for (j=jz1;j | 772 Chapter 17. Two Point Boundary Value Problems for j jz1 j jz2 j Loop over columns to be zeroed. for l jm1 l jm2 l Loop over columns altered. vx c ic l loff kc for i iz1 i iz2 i s i l - s i j vx Loop over rows. vx c ic jcf kc for i iz1 i iz2 i s i jmf - s i j vx Plus final element. ic 1 Algebraically Difficult Sets of Differential Equations Relaxation methods allow you to take advantage of an additional opportunity that while not obvious can speed up some calculations enormously. It is not necessary that the set of variables yj k correspond exactly with the dependent variables of the original differential equations. They can be related to those variables through algebraic equations. Obviously it is necessary only that the solution variables allow us to evaluate the functions y g B C that are used to construct the FDEs from the ODEs. In some problems g depends on functions of y that are known only implicitly so that iterative solutions are necessary to evaluate functions in the ODEs. Often one can dispense with this internal nonlinear problem by defining a new set of variables from which both y g and the boundary conditions can be obtained directly. A typical example occurs in physical problems where the equations require solution of a complex equation of state that can be expressed in more convenient terms using variables other than the original dependent variables in the ODE. While this approach is analogous to performing an analytic change of variables directly on the original ODEs such an analytic transformation might be prohibitively complicated. The change of variables in the relaxation method is easy and requires no analytic manipulations. CITED REFERENCES AND FURTHER READING Eggleton . 1971 Monthly Notices ofthe RoyalAstronomical Society vol. 151 pp. 351-364. 1 Keller . 1968 Numerical Methods for Two-Point Boundary-Value Problems Waltham MA Blaisdell . Kippenhan R. Weigert A. and Hofmeister E. 1968 in Methods in Computational Physics vol. 7 New York
