G'day Gordan,
The Jacobian matrix J is the differential of your objective equation
with respect to each parameter (corresponding to the second index)
at each data point (corresponding to the first index).
The relevant equation is [a * sin(b / t + c) + d - yi ] / sigma[i],
i.e. you're trying to minimize the difference between the model and
the data (yi), where some points may be more reliable than others
(given sigma[i]), by optimizing the amplitude, period, phase, and
offset of your sine wave. The Jacobian provides the analytic
equations to inform the system of the sensitivity of the residuals in
the evolving fit to changes in each of the parameters.
Taking the partial derivative of your equation with respect to each
of the floating parameters, and setting them appropriately into the
matrix J, assuming your vector of floating parameters elsewhere is
ordered (a,b,c,d), and retaining your s = sigma[i], I get:
gsl_matrix_set (J, i, 0, sin(b / t + c) / s); # derivative
with respect to a
gsl_matrix_set (J, i, 1, cos(b / t + c) * a/(t*s) ); # derivative
with respect to b
gsl_matrix_set (J, i, 2, cos(b / t + c) * a/s); # derivative
with respect to c
gsl_matrix_set (J, i, 3, 1/s); #
derivative with respect to d
The analytic derivatives are usually my problem, however, so please
confirm them!
Regards,
john
---------------------------------------------------------
Dr John Gehman
Research Fellow
School of Chemistry
University of Melbourne (Australia)
On 17/07/2007, at 9:46 PM, Gordan Bobic wrote:
Hi,
I have a question regarding the example 37.9 in the manual.
In the df function, there is the following code:
for (i = 0; i < n; i++)
{
/* Jacobian matrix J(i,j) = dfi / dxj, */
/* where fi = (Yi - yi)/sigma[i], */
/* Yi = A * exp(-lambda * i) + b */
/* and the xj are the parameters (A,lambda,b) */
double t = i;
double s = sigma[i];
double e = exp(-lambda * t);
gsl_matrix_set (J, i, 0, e/s);
gsl_matrix_set (J, i, 1, -t * A * e/s);
gsl_matrix_set (J, i, 2, 1/s);
}
I am trying to adapt this to fit a generic sine curve. So, I am
trying to
do something like this:
for (i = 0; i < n; i++)
{
/* Jacobian matrix J(i,j) = dfi / dxj, */
/* where fi = (Yi - yi)/sigma[i], */
/* Yi = a * sin(b / t + c) + d */
/* and the xj are the parameters (a,b,c,d) */
double t = i;
double s = sigma[i];
double e = sin(b / t + c);
gsl_matrix_set (J, i, 0, e/s);
gsl_matrix_set (J, i, 1, cos (b / t + c) * e/s);
gsl_matrix_set (J, i, 2, 1/s);
gsl_matrix_set (J, i, 3, 1/s);
}
Is this a correct adaptation, or am I misunderstanding what this is
supposed to do? Is the 4th parameter in this case the
differential/derivative of the Yi equation with respect to a, b, c
and d
respectively?
Thanks.
Gordan
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