Clarification needed on Jacobian for Levenberg Marquardt Least Squares function

[AeroStevie]

Hi,

I am trying to write a MATLAB script to estimate the damage to a turbofan engine component based on key engine parameters; spool speed (NH), high pressure compressor inlet pressure (P30) and high pressure compressor inlet temperature (T30). I have an empirical formula to base the fit on, contained in a function AlgoFit.m. This function has 22 coefficients, stored in a vector K.

As my university did not focus much on numerical analysis or statistics this is all fairly new to me and I am having a hard time understanding descriptions online. What I want to do is fit the AlgoFit.m function to experimental data I have by altering all 22 coefficients in K to reduce the residuals, but I cannot get clear in my head what I am doing.

I have read that I want the Jacobian, and i think this should be of the form J = [dNH/dK1,dP30/dK1,dT30/dK1; dNH/dK2,dP30/dK2,...,dT30/dK22], but i am not sure how to compute this jacobian from my parameters P = [NH),P30),T30)], coefficients K = [K1 K2 ... K22], experimental data SA40 = [time) Stress) temp)] and fitted approximation/residuals from AlgoFit.m.

If anyone can clear up how to get the Jacobian then hopefully that will be all I need to get going on this. I do have plenty more data as this is a small part of a larger project, i've given all that i think is relevant but if i need anything else I can probably find it quite easily.


Thank you,


Steve

 

[AeroStevie]

OK, after a bit more thinking I believe the Jacobian should actually be J = [F(P,K)/dK1; F(P,K)/dK2; ...; F(P,K)/dK22] where F(P,K) is the fit function I have written in MATLAB, P are the input parameters for spool speed, temperature and pressure. It was confusing me how to take the derivative of the individual parameters w.r.t. the coefficients because they wouldn't change. However I am still unsure how to take the derivative of the function w.r.t. values I only have one initial guess for. Any hints or prods in the right direction would be much appreciated.