Computing first-order zeros of analytic functions with large values of derivatives

Abstract

There are some practically important types of complex analytic functions whose zeros are narrowly surrounded by large function values. Zeros of this kind are said to be deep. Computation of deep zeros presents difficulty for commonly used methods because of large values of function derivatives. An efficient algorithm for computing deep zeros is proposed on the basis of contour integration of the function argument. Its variations along the contour are much smaller than variations of the function values, which makes the algorithm efficient. The location of the argument maxima along the contour of integration yields an initial approximation for the zero whose value is further refined by applying the Muller algorithm.