I work in related areas of math but don't know anything about the applied side of this stuff. The general rule of thumb though is that whatever algorithm the mathematicians are proving correct is chosen so that it is easy to analyze and isn't going to have the best convergence properties for actual use. After this result, though, expect others to tackle questions like faster converging algorithms, stability, etc. There are exceptions though: Newton's method was both ridiculously fast and easy to work with mathematically and is still used in both applied and pure contexts today.
