  In this paper, we develop efficient and accurate algorithms for evaluating
$\varphi(A)$ and $\varphi(A)b$, where $A$ is an $N\times N$ matrix, $b$ is an
$N$ dimensional vector and $\varphi$ is the function defined by
$\varphi(x)\equiv\sum\limits^{\infty}_{k=0}\frac{z^k}{(1+k)!}$. Such matrix
function (the so-called $\varphi$-function) plays a key role in a class of
numerical methods well-known as exponential integrators. The algorithms use the
scaling and modified squaring procedure combined with truncated Taylor series.
The backward error analysis is presented to find the optimal value of the
scaling and the degree of the Taylor approximation. Some useful techniques are
employed for reducing the computational cost. Numerical comparisons with
state-of-the-art algorithms show that the algorithms perform well in both
accuracy and efficiency.
