If you are new to the computation of persistent homology a good idea is to start with javaPlex, which is the new library of the Plex family:

After that you could try out Perseus, which implements Morse theoretic reductions to reduce the size of the complex:If you are familiar with R you might want to try out these two libraries:

....and if you are familiar with GAP this one:The following is a powerful toolbox of algorithms for the computation of barcodes from the boundary matrix (the first package to implement parallel computation in a shared setting): ....if you do not want to compute the boundary matrix yourself, you can use its spin-off library (which implements algorithms for parallel computations in a distributed setting):

Another powerful software is this:

This is the only software to implement, among other things, the computation of PH with Čech and alpha complexes, and vineyards:

...while the following library implements, among other things, a parallel computation of homology based on the Mayer-Vietoris principle and the computation of PH over finite fields:

The following is a module for the Kenzo library which implements computation of PH in a more general setting, i.e. with integer coefficients and for infinite spaces (provided their homology is effective): Kenzo module for PH (see the companion preprint for instructions)

Finally some more libraries: ....to do statistics on persistence diagrams: ....for simplicial maps: ....to compute the weight rank clique filtration for networks: ....and its older version: ....for fast computation of PH with alpha complexes in 1-D: