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====== Background Polynomial ======
Chebyshev polynomials of the first kind as described at http://mathworld.wolfram.com/ChebyshevPolynomialoftheFirstKind.html is used by default in Topas. The recurrences relations of Eq. (26) can be used with the x axis normalized between -1 and 1.
A Chebychev Polynomial, say order 5, written as:
bkg @ 359.793901` 140.27403` 67.3210027` 21.057843` 11.3443248` -4.46398819`
can be coded as a fit_obj as follows:
prm c0 359.79390`
prm c1 140.27403`
prm c2 67.32100`
prm c3 21.05784`
prm c4 11.34432`
prm c5 -4.46399`
prm !xm = X2 - X1;
prm !xp = X2 + X1;
prm x = (2 X - xp) / xm;
prm tn2 = 2 x x - 1;
prm tn3 = 2 x tn2 - x;
prm tn4 = 2 x tn3 - tn2;
prm tn5 = 2 x tn4 - tn3;
fit_obj =
c0 +
c1 x +
c2 tn2 +
c3 tn3 +
c4 tn4 +
c5 tn5
;
Note that X1 and X2 are reserved parameter names that correspond to the start and end of the x-axis.
--- //[[alan.coelho@bigpond.com|Alan Coelho]] 2011/08/25 19:12//