An automatic and guaranteed determination of the number of roots of an analytic function interior to a simple closed curve in the complex plane

A well known resuk from complex analysis allows us, under suitable drcumstances, to compute the number of roots of an analytic function, f(z), that lie inside a counterclockwise, simple dosed curve, C, by computing the integral, We employ interval arithmetic and automatic differentiation m give an auminatic and guaranteed bound on the integral. Furthermore, we explore the interplay of the choice of curve (7, the location of the roots relative m C, the number of subdivigons, and the arithmetic pre~ion used, upon the time necer~ry to obtain .~ltisfactm-~ bounds. l'IpHMe~s m-rrep~u~hm/~ apH~bMerHxy H as'roMa-m~ec~oe aNqtx~peam~posaaHe, M,-uoay,meM rapaHxHpoRaHmae rpaxlatu~ 3Toro HHrerpa.nm. KpoMe Toro, ~ccaealyerca aaH~raHe s~6opa xpH~oR C, pacuoaoaceHH~ xopHe/E no OTHomeHmo X. 6*, Ko.rlHqec'rBa pa361ieHHR H paap~anocrH HClIOnh3yeMofl apHd~bMeTHXH Ha BpeMR, Tp~yeMoe Ra~[ IIo.~ytleHHa yaoBaeTaopaTeaba~x rpanHu.