Does anyone have any tips on reparametrization of variables so as to set more complex forms of bounding on the variable for the levenberg-Marquardt fit.

I've managed to find a number for the simpler cases i.e single bound, double sided boundaries. However, I require more complex boundaries (or a complete rethink of my method).

Given a spectra which consists of a background (noise + 3rd degree polynomial +integrated background) and a number of voigt like peaks. I find that the fit takes a rather long time and it seem mainly due to problems determining the background (boundaries are way to large).

The model curve for the background is as follows

U=x_1+x_2*(x_5-X)+x_3*(x_5-X)^2+x_4*(x_5-X)^3


How do I constrain the entire equation so that the follwing is true


0<=x_1+x_2*(x_5-X)+x_3*(x_5-X)^2+x_4*(x_5-X)^3<=I2 (1)
0<=x_1<=I1 (2)
0<=(x_5-X) (3)

where I1 and I2 are the intensities at the lower and higher side of the background respectively.

(2) and (3) I can reparameterize, I am stumped in regards to (1).

Any help would be really apreciated (guess, suggestion, web link or book). Maybe this would make a good addition to the next book.

Mick