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- Victor Y. Pan, Ailong Zheng
- Computers & Mathematics with Applications
- 2011

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)

- Victor Y. Pan, Ailong Zheng
- Computers & Mathematics with Applications
- 2011

- Victor Y. Pan, Ailong Zheng, Xiaohan Huang, Olen Dias
- J. Complexity
- 1997

- Victor Y. Pan, Myong-Hi Kim, Akimou Sadikou, Xiaohan Huang, Ailong Zheng
- J. Complexity
- 1996

- Victor Y. Pan, Guoliang Qian, Ailong Zheng
- ISSAC
- 2011

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)

- Victor Y. Pan, Ailong Zheng
- ISSAC
- 2010

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)

- Victor Y. Pan, Guoliang Qian, Ailong Zheng
- CSR
- 2010

The eigenproblem for an n-by-n matrix A is the problem of the approximation (within a relative error bound 2 ?b) of all the eigenvalues of the matrix A and computing the associated eigenspaces of all these eigenvalues. We show that the arithmetic complexity of this problem is bounded by O(n 3 + (n log 2 n) log b). If the characteristic and minimum… (More)

- Victor Y. Pan, Brian Murphy, Rhys Eric Rosholt, Yuqing Tang, Xinmao Wang, Ailong Zheng
- Computers & Mathematics with Applications
- 2008

Highly effective polynomial root-finders have been recently designed based on eigen-solving for DPR1 (that is diagonal + rankone) 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)

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… (More)