Approximation of Polynomial Root Using a Single Input and the Corresponding Derivative Values

A new formula for the approximation of root of polynomials with complex coefficients is presented. For each simple root there exists a neighborhood such that given any input within this neighborhood, the formula generates a convergent sequence, computed via elementary operations on the input and the corresponding derivative values. Each element of the sequence is defined in terms of the quotient of two determinants, computable via a recursive formula. Convergence is proved by deriving an explicit error estimate. For special polynomials explicit neighborhoods and error estimates are derived that depend only on the initial error. In particular, the latter applies to the approximation of root of numbers. The proof of convergence utilizes a family of iteration functions, called the Basic Family; a nontrivial determinantal generalization of Taylor’s theorem; a lower bound on determinants; Gerschgorin’s theorem and Hadamard’s inequality; as well as several new key results. The convergence results motivate a new strategy for general rootfinding, where in each iteration one approximates a root of the Taylor polynomial of a desirable degree, via the above sequence. The results also motivate the development of new sequences of iteration functions.