On 2/2/07, Christian Boyer <cboyer@club-internet.fr> wrote:
Sorry, some mistakes in my previous hypothesis.
With the 5th-degree equation:
x(x² + a)(x² + b) = c
x, a, b, c real numbers a, b > 0 c > (1+a)(1+b) x > 1
what is the fastest method to get an approximated value of x? (very few steps)
How about iterating x -> c/((x^2 + a) (x^2 + b))? Nope, that seems awful. (In general, it seems to lead to a cycle of period 2, instead of converging, or maybe I made a mistake in my calculations.) OK, then, how about iterating x -> sqrt( (c/(x*(x^2 + a)) - b)? Nope, that leads to complex values of x, unless maybe I made a mistake in my calculations. Hm, this is harder than I thought it would be! OK, I think I'd just resort to plain old Newton's method then. Let f(x) = x*(x^2 + a)(x^2 + b) - c, and map x -> x - f(x)/f'(x). A dozen or two iterations seems to be enough ... --Joshua Zucker