马氏函数的有理逼近及其在Toeplitz矩阵马氏函数计算中的应用
On the rational approximation of Markov functions,with applications to the computation of Markovfunctions of Toeplitz matrices
摘要：
研究了用$r(A)$逼近矩阵函数$f(A)$的问题，其中$f$是马尔可夫函数，$r$是$f$的有理插值，$A$是对称Toeplitz矩阵.在第一步中，我们得到一个新的上界的相对插值误差$1-r/f$的频谱间隔$A$。通过在所有插值点上最小化这个上界，我们得到了一个新的，简单和尖锐的相对插值误差的先验界。然后，我们考虑三种不同的方法表示和计算的有理插值$r$。理论和数值证据表明，任何这些方法的标量参数允许实现高精度，即使在有限精度算术的存在。最后，我们调查的问题，有效地评估$r(A)$，事实证明，矩阵参数的相对误差是很小的，如果我们使用的部分分式分解$r$以下Antoulas和梅奥。一个重要的作用是由一个新的停止标准，确保自动找到程度的$r$导致一个小的错误，即使在存在的有限精度的算法。
英文摘要：
We investigate the problem of approximating the matrix function $f(A)$ by $r(A)$, with $f$ a Markov function, $r$ a rational interpolant of $f$, and $A$ a symmetric Toeplitz matrix. In a first step, we obtain a new upper bound for the relative interpolation error $1-r/f$ on the spectral interval of $A$. By minimizing this upper bound over all interpolation points, we obtain a new, simple and sharp a priori bound for the relative interpolation error. We then consider three different approaches of representing and computing the rational interpolant $r$. Theoretical and numerical evidence is given that any of these methods for a scalar argument allows to achieve high precision, even in the presence of finite precision arithmetic. We finally investigate the problem of efficiently evaluating $r(A)$, where it turns out that the relative error for a matrix argument is only small if we use a partial fraction decomposition for $r$ following Antoulas and Mayo. An important role is played by a new stopping criterion which ensures to automatically find the degree of $r$ leading to a small error, even in presence of finite precision arithmetic.
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