On the Solvability Condition and Numerical Algorithm for the Parameterized Generalized Inverse Eigenvalue Problem
Hua Dai, Zhong‐Zhi Bai, Ying Wei · SIAM Journal on Matrix Analysis and Applications · 2015
We discuss the parameterized generalized inverse eigenvalue problem (PGIEP): For given matrices $A_i$, $B_i \in {\bf C}^{n \times n}$ ($i=0, 1, \ldots, n$), find complex numbers $c_i \in {\bf C}$ ($i=1, 2, \ldots, n$) such that the generalized eigenvalue problem $(A_0+\sum_{i=1}^nc_iA_i)x =\lambda (B_0+\sum_{i=1}^nc_iB_i)x$ has the prescribed eigenvalues $\lambda_1, \lambda_2,\ldots, \lambda_n$. We show that this problem is equivalent to a multiparameter eigenvalue problem if the given eigenvalues $\lambda_1, \lambda_2,\ldots, \lambda_n$ are distinct. Applying the theory about the multiparameter eigenvalue problem, we obtain sufficient conditions that guarantee the existence of a solution of the PGIEP. In addition, we propose a smooth LU decomposition for a matrix depending on several parameters and discuss its algebraic property. Based on these theoretical results, we present a numerical algorithm for solving the PGIEP and prove its locally quadratic convergence. Numerical implementations show that the new algorithm is feasible and effective for solving the PGIEP.