  We present an algorithm to compute the Hodge ideals of $\mathbb{Q}$-divisors
associated to any reduced effective divisor $D$. The computation of the Hodge
ideals is based on an algorithm to compute parts of the $V$-filtration of
Malgrange and Kashiwara on $\iota_{+}\mathscr{O}_X(*D)$ and the
characterization of the Hodge ideals in terms of this $V$-filtration. In
particular, this gives a new algorithm to compute the multiplier ideals and the
jumping numbers of any effective divisor.
