﻿ Elementary Solutions
ISSAC 94 - Symbolic-Numeric Nonlinear Equation Solving
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 Elementary Solutions

Bundy and Welham[3] describe concepts of attraction, collection, and isolation which sometimes lead to solutions of nonlinear equations.  Taking a somewhat related algorithmic approach, suppose is an elementary expression where each is exponential, logarithmic, or algebraic.  Factoring, we suppose is irreducible in .  Apply rules

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

where and is an integer-valued piecewise constant function of

For rule (1), let and be exponentials.  Try to find integers , , such that (consider and degrees) and simplify to where

For rule (2), let and be logarithms.  Try to find a rational such that (consider , , and lcoeff's) and simplify to where , , and is chosen according to a given root (given by our numerical algorithm).

Rules (1)-(2) are applied first.  Rules (3)-(8) are applied with new factorization.  This scheme will solve some equations.