  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.
