Contributed talks on Eigenvalue Algorithms and Pseudospectra
Mon 15:00–15:30, Room AV 03.12, Chair: Kensuke AishimaInverse eigenvalue problems arise from a variety of applications. We focus on the following inverse eigenvalue problem. Let $A_{0},A_{1},\ldots ,A_{n}$ be $n\times n$ real symmetric matrices. For any real vector ${c}=(c_{1},\ldots ,c_{n})^{T}$, we define $A({c})=A_{0}+c_{1}A_{1}+\cdots +c_{n}A_{n}$. Also, we denote the eigenvalues of $A({c})$ by $\lambda_{1}({c})\le \cdots \le \lambda_{n}({c})$. For $n$ real numbers $\lambda_{1}^{*}\le \cdots \le \lambda_{n}^{*}$, our aim is to find a real vector ${c}$ such that $\lambda_{k}({c})=\lambda_{k}^{*}$ for $k=1,\ldots ,n$.
We propose a new quasi-newton method for such an inverse eigenvalue problem and prove quadratic convergence whenever $\lambda_{1}^{*}< \cdots < \lambda_{n}^{*}$. The proposed method is viewed as an improved version of a typical quasi-newton method by Friedland, Nocedal, and Overton [1, Method III]. This existing method requires the Cayley transform, which takes $O(n^{3})$ operations, to ensure the orthogonality of the iterative matrices. The proposed method can refine the orthogonality without the Cayley transform. As a result, if the approximate Jacobian matrix is formed in $O(n^{3})$ time, the proposed method is significantly faster than any quasi-Newton method using the Cayley transform.
- S. Friedland, J. Nocedal, and M. Overton, The formulation and analysis of numerical methods for inverse eigenvalue problems, SIAM Journal on Numerical Analysis, 24(3) (1987), pp. 634–667.