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background_polynomial [topas wiki]

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background_polynomial [2022/11/03 15:08] (current) – external edit 127.0.0.1
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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:
 +
 +<code topas>bkg @  359.793901`  140.27403`  67.3210027`  21.057843`  11.3443248` -4.46398819`</code>
 +
 +can be coded as a fit_obj as follows:
 +
 +<code topas>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 
 + ;</code>
 +
 +
 +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//