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  "body": "\n## Definition\n\nLet $a \u003c b$, let $\\alpha, \\beta \\in \\R$, and let $f \\colon [a, b] \\times \\R^2 \\to \\R$ be continuous and locally Lipschitz in its last two arguments.\nConsider the [[boundary value problem|TM8VQJWGHZ]]\n$$y'' = f(x, y, y'), \\qquad y(a) = \\alpha, \\qquad y(b) = \\beta.$$\nFor $s \\in \\R$, let $y(\\cdot\\,; s)$ be the solution of the [[initial value problem|0FXDW992DS]] $y'' = f(x, y, y')$, $y(a) = \\alpha$, $y'(a) = s$.\nLet $S$ be the set of $s$ for which this solution exists on all of $[a, b]$, and define $F \\colon S \\to \\R$ by\n$$F(s) = y(b; s) - \\beta.$$\nThe shooting method is the procedure that finds a root $s^*$ of $F$ and returns $y(\\cdot\\,; s^*)$.\nThe root is found, for example, by bisection, by the secant method or by [[Newton's method|07EXVNF9VP]].\n\n## Remarks\n\nEach root $s^*$ of $F$ gives a solution $y(\\cdot\\,; s^*)$ of the boundary value problem, and each solution arises this way from $s^* = y'(a)$.\nExample: for $y'' = 0$, $y(0) = 0$, $y(1) = 3$, one gets $y(x; s) = s x$ and $F(s) = s - 3$, so $s^* = 3$ and $y = 3x$.\nNon-example of success: for $y'' = -y$, $y(0) = 0$, $y(\\pi) = 1$, one gets $y(x; s) = s \\sin x$ and $F(s) = -1$ for every $s$.\nThen $F$ has no root, and indeed the boundary value problem has no solution.\nWhen the solutions of the initial value problems grow fast, small errors in $s$ blow up.\nThen one splits $[a, b]$ into pieces and shoots on each piece; this is called multiple shooting.\n\n## Explanations\n\n### L1 human\n\nThe method guesses the slope at the left end, solves the equation from there, and looks at where the solution ends up on the right.\nThe miss distance is a function of the guess.\nMaking the miss zero is a problem of finding a root of one function of one variable.\n",
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