Rational circular complex centered forms

摘要

A discussion is given of the problem of computing including estimates of the range of a complex rational function over a circular complex interval. For this purpose, rational circular complex centered forms are defined. Explicit formulas are given for the first few forms and these formulas are used to prove that forms of higher order are an improvement over the forms of lower order. The forms are furthermore shown to be quadratically convergent.A semi-centered form is also discussed. This form is shown to be quadratically convergent depending on some conditions on the coefficients of the polynomials defining the complex rational function.Finally, a number of numerical examples are given showing the improvements obtained using the circular centered forms as compared to simple circular complex rational function estimations.