I'll have to eat my words here, not for the first time. First off, secant method is absolutely dreadful, at any rate for this equation. [The literature ignores it completely, and now I know why --- I must just have been lucky earlier.] Secondly, a good starting value for iteration seems always to be min( c/(ab), c^(1/5) ) pace Gene --- it doesn't matter about the fifth root, because that of course needs computing only to low precision! Finally, an absolutely cracking iterative method of which I was previously ignorant is Halley third-order --- with function and derivatives for this case f0 = f(x) = x*(x^2+a)*(x^2+b) - c, f1 = df/dx = 5x^4 + 3(a+b)x^2 + ab, f2 = d^2f/dx^2 = 20x^3 + 6(a+b)x, we iterate x := x - f0 f1 / (f1^2 - f0 f2 / 2) ); getting at least 10 sig figs in 2 or maybe 3 iterations. By the way, the most awkward parameter set, where the curvature is maximal, is typified by a = 16.0, b = 16.0, c = 2560.0. And note that the problem can be reduced to two parameters by the substitution x' = x/sqrt(a), a' = 1, b' = b/a, c' = c/a^(5/2). I found Halley in L. Volpi "Non Linear Equations", on web at http://www.digilander.libero.it/foxes/poly/Polynomial_zeros.pdf He claims it is more "stable" than Newton, and backs this up with a discussion of the basins of attraction. Inter alia, he also surveys a wide selection of other methods, most of which --- like those in journal articles --- are more concerned with finding _all_ roots of a polynomial. Fred Lunnon