Math 2.0

Numerical analysis Definition

Shooting method

Builds on Boundary value problemInitial value problemNewton's method

Definition

Let a<ba < b, let α,β∈ℝ\alpha, \beta \in \R, and let f ⁣:[a,b]×ℝ2→ℝf \colon [a, b] \times \R^2 \to \R be continuous and locally Lipschitz in its last two arguments. Consider the boundary value problem y′′=f(x,y,y′),y(a)=α,y(b)=β.y'' = f(x, y, y'), \qquad y(a) = \alpha, \qquad y(b) = \beta. For s∈ℝs \in \R, let y(⋅ ;s)y(\cdot\,; s) be the solution of the initial value problem y′′=f(x,y,y′)y'' = f(x, y, y'), y(a)=αy(a) = \alpha, y′(a)=sy'(a) = s. Let SS be the set of ss for which this solution exists on all of [a,b][a, b], and define F ⁣:S→ℝF \colon S \to \R by F(s)=y(b;s)−β.F(s) = y(b; s) - \beta. The shooting method is the procedure that finds a root s∗s^* of FF and returns y(⋅ ;s∗)y(\cdot\,; s^*). The root is found, for example, by bisection, by the secant method or by Newton's method.

Remarks

Each root s∗s^* of FF gives a solution y(⋅ ;s∗)y(\cdot\,; s^*) of the boundary value problem, and each solution arises this way from s∗=y′(a)s^* = y'(a). Example: for y′′=0y'' = 0, y(0)=0y(0) = 0, y(1)=3y(1) = 3, one gets y(x;s)=sxy(x; s) = s x and F(s)=s−3F(s) = s - 3, so s∗=3s^* = 3 and y=3xy = 3x. Non-example of success: for y′′=−yy'' = -y, y(0)=0y(0) = 0, y(π)=1y(\pi) = 1, one gets y(x;s)=ssin⁡xy(x; s) = s \sin x and F(s)=−1F(s) = -1 for every ss. Then FF has no root, and indeed the boundary value problem has no solution. When the solutions of the initial value problems grow fast, small errors in ss blow up. Then one splits [a,b][a, b] into pieces and shoots on each piece; this is called multiple shooting.

Reading path

To understand this page from the ground up, read these 8 pages in order. The foundations come first.

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Boundary value problem, concept, no evidence levelBoundary value problemconceptInitial value problem, concept, no evidence levelInitial value problemconceptLimit point of a set, definition, no evidence levelLimit point of a setdefinitionLimit of a function, definition, no evidence levelLimit of a functiondefinitionDerivative, definition, no evidence levelDerivativedefinitionTangent line, concept, no evidence levelTangent lineconceptNewton's method, concept, no evidence levelNewton's methodconceptShooting method, definition, no evidence levelShooting methoddefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Boundary value problem concept
  2. Initial value problem concept
  3. Limit point of a set definition
  4. Limit of a function definition
  5. Derivative definition
  6. Tangent line concept
  7. Newton's method concept
  8. Shooting method definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Shooting_method
About this page
TagDKR8R8Y1RJ
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Kinddefinition
WritingW0 stub
MSC65L10, 34B15
DefinesShooting method
UsesBoundary value problem, Initial value problem, Newton's method
Changed2 November 2026

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