18.0 Introduction 789 is called a Fredholm equation of the second kind.usually written f(t)=入 K(t,s)f(s)ds+g(t) (18.0.4) Again,the notational conventions do not exactly correspond:A in equation(18.0.4) is 1/o in (18.0.3),while g is -g/A.If g (or g)is zero,then the equation is said to be homogeneous.If the kernel K(t,s)is bounded,then,like equation (18.0.3), equation (18.0.4)has the property that its homogeneous form has solutions for at most a denumerably infinite set A =An,n =1,2,...,the eigenvalues.The 8 corresponding solutions fn(t)are the eigenfunctions.The eigenvalues are real if the kernel is symmetric. In the inhomogeneous case of nonzero g (or g),equations (18.0.3)and (18.0.4) are soluble except when A (or o)is an eigenvalue-because the integral operator (or matrix)is singular then.In integral equations this dichotomy is called the Fredholm alternative. Fredholm equations of the first kind are often extremely ill-conditioned.Ap- RECIPES plying the kernel to a function is generally a smoothing operation,so the solution, which requires inverting the operator,will be extremely sensitive to small changes or errors in the input.Smoothing often actually loses information,and there is no way to get it back in an inverse operation.Specialized methods have been developed 93P for such equations,which are often called inverse problems.In general,a method must augment the information given with some prior knowledge of the nature of the solution.This prior knowledge is then used,in one way or another,to restore lost 、s是%o information.We will introduce such techniques in 818.4. Inhomogeneous Fredholm equations of the second kind are much less often ill-conditioned.Equation (18.0.4)can be rewritten as 61 K(t,s)-08(t-s)]f(s)ds=-ag(t) (18.0.5) where 6(t-s)is a Dirac delta function (and where we have changed from A to its 10.621 reciprocal o for clarity).If o is large enough in magnitude,then equation (18.0.5) is,in effect,diagonally dominant and thus well-conditioned.Only if o is small do Numerical 431 we go back to the ill-conditioned case. Recipes Homogeneous Fredholm equations of the second kind are likewise not partic- ularly ill-posed.If K is a smoothing operator,then it will map many f's to zero, or near-zero:there will thus be a large number of degenerate or nearly degenerate North eigenvalues around o=0(A-oo),but this will cause no particular computational difficulties.In fact,we can now see that the magnitude of o needed to rescue the inhomogeneous equation(18.0.5)from an ill-conditioned fate is generally much less than that required for diagonal dominance.Since the o term shifts all eigenvalues, it is enough that it be large enough to shift a smoothing operator's forest of near- zero eigenvalues away from zero,so that the resulting operator becomes invertible (except,of course,at the discrete eigenvalues). Volterra equations are a special case of Fredholm equations with K(t,s)=0 for s>t.Chopping off the unnecessary part ofthe integration,Volterra equations are written in a form where the upper limit of integration is the independent variable t.18.0 Introduction 789 Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copyin Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) g of machine￾readable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America). is called a Fredholm equation of the second kind, usually written f(t) = λ
b a K(t, s)f(s) ds + g(t) (18.0.4) Again, the notational conventions do not exactly correspond: λ in equation (18.0.4) is 1/σ in (18.0.3), while g is −g/λ. If g (or g) is zero, then the equation is said to be homogeneous. If the kernel K(t, s) is bounded, then, like equation (18.0.3), equation (18.0.4) has the property that its homogeneous form has solutions for at most a denumerably infinite set λ = λn, n = 1, 2,... , the eigenvalues. The corresponding solutions fn(t) are the eigenfunctions. The eigenvalues are real if the kernel is symmetric. In the inhomogeneous case of nonzero g (or g), equations (18.0.3) and (18.0.4) are soluble except when λ (or σ) is an eigenvalue — because the integral operator (or matrix) is singular then. In integral equations this dichotomy is called the Fredholm alternative. Fredholm equations of the first kind are often extremely ill-conditioned. Ap￾plying the kernel to a function is generally a smoothing operation, so the solution, which requires inverting the operator, will be extremely sensitive to small changes or errors in the input. Smoothing often actually loses information, and there is no way to get it back in an inverse operation. Specialized methods have been developed for such equations, which are often called inverse problems. In general, a method must augment the information given with some prior knowledge of the nature of the solution. This prior knowledge is then used, in one way or another, to restore lost information. We will introduce such techniques in §18.4. Inhomogeneous Fredholm equations of the second kind are much less often ill-conditioned. Equation (18.0.4) can be rewritten as
b a [K(t, s) − σδ(t − s)]f(s) ds = −σg(t) (18.0.5) where δ(t − s) is a Dirac delta function (and where we have changed from λ to its reciprocal σ for clarity). If σ is large enough in magnitude, then equation (18.0.5) is, in effect, diagonally dominant and thus well-conditioned. Only if σ is small do we go back to the ill-conditioned case. Homogeneous Fredholm equations of the second kind are likewise not partic￾ularly ill-posed. If K is a smoothing operator, then it will map many f’s to zero, or near-zero; there will thus be a large number of degenerate or nearly degenerate eigenvalues around σ = 0 (λ → ∞), but this will cause no particular computational difficulties. In fact, we can now see that the magnitude of σ needed to rescue the inhomogeneous equation (18.0.5) from an ill-conditioned fate is generally much less than that required for diagonal dominance. Since the σ term shifts all eigenvalues, it is enough that it be large enough to shift a smoothing operator’s forest of near￾zero eigenvalues away from zero, so that the resulting operator becomes invertible (except, of course, at the discrete eigenvalues). Volterra equations are a special case of Fredholm equations with K(t, s)=0 for s>t. Chopping off the unnecessary part of the integration, Volterra equations are written in a form where the upper limit of integration is the independent variable t