# Shooting method

In numerical analysis, the **shooting method** is a method for solving a boundary value problem by reducing it to the system of an initial value problem. Roughly speaking, we 'shoot' out trajectories in different directions until we find a trajectory that has the desired boundary value. The following exposition may be clarified by this illustration of the shooting method.

For a boundary value problem of a second-order ordinary differential equation, the method is stated as follows. Let

be the boundary value problem.
Let *y*(*t*; *a*) denote the solution of the initial value problem

Define the function *F*(*a*) as the difference between *y*(*t*_{1}; *a*) and the specified boundary value *y*_{1}.

If *F* has a root *a* then the solution *y*(*t*; *a*) of the corresponding initial value problem is also a solution of the boundary value problem.
Conversely, if the boundary value problem has a solution *y*(*t*), then *y*(*t*) is also the unique solution *y*(*t*; *a*) of the initial value problem where *a = y*'(*t*_{0}), thus *a* is a root of *F*.

The usual methods for finding roots may be employed here, such as the bisection method or Newton's method.

## Linear shooting method

The boundary value problem is linear if *f* has the form

In this case, the solution to the boundary value problem is usually given by:

where is the solution to the initial value problem:

and is the solution to the initial value problem:

See the proof for the precise condition under which this result holds.

## Example

A boundary value problem is given as follows by Stoer and Burlisch[1] (Section 7.3.1).

was solved for *s* = −1, −2, −3, ..., −100, and *F*(*s*) = *w*(1;*s*) − 1 plotted in the first figure.
Inspecting the plot of *F*,
we see that there are roots near −8 and −36.
Some trajectories of *w*(*t*;*s*) are shown in the second figure.

Stoer and Burlisch[1] state that there are two solutions,
which can be found by algebraic methods.
These correspond to the initial conditions *w*′(0) = −8 and *w*′(0) = −35.9 (approximately).

## Notes

- Stoer, J. and Burlisch, R.
*Introduction to Numerical Analysis*. New York: Springer-Verlag, 1980.

## References

- Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007). "Section 18.1. The Shooting Method".
*Numerical Recipes: The Art of Scientific Computing*(3rd ed.). New York: Cambridge University Press. ISBN 978-0-521-88068-8.

## External links

- Brief Description of ODEPACK
*(at Netlib; contains LSODE)* - Shooting method of solving boundary value problems – Notes, PPT, Maple, Mathcad, Matlab, Mathematica at
*Holistic Numerical Methods Institute*