for (i=1;i | 762 Chapter 17. Two Point Boundary Value Problems for i 1 i n i f i f1 i -f2 i free_vector y 1 nvar free_vector f2 1 nvar free_vector f1 1 nvar There are boundary value problems where even shooting to a fitting point fails the integration interval has to be partitioned by several fitting points with the solution being matched at each such point. For more details see 1 . CITED REFERENCES AND FURTHER READING Acton . 1970 Numerical Methods That Work 1990 corrected edition Washington Mathematical Association of America . Keller . 1968 Numerical Methods for Two-Point Boundary-Value Problems Waltham MA Blaisdell . Stoer J. and Bulirsch R. 1980 Introduction to NumericalAnalysis New York Springer-Verlag . 1 Relaxation Methods In relaxation methods we replace ODEs by approximate finite-difference equations FDEs on a grid or mesh of points that spans the domain of interest. As a typical example we could replace a general first-order differential equation X gx y with an algebraic equation relating function values at two points k k 1 yk yk-i xk xk-i g xk xk-i 1 yk yk-i 0 The form of the FDE in illustrates the idea but not uniquely There are many ways to turn the ODE into an FDE. When the problem involves N coupled first-order ODEs represented by FDEs on a mesh of M points a solution consists of values for N dependent functions given at each of the M mesh points or N x M variables in all. The relaxation method determines the solution by starting with a guess and improving it iteratively. As the iterations improve the solution the result is said to relax to the true solution. While several iteration schemes are possible for most problems our old standby multidimensional Newton s method works well. The method produces a matrix equation that must be solved but the matrix takes a special block diagonal form that allows it to be inverted far more economically both in time and storage than would be possible for a general matrix of size MN x
