Fast Cauchy-like and Singular Toeplitz-like Matrix Computations

computes the solution ~ x = T ?1 ~ b to a nonsingular Toeplitz or Toeplitz-like linear system T~ x = ~ b, a short displacement generator for the inverse T ?1 of T, and det T. We extend this algorithm to the similar computations with nn Cauchy (generalized Hilbert) and Cauchy-like matrices. Recursive triangular factorization of such a matrix can be computed by our algorithm at the cost of executing O(nr 2 log 3 n) arithmetic operations, where r is the scaling rank of the input Cauchy-like matrix C (r = 1 if C is a Cauchy matrix). Consequently, the same cost bound applies to the computation of the determinant of C, a short scaling generator of C ?1 , and the solution to a nonsingular linear system of n equations with such a matrix C. Our algorithm does not use the reduction to Toeplitz-like computations. We avoid singularity in this algorithm and run it in an arbitrary eld by using randomization. We also ameliorate slightly Kaltofen's solver of Toeplitz-like singular linear systems in an arbitrary eld.