The Secant method for simultaneous nonlinear equations

A procedure for the simultaneous sohition of a sys{)em of not-necessari ly-l inear equations, t~ generalization of the secanl, method for a single function of one w~ri~fi)le, is given. 1. S e c a n t M e t h o d for n E q u a t i o n s This note is concerned with a eomputa t ionM procedure for the solution of the s imultaneous equat ions (1) f~(x) = 0, i = 1, . . , n , where x = (x~ , . . . , x,) ~ I/:,, and each f i is a computab le funct ion of x. (Fo r the later theoretical discussion of the procedure, the f~ will be supposed to have second derivatives, bu t no derivatives are calculated.) The procedure consists in the i teration of the following step, at the beginning of which it is assumed thai; the n + 1 "trial solutions" x ~, . . . , x '~ ~1 are at hand. F ind rrt, . . . , rr~,+l such tha t r~+t ~ r : = 1 j = l (2 )