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192 chapter 8Your problem here is to determine what values for the parameters E r ,f r , and Γ in(8.70) provide the best fit to the data in Table 8.1.Because (8.70) is not a linear function of the parameters (E r , Σ 0 ,Γ), the threeequations that result from minimizing χ 2 are not linear equations and so cannotbe solved by the techniques of linear algebra (matrix methods). However, in ourstudy of the masses on a string problem in Unit I, we showed how to use theNewton–Raphson algorithm to search for solutions of simultaneous nonlinearequations. That technique involved expansion of the equations about the previousguess to obtain a set of linear equations and then solving the linear equations withthe matrix libraries. We now use this same combination of fitting, trial-and-errorsearching, and matrix algebra to conduct a nonlinear least-squares fit of (8.70) tothe data in Table 8.1.Recollect that the condition for a best fit is to find values of the M P parametersa m in the theory g(x, a m ) that minimize χ 2 = ∑ i [(y i − g i )/σ i ] 2 . This leads to theM P equations (8.53) to solve∑N Di=1[y i − g(x i )] ∂g(x i )σi2 =0, (m =1,M P ). (8.71)∂a mTo find the form of these equations appropriate to our problem, we rewrite ourtheory function (8.70) in the notation of (8.71):a 1 = f r , a 2 = E R , a 3 =Γ 2 /4, x= E, (8.72)⇒g(x)=a 1(x − a 2 ) 2 + a 3. (8.73)The three derivatives required in (8.71) are then∂g 1=∂a 1 (x − a 2 ) 2 ,+ a 3∂g= −2a 1(x − a 2 )∂a 2 [(x − a 2 ) 2 + a 3 ] 2 ,∂g −a 1=∂a 3 [(x − a 2 ) 2 + a 3 ] 2 .Substitution of these derivatives into the best-fit condition (8.71) yields three simultaneousequations in a 1 , a 2 , and a 3 that we need to solve in order to fit the N D =9data points (x i ,y i ) in Table 8.1:9∑i=1y i − g(x i ,a)(x i − a 2 ) 2 + a 3=0,9∑i=1y i − g(x i ,a)[(x i − a 2 ) 2 + a 3 ] 2 =0,9∑i=1{y i − g(x i ,a)} (x i − a 2 )[(x i − a 2 ) 2 + a 3 ] 2 =0. (8.74)Even without the substitution of (8.70) for g(x, a), it is clear that these three equationsdepend on the a’s in a nonlinear fashion. That’s okay because in §8.2.2 we−101COPYRIGHT 2008, PRINCET O N UNIVE R S I T Y P R E S SEVALUATION COPY ONLY. NOT FOR USE IN COURSES.ALLpup_06.04 — 2008/2/15 — Page 192