  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.
