When one is memorizing just a verse of Scripture, or even some chapters, it can be relatively simple to tell if one has memorized the text perfectly or not. When one progresses along an extended journey into reciting multiple chapters as part of memorizing a book of the Bible, it can be harder to tell if one is reciting it perfectly, or if one is missing small phrases here and there. (Of course, large phrases or even sections can get missed, too.)
For these longer stretches, I recommend using a voice recorder of some kind. In the old days this meant buying a separate device like a digital voice recorder. (In the really old days this meant something like a tape recorder or a micro-cassette recorder.) These days, one may already have everything needed.
Any smartphone (or iPod touch) with a camera already likely takes video and not just photos, so that means it already has a microphone. Given speakerphone or video playing capabilities, that usually means there is also an external speaker or a headphone jack one can use for listening to the recording. (Strangely, Apple seems to think removing the headphone jack on its devices will be without consequence.)
iOS already comes with a Voice Memos app, and one can also download other software in which one can customize settings. I use Voice Record Pro 7 which has low-usage (low-quality) presets, and one can tweak things further from there. I recommend some test recording and listening to find settings one likes and finds sufficient.
Voice audio does not require high quality (32 kbps max), so one can also dial down the settings in another audio recording app to make its audio files take less space. Approximately 1 MB per minute is a good range for audio voice recordings. These files are still large when one is reciting Romans, for instance, which takes at least 36 minutes.
An alternative to voice recording is having a friend check your progress. I have not had much success with this. Once I was in my final weeks of memorizing Colossians, and I had a coworker read along while I was reciting. Having never heard me do that before, my listener was a bit distracted by what I was doing, and was not very helpful at inspecting how well I was doing it. If one has a memorizing partner, this method may work better.
One way or the other, reciting with another person is useful at times for building relationships, and encouraging others to read the Word and to memorize it. For the majority of the times when no one else wants to listen, a voice recorder is an excellent tool for ensuring that one is meeting his aim for perfection.