Complexity of Bezout’s Theorem VI: Geodesics in the Condition (Number) Metric

Authors:
Michael Shub
Affiliations:
University of Toronto, Department of Mathematics, 40 St. George Street, M5S 2E4, Toronto, Ontario, Canada
Venue:
Foundations of Computational Mathematics
Year:
2009

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Abstract

We introduce a new complexity measure of a path of (problems, solutions) pairs in terms of the length of the path in the condition metric which we define in the article. The measure gives an upper bound for the number of Newton steps sufficient to approximate the path discretely starting from one end and thus produce an approximate zero for the endpoint. This motivates the study of short paths or geodesics in the condition metric.