for x^+(1/n), fitting optimal initial guess is 0.0436864*n - 0.0294222
for x^-(1/n), fitting optimal initial guess is 0.0439341*n + 0.0505897

for x^-(1/n), the Lagny iterate is:
x = x + f/f'(1 - 0.5 * f/f' * f''/f')
and the Newton iterate is:
x = x + f/f'

f   = (x^-n - a)
f'  = -n*x^-(n+1)
f'' = n*(n+1)*x^-(n+2)

from this:
f/f' = x^(n+1)*(x^-n - a)*(1/n)
     = x*(1 - x^n*a)*(1/n)
f''/f' = n*(n+1)*x^-(n+2)*x^(n+1)*(1/n)
       = (n+1)*x^-1

let r = (1 - x^n*a)

so we get:
x = x + x*r/n
and
x = x + x*r/n*(1 - (n+1)*r/n)
  = x + x*r/n*(1 - r - r/n)