Toeplitz矩阵函数的快速逼近算法及其应用

项目名称： Toeplitz矩阵函数的快速逼近算法及其应用

项目编号： No.11201192

项目类型： 青年科学基金项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 庞宏奎

作者单位： 江苏师范大学

项目金额： 23万元

中文摘要： 矩阵函数有着非常广泛的应用背景，考虑矩阵函数的快速逼近算法是矩阵函数问题中的一个重要研究方向。本项目主要研究Toeplitz矩阵函数与向量乘积的快速逼近算法，研究内容包括：充分利用Toeplitz矩阵的性质和特征，设计新的快速有效的有理函数逼近算法；构建和分析新的Krylov子空间重启动算法；研究Toeplitz矩阵函数的有理Krylov子空间算法；考虑用围道积分法逼近Toeplitz矩阵函数与向量的乘积；拟用Toeplitz矩阵相关的工具、数值域、扰动分析或逼近论中的结果等对算法的收敛性和稳定性做细致的分析。本项目旨在促进结构矩阵函数与向量乘积的算法的研究，为更快更精确的逼近矩阵函数与向量的乘积提供更多好的算法和理论。本项目的开展将会极大的丰富现有的研究方法和研究手段，并对矩阵理论，数值分析，逼近理论等相关领域发展提供丰富的结果和研究课题。

中文关键词： Toeplitz矩阵；矩阵指数函数；Krylov子空间；期权定价；分数阶微分方程

英文摘要： Functions of matrices play an important role in many applications. The problem of fast approxiamting matrix functions is one of the major topics in the study of matrix functions. In this project, we consider fast computing the action of the Toeplitz matrix funcion on a vector. Research contents include the following several aspects: design of fast and efficient rational approximation algorithms by taking full advantage of Toeplitz structure and proporties; construction of new restarted Krylov subspace methods; investigation of the rational Krylov subspace method from the Toeplitz point of view; as well as fast approximation of the action of the Toepltiz matrix functions and a vector by the contour integration. We would also give a detailed discussion on the stability and convergence of the proposed algorithms by utilizing the tools related to the Toepltiz matrix, field of values, perturbation analysis, or results in theory of approximation. The aim of the project is to promote the study of the action of matrix functions with structured matrices and a vector, and provide more elegant algorithms and results for fast and accurate approximating matrix functions. The study on this project would greatly enrich the current research methods and research means, and provide fruitful results and research topics for the fie

英文关键词： Toeplitz matrix；Matrix exponential；Krylov subspace；Option pricing；Fractional differential equations