We present a massively parallel algorithm for computing persistent homology, a concept within the field of topological data analysis, and we implement it in the purely functional array-based language Futhark, which has an efficient compiler targeting GPUs. Computing persistent homology consists of bringing a certain sparse matrix to a reduced form. We compare our implementation with OpenPH, an existing library for computing persistent homology on GPUs, and on large matrices we achieve speedups of 2.3 to 5. Our work shows both that persistent homology can be computed efficiently entirely on GPU hardware, and that Futhark can be used for this kind of sparse matrix manipulation.