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From Luc Maisonobe <Luc.Maison...@free.fr>
Subject Re: [Math] LevenbergMarquardtOptimizerTest Question
Date Fri, 04 Feb 2011 08:25:03 GMT
```Le 04/02/2011 03:13, Ole Ersoy a écrit :
> Hi,

Hi Ole,

>
> I have a few questions regarding the implementation of the quadratic
> problem in the
> org.apache.commons.math.optimization.general.LevenbergMarquardtOptimizerTest.
>
>
> I assume the quadratic is defined as:
> f(x) = a*x^2 + b*x + c
>
> This is the implementation of the jacobian function:
>
>         private double[][] jacobian(double[] variables) {
>             double[][] jacobian = new double[x.size()][3];
>             for (int i = 0; i < jacobian.length; ++i) {
>                 jacobian[i][0] = x.get(i) * x.get(i);
>                 jacobian[i][1] = x.get(i);
>                 jacobian[i][2] = 1.0;
>             }
>             return jacobian;
>         }
>
> It seems like the lines
>
>                 jacobian[i][0] = x.get(i) * x.get(i);
>                 jacobian[i][1] = x.get(i);
>                 jacobian[i][2] = 1.0;
>
> Really should be:
>
>                 jacobian[i][0] = 2 * x.get(i) * variables[0];
>                 jacobian[i][1] = variables[1];
>                 jacobian[i][2] = 0;
>
> Does that make sense?

No. The Jacobian is the partial derivatives of the function with respect
to the parameters a, b and c, not with respect to the free variable x.
So d(ax^2+bx+c)/da = x^2, d(ax^2+bx+c)/db = x, d(ax^2+bx+c)/dc = 1.

>
> My second question has to do with the value function below:
>
>         public double[] value(double[] variables) {
>             double[] values = new double[x.size()];
>             for (int i = 0; i < values.length; ++i) {
>                 values[i] = (variables[0] * x.get(i) + variables[1]) *
> x.get(i) + variables[2];
>             }
>             return values;
>         }
>
> Should this:
>
> values[i] = (variables[0] * x.get(i) + variables[1]) * x.get(i) +
> variables[2];
>
> Really be:
>
> values[i] = (variables[0] * x.get(i)*x.get(i) + variables[1]) * x.get(i)
> + variables[2];

The two expressions compute exactly the same value. We have written it
using Hörner's rule. This is a classical method to evaluate polynomials
that save some multiplications. Look at the parentheses and you will see
that indeed variables[0] is multiplied twice by x.get(i), once inside
the parenthesis, and once outside, after variables[1] has been added.

best regards,
Luc

>
> Thanks,
> - Ole
>
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