Editing Shooting method (section)

== Mathematical description ==
Suppose one wants to solve the boundary-value problem<math display="block"> y''(t) = f(t, y(t), y'(t)), \quad y(t_0) = y_0, \quad y(t_1) = y_1. </math>Let <math> y(t; a) </math> solve the initial-value problem<math display="block"> y''(t) = f(t, y(t), y'(t)), \quad y(t_0) = y_0, \quad y'(t_0) = a. </math>If <math> y(t_1; a) = y_1 </math>, then <math> y(t; a) </math> is also a solution of the boundary-value problem.

The shooting method is the process of solving the initial value problem for many different values of <math> a </math> until one finds the solution <math> y(t; a) </math> that satisfies the desired boundary conditions. Typically, one does so [[Numerical methods for ordinary differential equations|numerically]]. The solution(s) correspond to root(s) of <math display="block"> F(a) = y(t_1; a) - y_1.</math>To systematically vary the shooting parameter <math> a </math> and find the root, one can employ standard root-finding algorithms like the [[bisection method]] or [[Newton's method]].

Roots of <math> F </math> and solutions to the boundary value problem are equivalent. If <math> a </math> is a root of <math> F </math>, then <math> y(t; a) </math> is a solution of the boundary value problem. Conversely, if the boundary value problem has a solution <math> y(t) </math>, it is also the unique solution <math> y(t; a) </math> of the initial value problem where <math> a = y'(t_0) </math>, so <math> a </math> is a root of <math> F </math>.