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--- background_polynomial [2011/08/25 18:12] – alancoelho
+++ background_polynomial [2025/09/19 15:18] (current) – external edit 127.0.0.1
@@ Line 1: / Line 1: @@
+====== 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//