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systems of equations with matrices; data fitting 191Here the definitions of the S’s are simple extensions of those used in(8.56)–(8.58) and are programmed in JamaFit.java shown in Listing 8.2. Afterplacing the three parameters into a vector a and the three RHS terms in (8.67)into a vector s, these equations assume the matrix form:⎡⎢[α]= ⎣S xx S xxx S xxxx[α]a = S, (8.68)S S x S xx⎤ ⎡a 1⎤ ⎡S y⎤S x S xx S xxx⎥⎦,⎢a = ⎣ a 2⎥⎦,⎢S = ⎣ S xy⎥⎦.a 3S xxyThis is the exactly the matrix problem we solved in § 8.3.3 with the codeJamaFit.java given in Listing 8.2. The solution for the parameter vector a isobtained by solving the matrix equations. Although for 3 × 3 matrices we canwrite out the solution in closed form, for larger problems the numerical solutionrequires matrix methods.8.7.5 Linear Quadratic Fit Assessment1. Fit the quadratic (8.63) to the following data sets [given as (x 1 ,y 1 ),(x 2 ,y 2 ),...]. In each case indicate the values found for the a’s, the number ofdegrees of freedom, and the value of χ 2 .a. (0, 1)b. (0, 1), (1, 3)c. (0, 1), (1, 3), (2, 7)d. (0, 1), (1, 3), (2, 7), (3, 15)2. Find a fit to the last set of data to the functiony = Ae −bx2 . (8.69)Hint: A judicious change of variables will permit you to convert this to alinear fit. Does a minimum χ 2 still have meaning here?8.7.6 Nonlinear Fit of the Breit–Wigner Formulato a Cross SectionProblem: Remember how we started Unit II of this chapter by interpolating thevalues in Table 8.1, which gave the experimental cross section Σ as a function ofenergy. Although we did not use it, we also gave the theory describing these data,namely, the Breit–Wigner resonance formula (8.31):f(E)=f r(E − E r ) 2 +Γ 2 /4 . (8.70)−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 191