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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