Optimal Starting Approximations for Newton's Method

Best Rational Starting Approximations and Improved Newton Iteration for the Square Root

“…Recently, Sterbenz and Fike [9], King and Phillips [8], and the present author [10], discovered independently a surprisingly simple relationship, Theorem 2 in Section 2, among the three criteria. This paper presents two new contributions concerning the computation of the square root.…”

Chebyshev approximations for the Riemann zeta function

“…where î(s) is the "logarithmic error" discussed by King and Phillips [3] and Sterbenz and Fike [6]. The logarithmic error has been associated primarily with obtaining starting values for various Newton iteration schemes.…”

Minimax approximations subject to a constraint

“…In this case, f*(x) and fix) may be noticeably different, since the approximation is not very accurate. But we showed in [2] that f(x) minimizes the maximum relative error after one or more iterations, so we would prefer to have f(x) instead of f*(x). Then the constraint may be used to simplify the computation as in [4].…”