2(x)$, and so forth. The variable $x$ is the vector of your variables $x_1,x_2,...,x_n$.

Then define a function $\Phi(x)=(F_1(x)-1)^2+(F_2(x)-0)^2+... +(F_n(x)-0)^2$.

Now, use an algorithm (Newton-Raphson, gradient decent eg.) to minimize $\Phi(x)$. For small systems, this can be achieved even with MS Excel. You could think of other functions $\Phi(x)$, I just used the simple least squares approach.