Comparison of a pure greedy algorithm with a pure greedy one in a pair of dictionaries

In this paper, we compare the standard Pure Greedy Algorithm (PGA) with its modification — PGA in a pair of dictionaries. We show that PGA in a pair of dictionaries converges in certain cases in a finite number of steps, while the standard PGA for each individual dictionary has non-zero remainders at each step; at the same time, in certain cases the opposite holds true. Similarly, for the comparison of PGA in a pair of dictionaries vs standard PGA in a union of these dictionaries both situations are possible. For the convergence rate, we also prove that in certain cases PGA in a pair of dictionaries can be faster, and in certain cases can be slower than the standard PGA in individual dictionaries.