Picard iteration involves the theory that metric space has "existence" and "uniqueness" and that, if specified regional coordinates are continuous, there is always a single solution to the initial value. There are two basic types of Picard iteration. The first uses computers to generate random, numerical sequences which "converge or reduce, into a single mathematical solution. Computers are essential in speeding up the process. A huge number of different computations can be involved each time.
The second application of Picard iteration is the generation of a sequence of "functions" which converge to a solution. Computers are essential, as before. There are different software applications that can be used to perform the task of function generation, including "Maple", MuPad" and "Derive." The software makes it easier for students to control and assess the output of the Picard iterative process and to effectively discover the unique solution to some first order differentials.
Rate of convergence during iteration can be increased by the use of a "convergence acceleration" method. One example of such a procedure is "Aitken's delta-squared process." When Aitken's is applied to fixed point iteration -- such as Picard's -- it is called the "Steffensen's method" and it has been demonstrated to produce a rate of convergence that is at least quadratic, which is rare, according to Joe Mahaffy at San Diego State University.
Functional analysis is a branch of mathematics that draws together the ideas of differential and integral equations, quantum mechanics and variation calculus. In 2007, Bent E. Petersen, an expert at Oregon State University, explained that basic existence theorem for differentials can evolve from abstract functional analysis. He worked using the well-known Banach Contraction Mapping Principle, the theory of Complete Metric Space and the characterization of what are called "compact sets of total boundedness."