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Unformatted text preview: CHAPTER 7 LEAST-SQUARES FIT TOA POLYNOMIAL 7.1 DETERMINANT SOLUTION So far we have discussed fitting a straight line to a group of data points. However, slIppose our data (Xi' y) were not consistent with a straight line fit. We might con- xtrucl a more complex function with extra parameters and try varying the parame- tns 01" this function to fit the data more closely. A very useful function for such a fit is a power-series polynomial y(X) = al + (l2X + a3x2 + a4x' + ... + amx m- I (7.1) where the dependent variable y is expressed as a sum of power series of the inde- prudent variable x with coefficients ai' (12) a3, a4, and so forth. For problems in which the fitting function is linear in the parameters, the method of least squares is readily extended to any number of terms m, limited only hy our ability to solve m linear equations in m unknowns and by the precision with which calculations can be made. We can rewrite Equation (7.1) as 111 y(x) = 2: ak xk- 1 k=! (7.2) lie, where till' index k runs lrou: I lOllI, III luct. W l' cun gcncrulizc the method even fur- ther by writing I':qualillil (7,2) as I' f /1/ y(x) = 2: akfi:(x) k=l (7.3) where the functions fk(X) could be the powers of x as in Equation (7 .2),f1 (x) = 1, .fi(x) = x,f-b) = X2, and so forth, or they could be other functions of x as long as...
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- Fall '07