Analytic Root Clustering: A Complete Algorithm Using Soft Zero Tests

A challenge to current theories of computing in the continua is the proper treatment of the zero test. Such tests are critical for extracting geometric information. Zero tests are expensive and may be uncomputable. So we seek geometric algorithms based on a weak form of such tests, called soft zero tests. Typically, algorithms with such tests can only determine the geometry for “nice” (e.g., non-degenerate, non-singular, smooth, Morse, etc) inputs. Algorithms that avoid such niceness assumptions are said to be complete. Can we design complete algorithms with soft zero tests? We address the basic problem of determining the geometry of the roots of a complex analytic function f. We assume effective box functions for f and its higher derivatives are provided. The problem is formalized as the root clustering problem, and we provide a complete (δ,e)-exact algorithm based on soft zero tests.

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