DerivativeStructure for symbolic derivatives

[axkr]

Can you show an example/outline, how to implement a DerivativeStructure based on symbolic derivation?

At the moment I've used my own NewtonSolver in FindRoot, but it would be nice to have a general solution which I can use with hipparchus.

[maisonobe]

In most cases, the idea behind DerivativeStructure is to remove the need for symbolic differentiation: user just implement f(x) with x being a DerivativeStructure instead of a primitive double, and the derivatives come for free.

If user already has expressions for both f(x) and df(x)/dx (and possibly higher order derivatives), then using DerivativeStructure can still be used, but in this case the DerivativeStructure will only be used to store the derivatives, not to compute them. Here is one way to build a DifferentiableUnivariateFunction from two UnivariateFunction implementing f(x) and df(x)/dx in the case symbolic differentiation allowed the user to implement both

  public static void DifferentiableUnivariateFunction toDifferentiableUnivariateFunction(
   final  UnivariateFunction f, final UnivariateFunction fPrime) {
    return new DifferentiableUnivariateFunction {

      public double value(final double x) {
        return f.value(x);
      }

      public DerivativeStructure value(final DerivativeStructure x) {
        return x.getFactory().build(f.value(x),
                                                  fPrime.value(x) * x.getPartialDerivative(1));
      }

    };
  }

Beware I wrote this just off the top of my head, you'll have to adjust to suit your needs. This also will
work only if the DerivativeStructure variable x passed as the argument of the second method is a
derivative with one parameter and order one. If the variable is more rich, we should provide more derivatives to the factory build method.

Also be aware that Newton method which requires derivatives is in fact less efficient than for example BracketingNthOrderBrentSolver. The first method is only order 2 whereas the other method can be alomst any order. The default is to set it to order 5 and it work very well in this configuration, even without derivatives.

Is this what you were looking for?