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From Ole Ersoy <ole.er...@gmail.com>
Subject Re: [Math] LevenbergMarquardtOptimizerTest Question
Date Fri, 04 Feb 2011 18:21:54 GMT
```Hi Luc,

It all makes sense now - Thanks!

Cheers,
- Ole

On 02/04/2011 02:25 AM, Luc Maisonobe wrote:
> 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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>
>
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