Efficient computation of an implicitly defined function

Question

I have a scalar function f(u) defined implicitly as follows:

pow( u, -f(u) ) + pow( u, f(u) ) = u

The function is approximately 1, but evidently not quite so. I am scratching my head for an efficient means of numerically computing values of this function. Any suggestions?

I hope the notation is clear btw, pow( a, b) = a^b is a raised to the power b.

Answer 1

If we write

 u = exp( log(u))   

and remember that

 cosh( x) = (exp(x) + exp(-x))/2

then your equation turns into

 cosh( log(u)*f(u)) = u/2

since cosh(x) >= 1 for all real x, there can be no real solution for u<2, while for u>=2

f(u) = acosh( u/2) / log(u)

where acosh is the inverse of cosh.