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The linearization methods as a basis to derive the relaxation and the shooting methods

This chapter investigates numerical solution of nonlinear two-point boundary value problems. It establishes a connection between three important, seemingly unrelated, classes of iterative methods, namely: the linearization methods, the relaxation methods (finite difference methods), and the shooting methods. It has recently been demonstrated that using finite differences to discretize the sequence of linear problems obtained by quasi-linearization, Picard linearization, or constant-slope linearization, leads to the usual iteration formula of the respective relaxation method. Thus, the linearization methods can be used as a basis to derive the relaxation methods. In this work we demonstrate that the shooting methods can be derived from the linearization methods, too. We show that relaxing a shooting trajectory, i.e. an initial value problem solution, is in fact a projection transformation. The obtained function, called projection trajectory, can be used to correct the initial condition. Using the new initial condition, we can find a new shooting trajectory, and so on. The described procedure is called shooting-projection iteration (SPI). We show that using the quasi-linearization equation to relax (project) the shooting trajectory leads to the usual shooting by Newton method, the constant-slope linearization leads to the usual shooting by constant-slope method, while the Picard linearization leads to the recently proposed shooting-projection method. Therefore, the latter method can rightfully be called shooting by Picard method. A possible application of the new theoretical results is suggested and numerical computer experiments are presented. MATLAB codes are provided.