具有带状或分层低秩结构矩阵函数的分治法
Divide and conquer methods for functions of matrices with banded or hierarchical low-rank structure
摘要：
本文研究带状矩阵、分层半可分矩阵及相关结构的近似矩阵函数。我们开发了一种新的分治方法的基础上（理性）Krylov子空间方法进行低秩更新的矩阵函数。我们的收敛性分析的新提出的方法收益建立关系，最佳多项式和有理逼近。当只有跟踪或矩阵函数的对角线是感兴趣的，我们证明-在实践中和理论上-收敛可以更快。对于带状矩阵的特殊情况下，我们表明，分治法减少到一个更简单的算法，通过计算小的子矩阵的矩阵函数。数值实验证实了新开发的算法计算大规模矩阵函数的各种应用的有效性。
英文摘要：
This work is concerned with approximating matrix functions for banded matrices, hierarchically semiseparable matrices, and related structures. We develop a new divide-and-conquer method based on (rational) Krylov subspace methods for performing low-rank updates of matrix functions. Our convergence analysis of the newly proposed method proceeds by establishing relations to best polynomial and rational approximation. When only the trace or the diagonal of the matrix function is of interest, we demonstrate -- in practice and in theory -- that convergence can be faster. For the special case of a banded matrix, we show that the divide-and-conquer method reduces to a much simpler algorithm, which proceeds by computing matrix functions of small submatrices. Numerical experiments confirm the effectiveness of the newly developed algorithms for computing large-scale matrix functions from a wide variety of applications.
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