Category talk:Root-finding algorithms
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Why is the definition of root-finding algorithms restricted to the case where x is scalar? Newton's method can also be used to find zeros of vector functions. I also don't understand why root-finding algorithms should be a subcategory of optimization algorithms. None of the root-finding algorithms is in common use in optimization, as far as I know. -- Jitse Niesen 19:49, 25 Jun 2004 (UTC)
- That's the definition of a root --- a scalar where a scalar-valued function reaches zero. Root-finding is much much easier than general non-linear equation solving. See, e.g., [1] Newton's method is both a root-finding algorithm and an optimization algorithm, depending on which version of Newton's method is used. The article Newton's method focuses almost exclusively on the scalar version, so it seemed like it belongs here.
- Root finding can be considered a form of optimization. If you apply root-finding to f'(x) = 0, where f is scalar-valued and x is a scalar, then you find the extremal points of f(x). But, this is not a strong association. If you'd like to move root-finding back to Category:Numerical analysis, I wouldn't be upset.
- -- hike395 01:50, 26 Jun 2004 (UTC)
You're probably right that root is usually defined for scalar functions only. It's just that root-finding algorithm says "x may be a single real number or a vector called the root." I also agree that Newton's method at the moment is about solving scalar equations (though I hope that this changes at some point).
- Tried to fix these problems with an extra explanatory paragraph over in root-finding algorithm, and extra sentences at Newton's method -- hike395
However, I do think of optimization and solving equations as different things, even though they are obviously related (particularly when restricting ourselves to scalar functions and unconstrained optimization). I checked some references: Atkinson's An Introduction to Numerical Analysis has a chapter on rootfinding for nonlinear equations, with sections on systems of equations and unconstrained optimization; Numerical Recipes has separated chapters on root finding and nonlinear sets of equations, and on minimization or maximization of functions; the Mathematics Subject Classification (MSC) of the American Mathematical Society uses 65H for solving nonlinear equations, and 65K for mathematical programming, optimization and variational techniques. So I did move root-finding back.
-- Jitse Niesen 14:06, 26 Jun 2004 (UTC)
- That's fine with me. -- hike395 16:13, 26 Jun 2004 (UTC)