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Date: Wed, 22 May 2013 13:52:42 +0200
From: =?UTF-8?B?Q2hyaXN0b3BoIEjDtmdlcg==?=
 <christoph.hoeger@tu-berlin.de>
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Subject: [math] Total derivative of a MultivariateVectorFunction
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Dear all,

I am currently working with the DerivativeStructure-based AD framework
integrated into math 3.2.

Calculating the n-th order partial derivatives works fine, but I am
facing some trouble calculating the n-th order total derivative of a
MultivariateVectorFunction.

My Idea was to have a class Derivative (implementing
MultivariateVectorFunction) that delegates to its base
MultivariateVectorFunction and returns the first order total derivative
of that function. Chaining many such classes should create the n-th
order total derivative (direct creation would be an optimization).

The class looks like this:

public final class Derivative implements
MultivariateDifferentiableFunction {

    private final MultivariateDifferentiableFunction base;

    @Override
    public double value(double[] point) {
        final int dim = 1 + point.length / 2;

        final DerivativeStructure[] dpoint = new DerivativeStructure[dim];
        for (int i = 0; i < dim; i++)
            dpoint[i] = new DerivativeStructure(dim, 1, i, point[i]);

        final double[] dvalue = base.value(dpoint).getAllDerivatives();

        double ret = dvalue[0]; // 𝛿base/𝛿t

        for (int i = 1; i < dvalue.length; i++)
            ret = dvalue[i] * point[i + dim]; // 𝛿base/𝛿point[i] *
                                              // dpoint[i]/dt

        return ret;
    }

    @Override
    public DerivativeStructure value(DerivativeStructure[] point)
            throws MathIllegalArgumentException {
       //??
        return null;
    }

}

As you can see, the Derivative takes _more_ parameters than the base
function. Those additional parameters are the total derivatives of the
original parameters. The first parameter is the independent variable.

This function is highly regular, and I can probably just calculate the
partial derivatives directly:

The partial derivative of a parameter from 1 to (dim-1) is the second
order partial derivative of that parameter in base. The partial
derivative of a parameter from dim upwards is the corresponding partial
derivative of base.


My problem is: What shall I do with the DerivativeStructure[] point I am
handed from the outside? How to get them into the equation?


-- 
Christoph Höger

Technische Universität Berlin
Fakultät IV - Elektrotechnik und Informatik
Übersetzerbau und Programmiersprachen

Sekr. TEL12-2, Ernst-Reuter-Platz 7, 10587 Berlin

Tel.: +49 (30) 314-24890
E-Mail: christoph.hoeger@tu-berlin.de

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