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In [[numerical analysis]], the '''shooting method''' is a method for solving a [[boundary value problem]] by reducing it to the solution of an [[initial value problem]]. The following exposition may be clarified by this [[#Example|illustration of the shooting method]].
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For a boundary value problem of a second-order [[ordinary differential equation]], the method is stated as follows.
Let
 
:<math> y''(t) = f(t, y(t), y'(t)), \quad y(t_0) = y_0, \quad y(t_1) = y_1 </math>
 
be the boundary value problem.
Let ''y''(''t''; ''a'') denote the solution of the initial value problem
 
:<math> y''(t) = f(t, y(t), y'(t)), \quad y(t_0) = y_0, \quad y'(t_0) = a </math>
 
Define the function ''F''(''a'') as the difference between ''y''(''t''<sub>1</sub>; ''a'') and the specified boundary value ''y''<sub>1</sub>.
 
:<math> F(a) = y(t_1; a) - y_1 \,</math>
 
If the boundary value problem has a solution, then ''F'' has a [[root of a function|root]],
and that root is just the value of ''y''<nowiki>'</nowiki>(''t''<sub>0</sub>) which yields a solution ''y''(''t'') of the boundary value problem.
 
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
:<math> f(t, y(t), y'(t))=p(t)y'(t)+q(t)y(t)+r(t). \, </math>
In this case, the solution to the boundary value problem is usually given by:
:<math>y(t) = y_{(1)}(t)+\frac{y_1-y_{(1)}(t_1)}{y_{(2)}(t_1)}y_{(2)}(t)</math>
where <math>y_{(1)}(t)</math> is the solution to the initial value problem:
:<math>y_{(1)}''(t) = p(t)y_{(1)}'(t)+q(t)y_{(1)}(t)+r(t),\quad y_{(1)}(t_0) = y_0, \quad y_{(1)}'(t_0) = 0, </math>
and <math>y_{(2)}(t)</math> is the solution to the initial value problem:
:<math>y_{(2)}''(t) = p(t)y_{(2)}'(t)+q(t)y_{(2)}(t),\quad y_{(2)}(t_0) = 0, \quad y_{(2)}'(t_0) = 1. </math>
See [http://math.fullerton.edu/mathews/n2003/shootingmethod/ShootingProof.pdf the proof] for the precise condition under which this result holds.
 
== Example ==
 
A [[boundary value problem]] is given as follows by Stoer and Burlisch (Section 7.3.1).
 
:<math> w''(t) = \frac{3}{2} w^2, \quad w(0) = 4, \quad w(1) = 1 </math>
 
The [[initial value problem]]
 
:<math> w''(t) = \frac{3}{2} w^2, \quad w(0) = 4, \quad w'(0) = s</math>
 
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.
 
Solutions of the initial value problem were computed by using the LSODE algorithm, as implemented in the mathematics package [[GNU Octave]].
 
Stoer and Bulirsch 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).
 
[[Image:Shooting_method_error.png|thumb|center|400px|The function ''F''(''s'') = ''w''(1;''s'') − 1.]]
[[Image:Shooting_method_trajectories.png|thumb|center|400px|Trajectories ''w''(''t'';''s'') for ''s'' = ''w''<nowiki>'</nowiki>(0) equal to −7, −8, −10, −36, and −40 (red, green, blue, cyan, and magenta, respectively). The point (1,1) is marked with a red diamond.]]
<br clear="all" />
 
== See also ==
* [[Direct multiple shooting method]]
* [[Computation of radiowave attenuation in the atmosphere]]
 
== References ==
 
* Josef Stoer and Roland Bulirsch. ''Introduction to Numerical Analysis''. New York: Springer-Verlag, 1980. ''(See Section 7.3.)''
*{{Cite book | last1=Press | first1=WH | last2=Teukolsky | first2=SA | last3=Vetterling | first3=WT | last4=Flannery | first4=BP | year=2007 | title=Numerical Recipes: The Art of Scientific Computing | edition=3rd | publisher=Cambridge University Press |  publication-place=New York | isbn=978-0-521-88068-8 | chapter=Section 18.1. The Shooting Method | chapter-url=http://apps.nrbook.com/empanel/index.html#pg=959}}
 
== External links ==
* [http://www.netlib.org/odepack/opks-sum Brief Description of ODEPACK] ''(at [[Netlib]]; contains LSODE)''
* [http://numericalmethods.eng.usf.edu/topics/shooting_method.html Shooting method of solving boundary value problems – Notes, PPT, Maple, Mathcad, Matlab, Mathematica] at ''Holistic Numerical Methods Institute'' [http://numericalmethods.eng.usf.edu]
* [http://math.fullerton.edu/mathews/n2003/ShootingMod.html Shooting Method for Boundary Value Problems]
 
[[Category:Numerical differential equations]]

Revision as of 09:17, 5 March 2014

Then in a pair connected with a lovely island where your amazing peaceful village is part way through beaches and woods right up until the enemies known as the BlackGuard led by Lieutenant Hammerman invades your remote island. After managing to guard against a minuscule invasion force, he offers to avenge his loss in battle.

Yet unfortunately Supercell, by allowing currently the illusion on the multi player game, taps into instinctual male drive to assist you to from the status hierarchy, and even though its unattainable to the surface of your hierarchy if there's no need been logging in morning because the game was launched plus you invested fundamental money in extra builders, the drive for getting a small bit further obliges enough visitors to make investments a real income on virtual 'gems'" that hobby could be the top-grossing app within the Practical application Store.

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