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On input, a and b are the limits ofintegration, and n is the number of points to use in the Gaussian quadrature. g and ak areuser-supplied external functions that respectively return g(t) and λK(t, s). The routine returnsarrays t[1..n] and f[1..n] containing the abscissas ti of the Gaussian quadrature and thesolution f at these abscissas.

Also returned is the array w[1..n] of Gaussian weights for usewith the Nystrom interpolation routine fredin.{void gauleg(float x1, float x2, float x[], float w[], int n);void lubksb(float **a, int n, int *indx, float b[]);void ludcmp(float **a, int n, int *indx, float *d);int i,j,*indx;float d,**omk;794Chapter 18.Integral Equations and Inverse TheoryK · D · f = σfMultiplying by D1/2 , we getD1/2 · K · D1/2 · h = σh(18.1.8)where h = D1/2 · f. Equation (18.1.8) is now in the form of a symmetric eigenvalueproblem.Solution of equations (18.1.7) or (18.1.8) will in general give N eigenvalues,where N is the number of quadrature points used.

For square-integrable kernels,these will provide good approximations to the lowest N eigenvalues of the integralequation. Kernels of finite rank (also called degenerate or separable kernels) haveonly a finite number of nonzero eigenvalues (possibly none). You can diagnosethis situation by a cluster of eigenvalues σ that are zero to machine precision. Thenumber of nonzero eigenvalues will stay constant as you increase N to improvetheir accuracy.

Some care is required here: A nondegenerate kernel can have aninfinite number of eigenvalues that have an accumulation point at σ = 0. Youdistinguish the two cases by the behavior of the solution as you increase N . If yoususpect a degenerate kernel, you will usually be able to solve the problem by analytictechniques described in all the textbooks.CITED REFERENCES AND FURTHER READING:Delves, L.M., and Mohamed, J.L.

1985, Computational Methods for Integral Equations (Cambridge, U.K.: Cambridge University Press). [1]Atkinson, K.E. 1976, A Survey of Numerical Methods for the Solution of Fredholm IntegralEquations of the Second Kind (Philadelphia: S.I.A.M.).18.2 Volterra EquationsLet us now turn to Volterra equations, of which our prototype is the Volterraequation of the second kind,ZtK(t, s)f(s) ds + g(t)f(t) =(18.2.1)aMost algorithms for Volterra equations march out from t = a, building up the solutionas they go. In this sense they resemble not only forward substitution (as discussedSample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software.Permission is granted for internet users to make one paper copy for their own personal use.

Further reproduction, or any copying of machinereadable files (including this one) to any servercomputer, is strictly prohibited. To order Numerical Recipes books,diskettes, or CDROMsvisit website http://www.nr.com or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk (outside North America).is symmetric.

However, since the weights wj are not equal for most quadraturee (equation 18.1.5) is not symmetric. The matrix eigenvaluerules, the matrix Kproblem is much easier for symmetric matrices, and so we should restore thesymmetry if possible. Provided the weights are positive (which they are for Gaussianquadrature), we can define the diagonal matrix D = diag(wj ) and its square root,√D1/2 = diag( wj ). Then equation (18.1.7) becomes.

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