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

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

- 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)

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, Guoliang Qian, Ailong Zheng
- CSR
- 2010

- 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 + 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)

- Victor Y. Pan, Guoliang Qian, Ailong Zheng
- SNC
- 2009

We propose novel randomized preprocessing techniques for solving linear systems of equations and eigen-solving with extensions to the solution of polynomial and secular equations. According to our formal study and extensive experiments, the approach turns out to be effective, particularly in the case of structured input matrices.