A Rapid Robust Rootf inder

A numerical algorithm is presented for solving one nonlinear equation in one real variable. Given Fx with brackets A and B, i.e., sign(T^) = -sign^g) * 0, the algorithm finds a zero of F, A < X < B. Alternately, a crossover pair is found, i.e., (X, Y): sign(Fx) = signify) * 0 where there is no floating-point number in the system between X and Y. This feature allows full use of machine precision. Optionally, a tolerance TOL > 0 may be given, to permit termination when \Y X\ < TOL. The method, once rapid convergence sets in, is alternation of one linear interpolation or extrapolation with one inverse quadratic interpolation. The resulting asymptotic convergence rate is competitive with other methods that refine both brackets and do not require dF/dX. Other merits of the algorithm are: robust calculation, efficient three-point interpolation, and superior behavior in bad cases. The algorithm is tested and compared with others. 1. The Problem, and an Outline of the Paper. Given a real-valued function F of one real variable X, with brackets A and 73 such that sign(i^) = sign(77B) ¥= 0, (1) find X such that Fx = 0 (we call such X a machine zero of F), or (2) find X and Y such that sign(Fx) = sign(Fy) # 0 and either (a) \X — Y\ < TOL, where TOL is a user-supplied tolerance, or (b) X and Y have no representable number between them (we call such X and Y a crossover pair). The rootfinder has no access to derivatives of F. It is to be superlinearly convergent for smooth F with simple roots, and reasonably efficient even in the worst of cases. The root is to be found by successively refining the bracket [A, B] until one of the termination criteria (1, 2a, or 2b, above) is met. The contents of the paper are as follows: Section 2: Background. Some current algorithms of Bus, Dekker, and Brent are discussed. Section 3: Structure and correctness. The general flow of Algorithm S is used to show that it must terminate. Section 4: Robust calculation. Overflow never occurs and underflow is safeguarded, permitting X and F to span the range of the machine. Section 5 : Machine issues. Floating-point numbers can be specified precisely with standard, civilized computer expressions. Section 6: Good cases. The various strategy sequences during rapid convergence are described. Received February 8, 1982; revised May 16, 1983 and February 3, 1984. 1980 Mathematics Subject Classification. Primary 65H05, 65-04, 65D05, 65G05, 68E05.