An efficient solution for Cauchy-likesystems of linear equations

Using the displacement approach and transformation of vectors, the complexity ofsolving strongly nonsingular Cauchy-like systems of linear equations will be improved off a factor of log n from the known algorithms of the recursive triangular factorization, where F is a field and v¯ Fn×1 is a vector and C Fn×n is a strongly nonsingular Cauchy-like matrix. The efficient algorithms were presented by Morf, Bitmead and Anderson to solve strongly nonsingular Toeplitz-like equations of linear systems by using the recursive triangular factorization in 1980. Recently, this approach of the recursive triangular factorization was extended to solve Cauchy-like systems with the complexity of O(n log3 n) operations by Pan et al. 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, the algorithms of the recursive triangular factorization are modified 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 Toeplitz-like matrices. Our improvement of the direct recursive approach is by a factor of log n due to changing the original vectors 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.