What is the purpose for having the firstDerivative() and secondDerivative() methods on a UnivariateRealFunction ?
The first derivative could be used in a Newton solver. The second derivative could be used in yet another higher order root finder, or for providing error estimates and guidance in modified, more robust Newton root finders (sorry, I'm too lazy to look up method names).
The reason they are in the function class is that partial results might be shared for calculating the values of the function and its derivatives. Reusing intermediate results can bring Newton in front of the Brent-Dekker solver in terms of efficiency again.
J.Pietschmann
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