Re: cool giant exactly solvable matrix equation!

> Let i and j be integers (indices) that range from
> {0,...,H}.
>
> Let M be an (H+1)x(H+1) matrix with the following
> matrix elements:
> M(i,j) = g_i^j = "g_i to the power of j"
> where {g_0, g_1, ..., g_H} are H+1 positive real
> numbers inside the
> range (0,R).
>
> Let v be an (H+1) vector with matrix elements
> v(i) = g_i / (R-g_i)
>
> Let z be the (H+1) vector determined by solving the
> matrix equation:
> M z = v
>
> Find the solution for the H+1 matrix elements of z
> for arbitrary H.
>
> (Hint: Using mathematica, I have examined the
> solutions for H=2, 3, 4,
> 5.
> It seems the solutions fall into a simple
> pattern that should allow one to write down a
> closed-form solution.
> Anybody recognize what that would be?)
>


Are the g_i's distinct? Otherwise, M would be singular and there would not be a unique solution.

Jack
.