TY - JOUR
T1 - On the Parameterized Polynomial Inverse Eigenvalue Problem
AU - Xiang , Meiling
AU - Dai , Hua
JO - East Asian Journal on Applied Mathematics
VL - 1
SP - 185
EP - 204
PY - 2025
DA - 2025/01
SN - 15
DO - http://doi.org/10.4208/eajam.2023-234.231223 
UR - https://global-sci.org/intro/article_detail/eajam/23746.html
KW - Polynomial inverse eigenvalue problem, multiparameter eigenvalue problem, Newton method, Newton-like method.
AB - <p style="text-align: justify;">The paper focuses on the solvability and computability of the parameterized
polynomial inverse eigenvalue problem (PPIEP). Employing multiparameter eigenvalue
problems, we establish a sufficient solvability condition for the PPIEP. Three numerical
methods are used to solve PPIEPs. The first one is the Newton method based on locally
smooth $QR$-decomposition with the column pivoting and the second the Newton method
based on the smallest singular value. In order to reduce the computational cost of computing the smallest singular values and the corresponding unit left and right singular
vectors in each iteration, we approximate these values by using one-step inverse iterations. Subsequently, we introduce another method — viz. a Newton-like method based
on the smallest singular value. Each of three methods exhibits locally quadratic convergence under appropriate conditions. Numerical examples demonstrate the effectiveness
of the methods proposed.</p>