is symmetric. However, since the weights wj are not equal for most quadrature rules, the matrix K (equation ) is not symmetric. The matrix eigenvalue problem is much easier for symmetric matrices, and so we should restore the symmetry if possible. | 794 Chapter 18. Integral Equations and Inverse Theory is symmetric. However since the weights wj are not equal for most quadrature rules the matrix K equation is not symmetric. The matrix eigenvalue problem is much easier for symmetric matrices and so we should restore the symmetry if possible. Provided the weights are positive which they are for Gaussian quadrature we can define the diagonal matrix D diag wj and its square root D1 2 diag pwj . Then equation becomes K D f of Multiplying by D1 2 we get D1 2 K D1 2 h oh where h D1 2 f. Equation is now in the form of a symmetric eigenvalue problem. Solution of equations or 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 integral equation. Kernels of finite rank also called degenerate or separable kernels have only a finite number of nonzero eigenvalues possibly none . You can diagnose this situation by a cluster of eigenvalues that are zero to machine precision. The number of nonzero eigenvalues will stay constant as you increase N to improve their accuracy. Some care is required here A nondegenerate kernel can have an infinite number of eigenvalues that have an accumulation point at o 0. You distinguish the two cases by the behavior of the solution as you increase N. If you suspect a degenerate kernel you will usually be able to solve the problem by analytic techniques described in all the textbooks. CITED REFERENCES AND FURTHER READING Delves . and Mohamed . 1985 Computational Methods for Integral Equations Cambridge . Cambridge University Press . 1 Atkinson . 1976 A Survey of Numerical Methods for the Solution of Fredholm Integral Equations of the Second Kind Philadelphia . . Volterra Equations Let us now turn to Volterra equations of which our prototype is the Volterra equation of the second kind Sample page .
