Why computer algebra systems sometimes can't solve simple equations

ABSTRACT

Among the basic equations one might wish a computer to solve symbolically is the inverse of the power function, solving y = z^w for z. (Note: z^w ≡ exp(w ln z)). While many special cases, easily solved, abound, the general question is fraught with implications: if this is so hard, how can we expect success in other ventures? Having solved this, we can naturally use it in a "composition" of solution methods for expressions of the form y = f(z)^w.Can't we already do this? Is it not the case that the solution of y = z^a+bi is trivially z = y^1/(a+bi)) ?Not so. if this were the case, then a plot of the function t(y) := y - (y^1/(1+i)))¹⁺ⁱ would be indistinguishable from t(y) ≡ 0. For many values, t(y) is (allowing for round-off error), zero. But if your computer system correctly computes with values in the complex plane, then, (to pick two complex points from a region described later), t(-10000 + 4000i) is not zero, but about -9981 + 3993i and t(-0.01 + 0.002i) is about 5.34 - 1.06i. These strange numbers are not the consequence of round-off error or some other numerical phenomena. The alleged solution is just not mathematically correct.Computer algebra systems as usually programmed lack the expressive capability to return the exact and complete set of solutions, in general.

INDEX TERMS

Primary Classification:
  I. Computing Methodologies
  I.1 SYMBOLIC AND ALGEBRAIC MANIPULATION

Additional Classification:
  C. Computer Systems Organization

General Terms:
Experimentation, Performance