A Machine Method for Solving Polynomial Equations

The problem of finding numerical approximations to the roots of a polynomial has a long and interesting history. The various methods have always been proposed in terms of the state of the art of computataon then current. Since the advent of high-speed computing systems there has been, the writer feels, an unusual lag m the development of new techniques for deahng with this timehonored problem, techmques bet ter stated to the capabihties and inadequacies of automatic computers. A recent survey of available methods [1] indicates tha t no one method is desirable for automatic computers. I t is true that methods such as those of Newton or Bernoulli for real roots and Graeff~ or Bairstow for complex roots have been programmed for automatm computers. Each of these classic methods requires a good deal of judgment in connection with the isolation or separation of roots. These judicial decasions are relatively easy for a human being to make when operating a desk calculator but are more difficult to anticipate and furnish to the machine's program. On the other hand, the machine is prepared to undertake thousands of times more arithmetical activity than was ever contemplated by the inventors of the classmal methods. Hence it is tame to consider methods of more uniform applmcabihty, ~dth possibly slower convergence rates, that may be far too laborious to carry out by hand but which nevertheless are sufficiently easy for an automatic computer. Such a method should be applicable to polynomials with complex coefficients whose roots are therefore any arbitrary finite set of points an the complex plane, distinct or not. I t thus becomes a problem of searching the complex plane for roots. One such method has already been tried by J. A. Ward [1]. I t seeks to minimize