非单项式基底的非线性特征值问题误差分析及相关预处理技术研究

求解非线性特征值问题的一种常用解法是首先通过矩阵多项式插值理论或者矩阵有理函数线性逼近得到一个近似特征值问题,然后将近似特征值问题转化为线性形式,进而用标准化算法求解基于线性形式的广义特征值问题.我们拟利用基于CORK线性形式的线性化方法求解几类典型的非单项式基底的非线性特征值问题,如泰勒,牛顿,拉格朗日,正交以及有理基底.随后,拟建立非单项式基底的非线性特征值问题的向后误差分析框架,进而构造CORK线性形式与非线性特征值问题的向后误差界,根据这些误差界分析基于CORK线性形式的线性化方法的数值稳定性,同时依据误差界设计新型预处理技术改进求解非线性特征值问题的向后误差.最后,通过数值实验验证我们的理论分析与预处理技术改进向后误差的效果.

Solving a nonlinear eigenvalue problem (NEP) usually begins with approximating the NEP by a polynomial (PEP) or rational (REP) eigenvalue problem. One of the most successful methods for solving such a PEP or REP proxy is to recast it, by linearization, as an equivalent but larger generalized eigenvalue problem which can be solved by standard eigensolvers. We propose to investigate the backward errors of the computed eigenpairs incurred by the application of the recently developed and well-received compact rational Krylov (CORK) linearization. We will aim at unifying the treatment for the original PEPs or REPs expressed in various commonly used bases, including Taylor, Newton, Lagrange, orthogonal, and rational basis functions. In addition, we will try to establish upper bounds for the backward error of an approximate eigenpair of the original eigenvalue problem relative to the backward error of an approximate eigenpair of the CORK linearization. These bounds may shed light on preconditioning strategies for improving the accuracy of the computed eigenpairs. Our investigation will be accompanied by adequate numerical experiments to justify our theoretical findings.