An Automatic and Guaranteed Determination of
the Number of Roots of an Analytic Function
Interior to A Simple Closed Curve in the
Complex Plane

Jonathan L. Herlocker
Department of Mathematical Sciences
Lewis and Clark College
Portland, OR 97219 U.S.A.
herlock@lclark.edu
Jeffrey S. Ely
Department of Mathematical Sciences
Lewis and Clark College
Portland, OR 97219 U.S.A.
jeff@lclark.edu


Abstract
A well known result from complex analysis allows us, under suitable circumstances, 
to compute the number of roots of an analytic function, f(z), that lie
inside a counterclockwise, simple closed curve, C, by computing the integral,
We employ interval arithmetic and automatic differentiation to give an automatic 
and guaranteed bound on the integral. Furthermore, we explore the
interplay of the choice of curve C, the location of the roots relative to C, the
number of subdivisions, and the arithmetic precision used, upon the time necessary 
to obtain satisfactory bounds.



References
[1] Churchill, R., Brown, J., Verhey, R., Complex Variables and Applications,
McGrawHill, New York. (1976).
[2] Ely, J., The VPI Software Package for Variable Precision Interval Arithmetic,
(to appear in Interval Computations).
[3] Henrici, P., Applied and Computational Complex Analysis, Vol.1. Wiley,
New York. (1974).
[4] Kagiwada, H., Kalaba, R., Rasakhoo, N., Spingarn, K., Numerical Derivatives 
and Nonlinear Analysis, Plenum, New York (1986).
[5] Moore, R.E., Methods and Applications of Interval Analysis, SIAM,
Philadelphia. (1979).
[6] Press, W., Flannery, B., Teukolsky, S., Vetterling, W., Numerical Recipes
in C, Cambridge University Press, Cambridge, MA. (1988).
[7] Stroustrup, B., The C++ Programming Language, AddisonWesley, Reading, Mass. (1986).
