Solution of nonlinear equations with space filling curves

Space filling curves provide a means of finding solutions of sets of nonlinear equations by exhaustive search, and hence appear to be useful for determining the approximate solutions usually required as starting points when using classical methods, for determining the nonexistence of solutions, and for determining all of a finite number of solutions. This paper extends the theory and techniques of employing space filling curves to accomplish these things. One result is the refinement of methods for determining the nonexistence of solutions and for determining all of a finite number of solutions. The results are, however, incomplete with respect to the latter problem; more work is required. Two basic methods, a first-order method and a second-order method, are proposed and it is shown that the two may profitably be combined into a hybrid method. A second result is an economical method of determining single (coarse) solutions. A third result is the development of a theory for studying asymptotic convergence rates. It is shown that both the first- and second-order methods, as well as the hybrid method, converge geometrically under usual conditions, provided the set of equations has a solution, and provided the "prime member" of the solution set is in a certain set T_n. The set T_n is dense, uncountable, of measure zero, and includes the rational vectors. An unanswered question is what sorts of equations have solutions in T_n; neither the difficulty nor the importance of this question is known at present. Experimental evidence is presented wherever necessary.