The two-parameter Mittag-Leffler function $E_{\alpha, \beta}$ is of fundamental importance in fractional calculus. It appears frequently in the solutions of fractional differential and integral equations. Nonetheless, this vital function is often expensive to compute. Several attempts have been made to construct cost-effective and accurate approximations. These attempts focus mainly on the completely monotone Mittag-Leffler functions. However, when $\alpha > 1$ the monotonicity property is largely lost and as such roots and oscillations are exhibited. Consequently, existing approximants constructed mainly for $\alpha \in (0,1)$ often fail to capture this oscillatory behavior. In this paper, we construct computationally efficient and accurate rational approximants for $E_{\alpha, \beta}(-t)$, $t \ge 0$, with $\alpha \in (1,2)$. This construction is fundamentally based on the decomposition of Mittag-Leffler function with real roots into one without and a polynomial. Following which new approximants are constructed by combining the global Pad\'e approximation with a polynomial of appropriate degree. The rational approximants are extended to approximation of matrix Mittag-Leffler and different approaches to achieve efficient implementation for matrix arguments are discussed. Numerical experiments are provided to illustrate the significant accuracy improvement achieved by the proposed approximants.

翻译：双参数Mittag-Leffler函数 $E_{\alpha, \beta}$ 在分数阶微积分中具有基础性重要地位，频繁出现在分数阶微分与积分方程的解中。然而，这一关键函数的计算往往代价高昂。已有若干研究致力于构建经济高效的精确逼近方法，但这些研究主要集中于完全单调的Mittag-Leffler函数。当 $\alpha > 1$ 时，该函数的单调性基本丧失，从而表现出根与振荡特性。因此，现有主要针对 $\alpha \in (0,1)$ 构建的逼近方法难以捕捉这种振荡行为。本文针对 $\alpha \in (1,2)$ 情形下 $t \ge 0$ 时的 $E_{\alpha, \beta}(-t)$ 函数，构建了计算高效且精确的有理逼近方法。该构建方法基于将具有实根的Mittag-Leffler函数分解为无根部分与多项式，进而通过结合全局Padé逼近与适当次数的多项式构造新的逼近形式。我们将有理逼近拓展至矩阵Mittag-Leffler函数的逼近，并讨论了针对矩阵参数实现高效计算的不同方案。数值实验表明，本文所提议的逼近方法在精度上取得了显著提升。