论文标题
研究基于缺陷的错误估计值的Krylov近似值$φ$ functions的研究
A study of defect-based error estimates for the Krylov approximation of $φ$-functions
论文作者
论文摘要
致力于研究多项式Krylov技术的事先工作，用于近似矩阵指数$ {\ rm e}^{ta} v $的作用，扩展到相关的$φ$ functions（指数积分器中发生）的情况。特别是，基于Krylov近似的缺陷（残留）概念，后验误差界限和估计值。讨论和分析了可计算的误差范围和估计。这包括一个新的错误，该错误与特定情况下的现有错误界限有利相比。各种误差范围的准确性与$ a $的相应丽思值相关。与实际的启动矢量$ v $有关的$ a $ a $ a $ a频谱的属性属性（例如，对于Hermitian或偏斜的矩阵），可以计算。这给出了理论上的结果以及标准，以量化运行中达到的准确性。对于其他现有错误估计，通过类似技术研究了可靠性和性能。还考虑了有限精度（浮点算术）的影响。
Prior recent work, devoted to the study of polynomial Krylov techniques for the approximation of the action of the matrix exponential ${\rm e}^{tA}v$, is extended to the case of associated $φ$-functions (which occur within the class of exponential integrators). In particular, a~posteriori error bounds and estimates, based on the notion of the defect (residual) of the Krylov approximation are considered. Computable error bounds and estimates are discussed and analyzed. This includes a new error bound which favorably compares to existing error bounds in specific cases. The accuracy of various error bounds is characterized in relation to corresponding Ritz values of $A$. Ritz values yield properties of the spectrum of $A$ (specific properties are known a~priori, e.g. for Hermitian or skew-Hermitian matrices) in relation to the actual starting vector $v$ and can be computed. This gives theoretical results together with criteria to quantify the achieved accuracy on the run. For other existing error estimates the reliability and performance is studied by similar techniques. Effects of finite precision (floating point arithmetic) are also taken into account.
