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 background_polynomial [2011/01/07 14:20]johnsoevans created background_polynomial [2020/07/16 11:29] (current) 2015/06/16 13:20 alancoelho 2011/08/25 19:12 alancoelho 2011/08/25 07:36 alancoelho 2011/08/25 07:31 alancoelho 2011/08/25 07:28 alancoelho 2011/01/07 14:20 johnsoevans created Next revision Previous revision 2015/06/16 13:20 alancoelho 2011/08/25 19:12 alancoelho 2011/08/25 07:36 alancoelho 2011/08/25 07:31 alancoelho 2011/08/25 07:28 alancoelho 2011/01/07 14:20 johnsoevans created Line 1: Line 1: ====== Background Polynomial ====== ====== Background Polynomial ====== - A 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. Thus your T1(x) should start at -1 for the first data point and end at 1 for the last data point. + 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// - You can use your own polynomial function by using the fit_obj syntax.