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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 to be 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.


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