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550 chapter 20We note that the last term in (20.30) implements the principal-value prescriptionand cancels the singular behavior of the previous term. Equation (20.30) containsthe (N +1)unknowns R(k j ,k 0 ) for j =0,N. We turn it into (N +1)simultaneousequations by evaluating it for (N +1)k values on a grid (Figure 20.2) consisting ofthe observable momentum k 0 and the integration points:k = k i ={kj , j =1,N (quadrature points),k 0 , i=0 (observable point).(20.31)There are now (N +1)linear equations for (N +1)unknowns R i ≡ R(k i ,k 0 ):R i = V i + 2 πN∑j=1kj 2V ijR j w j(k0 2 − k2 j )/2µ − 2 ∑ Nπ k2 0V i0 R 0m=1w m(k 2 0 − k2 m)/2µ . (20.32)We express these equations in matrix form by combining the denominators andweights into a single denominator vector D:⎧⎪⎨ + 2 w ik 2 iπ (k0 2D i =− k2 i )/2µ,for i =1,N,⎪⎩ − 2 ∑(20.33)N w jk 2 0π j=1 (k0 2 − )/2µ, for i =0. k2 jThe linear equations (20.32) now assume that the matrix formR − DV R =[1− DV ] R = V, (20.34)where R and V are column vectors of length N +1:⎛ ⎞R 0,0R 1,0[R]=⎜ . ⎝ .. ⎟⎠ ,R N,0⎛ ⎞V 0,0[V ]= V 1,0⎜ . ⎝ .. ⎟⎠ . (20.35)V N,0We call the matrix [1 − DV ] in (20.34) the wave matrix F and write the integralequation as the matrix equation[F ][R]=[V ], F ij = δ ij − D j V ij . (20.36)With R the unknown vector, (20.36) is in the standard form AX = B, which can besolved by the mathematical subroutine libraries discussed in Chapter 8, “SolvingSystems of Equations with Matrices; Data Fitting.”−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 550