arXiv subjects

Wei-Qiang Huang

Publications and source records attributed to Wei-Qiang Huang.

3 recordsLinked to original sources

arXiv2026

On Circular Numerical Ranges of Companion Matrices with Repeated Eigenvalues

We prove that if an $n\times n\ (n > 3)$ companion matrix $A$ with the spectrum $σ(A) = \{ a \}$ has a circular numerical range, then $A$ is the Jordan block. This problem can be described by examining zeros of the Laurent polynomial arising from geometric properties of the numerical range. The difficulty is that the relevant Laurent coefficients involve both the repeated eigenvalue $a$ and the radius parameter $λ$, so direct coefficient comparison does not isolate $a$. We address this by decomposing the relevant matrix into a tridiagonal Toeplitz part plus a rank-two update and using Chebyshev polynomials of the second kind. This reduction yields an explicit Laurent-coefficient formula whose vanishing under the circularity condition gives $a=0$. Furthermore, we extend this result when the spectrum is $σ(A) = \{0, a\}$ with algebraic multiplicities $n-m$ and $m$, respectively.

math.FA↗

arXiv2017

iSIRA: Integrated Shift-Invert Residual Arnoldi Method for Graph Laplacian Matrices from Big Data

The eigenvalue problem of a graph Laplacian matrix $L$ arising from a simple, connected and undirected graph has been given more attention due to its extensive applications, such as spectral clustering, community detection, complex network, image processing and so on. The associated graph Laplacian matrix is symmetric, positive semi-definite, and is usually large and sparse. Computing some smallest positive eigenvalues and corresponding eigenvectors is often of interest. However, the singularity of $L$ makes the classical eigensolvers inefficient since we need to factorize $L$ for the purpose of solving large and sparse linear systems exactly. The next difficulty is that it is usually time consuming or even unavailable to factorize a large and sparse matrix arising from real network problems from big data such as social media transactional databases, and sensor systems because there is in general not only local connections. In this paper, we propose an eignsolver based on the inexact residual Arnoldi method together with an implicit remedy of the singularity and an effective deflation for convergent eigenvalues. Numerical experiments reveal that the integrated eigensolver outperforms the classical Arnoldi/Lanczos method for computing some smallest positive eigeninformation provided the LU factorization is not available.

math.NA↗

arXiv2013

A Novel Skew-Hamiltonian Isotropic Lanczos Algorithm for Spectral Conformal Parameterizations

Numerous methods for computing conformal mesh paramterizations has been developed due to the vast applications in the field of geometry processing. Spectral conformal parameterization (SCP) is one of these methods to computing a quality conformal parameterization based on the spectral technique. SCP focus on a generalized eigenvalue problem (GEP) $L_{C}\mathbf{f} = λB\mathbf{f}$ whose eigenvector(s) associated with the smallest positive eigenvalue(s) will provide the parameterization result. This paper devotes to study a novel eigensolver for this GEP. Based on the structures of matrix pair $(L_{C},B)$, we show that this GEP can be transformed into a small-scaled compressed deating standard eigenvalue problem with a symmetric positive definite skew-Hamiltonian operator. We then propose a skew-Hamiltonian isotropic Lanczos algorithm (SHILA) to solve the reducing problem. Numerical experiments show that our compressed deating skill remove the inuence of the kernel of $L_{C}$ and transform the original problem to a more robust system. The novel SHILA method can effective avoid the disturbance of duplicate eigenvalues. As a result, our numerical eigensolver can accurately and efficiently compute the conformal parameterization based on the spectral model of SCP.

math.NA↗