 If the FMM is then applied in a hierarchical manner, it can improve the complexity of matrix-vector products in an iterative solver from formula_1 to formula_2 in finite arithmetic, i.e., given a tolerance formula_3, the matrix-vector product is guaranteed to be within a tolerance formula_4 The dependence of the complexity on the tolerance formula_3 is formula_6, i.e., the complexity of FMM is formula_7.This has expanded the area of applicability of the MOM to far greater problems than were previously possible.The FMM, introduced by Rokhlin Jr. and Greengard has been said to be one of the top ten algorithms of the 20th century.