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Mathematics of Computation

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A solution to certain polynomial equations with applications to nonlinear fitting

Author: Chris Connell
Journal: Math. Comp. 74 (2005), 303-319
MSC (2000): Primary 65H10
Published electronically: July 9, 2004
MathSciNet review: 2085413
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Abstract: We present a combinatorial method for solving a certain system of polynomial equations of Vandermonde type in $2N$ variables by reducing it to the problem of solving two special linear systems of size $N$ and rooting a single univariate polynomial of degree $N$. Over $\mathbb{C}$, all solutions can be found with fixed precision using, up to polylogarithmic factors, $O(N^2)$bitwise operations in the worst case. Furthermore, if the data is well conditioned, then this can be reduced to $O(N)$ bit operations, up to polylogarithmic factors. As an application, we show how this can be used to fit data to a complex exponential sum with $N$ terms in the same, nearly optimal, time.

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Additional Information

Chris Connell
Affiliation: Department of Mathematics, University of Chicago, 5734 S. University Avenue, Chicago, Illinois 60637
Address at time of publication: Department of Mathematics, Rawles Hall, Indiana University, Bloomington, Indiana 47401

Received by editor(s): January 18, 2002
Received by editor(s) in revised form: January 6, 2003
Published electronically: July 9, 2004
Additional Notes: The author was supported in part by an NSF postdoctoral fellowship
Article copyright: © Copyright 2004 American Mathematical Society