A Modified Recursive Triangular Factorization for Cauchy-Like Systems

Abstract

We derive new algorithms for solving strongly nonsingular Cauchy-like systems of linear equations \(C\tilde x = \tilde v{\kern 1pt} in{\kern 1pt} O(n{\log ^2}n)\) running time, where F is a field and \(\tilde v \in {F^{n \times 1}}\) is a vector, C ∈ F n×n is a strongly nonsingular Cauchy-like matrix. Morf, Bitmead and Anderson presented the efficient algorithms to solve strongly nonsingular Toeplitz-like equations of linear systems by using the Recursive Triangular Factorization in 1980. Recently, Pan and Zheng extended the Recursive Triangular Factorization to solve Cauchy-like systems with the complexity of O(n log3 n) operations. This is the best known complexity bound by using the direct approach of Recursive Triangular Factorization in Cauchy-like cases. However, these algorithms are still slower than the well known algorithms with the asymptotic bound of O(n log2 n) operations, which have been proposed by the means of reducing Cauchy-like matrices into Toeplitz-like matrices. In our present paper, we will modify the Recursive Triangular Factorization so that the complexity bound of the direct recursive approach can be decreased to O(n log2 n)operations. This matches the asymptotic bound without transforming to Toeplitzlike matrices. Our improvement of the direct recursive approach is by a factor off log n due to changing the original vectors which expressed in the given Cauchy-like matrix into the special vectors, where the entries are unit roots. The applications of structured matrices include Nevanlinna-Pick tangential interpolation problems.