TR-2013011: Fast Approximation Algorithms for Cauchy Matrices, Polynomials and Rational Functions

The papers [MRT05], [CGS07], [XXG12], and [XXCBa] have combined the advanced FMM techniques with transformations of matrix structures (traced back to [P90]) in order to devise numerically stable algorithms that approximate the solutions of Toeplitz, Hankel, Toeplitzlike, and Hankel-like linear systems of equations in nearly linear arithmetic time, versus classical cubic time and quadratic time of the previous advanced algorithms. We show that the power of these approximation algorithms can be extended to yield similar results for computations with other matrices that have displacement structure, which includes Vandermonde and Cauchy matrices, as well as to polynomial and rational evaluation and interpolation. The resulting decrease of the running time of the known approximation algorithms is again by order of magnitude, from quadratic to nearly linear. We present detailed description and analysis of the algorithms and provide new insights into the subject, formal complexity estimates, and the support for further advances in [Pa]. The techniques of our study can be of independent interest.