On calculating the zeros of polynomials by the method of lucas

Abstract

When/(#) is a polynomial of degree n and xiy i=0, 1, . . ., n, are any n-\-1 points at which fixi) 7*0, the zeros of fix) are known to be identical with the zeros of Soi/fx-a;,), where ai=fixi)/IL ixi—xi). Lucas proposed this principle for use in an electric analogue device for finding zeros. The present note evaluates this principle in digital computation for both real and complex zeros when the coefficients of fix) are given exactly (integral or rational) so that the zeros of fix) are identical with the zeros of 2Ai/ix—i), A{ integral. The chief advantages are (1) the saying of labor in tabulating XAiUx — i) instead of fix) in the neighborhood of the zero, especially for complex zeros, and (2) somewhat less work in the inverse interpolation for the zero. Three examples in locating a real root, and one example in locating a complex root were worked out in support of these findings.