The Art of Credit Derivatives: Demystifying the Black Swan

Appendix B

Generalized Gauss Laguerre Quadrature

We outline two algorithms for the computation of abscissas and weights for generalized Gauss Laguerre quadrature.

B.1 EIGENVALUE PROBLEM

The first algorithm is based on an eigenvalue problem. We refer to Golub and Welsch (1969) and Wilf (1962) for a detailed discussion. The abscissas of the quadrature formula can be found as the eigenvalues of the symmetric tridiagonal matrix T, given by

(B.1)

where α_i = 2i + 1 + s and β_i = i(i + s). The corresponding weights can be computed from the normalized eigenvectors. With

(B.2)

where = 1 we have that

(B.3)

where µ₀ = w(x ;s) = ┌(s + 1) .

The symmetric tridiagonal eigenvalue problem as in (B.2) can be solved efficiently using the classical algorithms for eigenvalues; see Golub and Van Loan (1989), Parlett (1980) or Wilkinson (1988) for details. This and other numerical algorithms are addressed in Press et al. (1992). Note that only the first component of the normalized eigenvector is required, hence an optimized eigenvalue solver can be used.

B.2 NEWTON-RAPHSON ITERATION

The second algorithm is based on the fact that good initial approximations for the zeros of the orthogonal polynomials are ...