Paper
Error Analysis of Krylov Subspace approximation Based on IDR($s$) Method for Matrix Function Bilinear Forms
Authors
Qianqian Xue, Xiaoqiang Yue, Xian-Ming Gu
Abstract
The bilinear form of a matrix function, namely , appears in many scientific computing problems, where , , and is a given analytic function. The Induced Dimension Reduction IDR() method was originally proposed to solve a large-scale linear system, and effectively reduces the complexity and storage requirement by dimension reduction techniques while maintaining the numerical stability of the algorithm. In fact, the IDR() method can generate an interesting Hessenberg decomposition, our study just applies this fact to establish the numerical algorithm and a posteriori error estimate for the bilinear form of a matrix function . Through the error analysis of the IDR() algorithm, the corresponding error expansion is derived, and it is verified that the leading term of the error expansion serves as a reliable posteriori error estimate. Based on this, in this paper, a corresponding stopping criterion is proposed. Numerical examples are reported to support our theoretical findings and show the utility of our proposed method and its stopping criterion over the traditional Arnoldi-based method.