Ailong Zheng

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Matrix methods are increasingly popular for polynomial root-finding. The idea is to approximate the roots as the eigenvalues of the companion or generalized companion matrix associated with an input polynomial. The algorithms also solve secular equation. QR algorithm is the most customary method for eigen-solving, but we explore the inverse Rayleigh(More)
Recent progress on root-finding for polynomial and secular equations largely relied on eigen-solving for the associated companion and diagonal plus rank-one generalized companion matrices. By applying to them Rayleigh quotient iteration, we could have already competed with the current best polynomial root-finders, but we achieve further speedup by applying(More)
MBA algorithm inverts a structured matrix in nearly linear arithmetic time but requires a serious restriction on the input class. We remove this restriction by means of randomization and extend the progress to some fundamental computations with polynomials, e.g., computing their GCDs and AGCDs, where most effective known algorithms rely on computations with(More)
For a Toeplitz or Toeplitz-like matrix T, we define a preconditioning applied to the symmetrized matrix TnT, which decreases the condition number compared to the one of TnT and even the one of T. This enables us to accelerate the conjugate gradient algorithm for solving Toepiltz and Toeplitz-like linear systems, thus extending the previous results of [1],(More)
Highly effective polynomial root-finders have been recently designed based on eigen-solving for DPR1 (that is diagonal + rank-one) matrices. We extend these algorithms to eigen-solving for the general matrix by reducing the problem to the case of the DPR1 input via intermediate transition to a TPR1 (that is triangular + rank-one) matrix. Our transforms use(More)