ABSTRACT
In the previous chapters I have emphasised my view that iterative or recursive techniques are well suited for calculations on a microcomputer with a limited RAM capacity. In this chapter I give several specific examples of how to set up an iterative method for handling problems. One common way to refer to successive estimates in an iterative process is to use the symbols x n and xn+1 , whereas I use x and y. My input-output point of view (§3.2) seems appropriate for computer work and the simple standard notation dy/dx or y’ can be used in discussions of convergence properties. I describe the simple theory in §3.2 and give Newton’s method, with a specific cubic equation example, in §3.3. An interesting example of a conflict between universality and accuracy for a program arises in that discussion. The iterative procedure for calculating inverses, described in §3.4, is one of the most useful simple methods in quantum mechanics, and has links with matrix theory, perturbation theory (§9.3), Padé approximant theory (§6.3) and operator resolvent theory (§12.6). In §§3.5 to 3.7 I outline how some matrix problems can be handled by an iterative approach. In particular I indicate how the Gauss-Seidel method can be rendered applicable even for matrices which are not diagonally dominant. The matrix folding method of §3.6 is really an application of Brillouin-Wigner perturbation theory to a numerical problem, and involves one of the nicest new programs which I have devised while writing this book. Exercise 4 introduces the Aitken process for treating sequences and series; this process plays a role in chapters 5 and 6.
