Krylov 子空间方法可用于计算大规模稀疏矩阵的矩阵函数乘向量 $f(A)v$. 与其他数值代数问题如求解线性方程组、计算特征值问题等不同的是，计算 $f(A)v$ 的 Krylov 子空间算法没有可计算的误差表达式或相应的残差的概念可以用来判定计算的精度、设计算法的停机准则. 对于 $e^{A}v$ 的Arnoldi 近似，Saad 曾于 1992 年建立了误差展开式，并从数值试验结果中观察到展开式中的第一项是一个较好的后验误差估计. 本文通过建立两个误差上界，首次给出了误差展开式中第一项是可靠的后验误差估计的理论依据. 更进一步，我们将 Saad 的结论进行了推广，针对足够光滑的函数 $f(z)$, 建立了其 Krylov-like 近似的误差展开式，同时从理论和数值实验两方面验证了误差展开式中的第一项是 $f(A)v$ 的 Krylov-like 近似的可靠的后验误差估计，据此理论设计出了相应 Krylov 子空间算法的可靠的停机准则.计算 $f(A)v$ 的 Krylov 子空间算法的误差可与求解线性方程组的对应方法的残差相联系. 标准 Arnoldi 方法对应于求解方程组的 Arnoldi 方法，一般没有任何最优性. 基于求解线性方程组的 GMRES 方法及其收敛性结果，本文提出了计算 $f(A)v$ 的调和 Ritz 方法，并给出了相应的收敛性分析. 同时根据标准 Arnoldi 算法的重启思想，设计出了重启的调和 Ritz 方法，解决了其计算量和存储量的问题. 调和 Ritz 方法与矩阵 $A$ 在目标点 $\xi$ 处的调和 Ritz 值密切相关.和 Arnoldi 方法相比，调和 Ritz 方法有两点显著的优势，一是收敛曲线相比于标准 Arnoldi 方法更为光滑，二是通过算法中参数 $\xi$ 的选取可以保证重启的调和 Ritz 方法对任意的重启步数均收敛，该点在实际中尤为重要. 本文给出了 $\xi$ 的三条选取准则及其显式表达式，并论证了调和 Ritz 算法对 $\xi$ 的选取并不敏感，这表明，我们的方法具有普适性. 数值实验表明了调和 Ritz 算法及其重启算法的有效性和光滑性，并且验证了算法对 $\xi$ 的选取并不敏感.隐式重启算法通过构造合适的多项式过滤算子，可以有效的选取重启算法的初始向量，提高重启后的子空间的质量，加速目标不变子空间的收敛. 基于该思想，本文给出了计算 $f(A)v$ 的隐式重启 Arnoldi 算法，且在选取精确位移（即 $A$ 的 Ritz 值）构造多项式过滤算子时，证明了计算 $f(A)v$ 的隐式重启 Arnoldi 算法数学上与带收缩技术的重启 Arnoldi 算法等价.
Krylov subspace methods for approximating a matrix function $f(A)$ times a vector $v$, which are further analyzed in this thesis, have been intensively studying over the years and constitute one of main topics in numerical algebra. Unlike in other linear algebra problems, such as the linear system, the eigenvalue problem and so on, this problem is not naturally equipped with an a posteriori error representation or a clear residual notion that can beused for determining the convergence and designing reliable stopping criteria for a Krylov subspace approximation to $f(A)v$. For the Arnoldi approximation to $e^{A}v$, Saad established an error expansion and empirically claimed that the first term of the error expansion can be a reliable a posteriori error estimate. However, its theoretical justification has not yet been given hitherto. In this thesis, some reliable a posteriori error estimates are derived from the new bounds and generalized error expansion we establish. We theoretically justify the empirical claim by Saad. Moreover, we establish the error expansion of the Krylov-like approximation for $f(z)$ sufficiently smooth, which generalizes Saad's result on the Arnoldi approximation to $e^{-\tau A}v$. Similarly, it is shown that the first term of the generalized error expansion can be used as a reliable a posteriori estimate for the Krylov-like approximation to $f(A)v$ for sufficiently smooth $f(z)$.Numerical examples are reported to demonstrate the effectiveness of the a posteriori error estimates for the Krylov-like approximations to $e^{-\tau A}v$, $\cos(A)v$ and $\sin(A)v$.The errors of the Krylov subspace approximations to $f(A)v$ are closely related to the residual of their counterparts for solving the linear systems.Based on the principle of the GMRES method,we propose a new method, called the harmonic Ritz method, for approximating $f(A)v$. The method is so named because it is associated with the harmonic Ritz values of $A$ with respect to a certain target $\xi$.We analyze the convergence of it and propose a restarted harmonic Ritz algorithm.Compared with the standard Arnoldi approximation, the new method has two remarkable advantages:firstly, it converges more smoothly;secondly, the restarted algorithm always converges with a suitable $\xi$.The second advantage is more important and has great practical value. We consider the crucial issue of selecting $\xi$ and show how to determine it robustly and cheaply. Importantly, we justify that the harmonic Ritz method is not sensitive to $\xi$ theoretically and numerically, demonstrating that our new method is robust and of generally.The choice of the restarting vector has great influence on the convergence of the restarted algorithms and can be elegantly dealt with the implicit restarting approach.We have adapted the implicit restarting technique to the Arnoldi method for approximating $f(A)v$.Furthermore, we have proved that the implicitly restarted Arnoldi method with exact shifts used is mathematically equivalent to the restarted Arnoldi method with deflation.
