c18-1 (779608)
Текст из файла
18.1 Fredholm Equations of the Second Kind791CITED REFERENCES AND FURTHER READING:Delves, L.M., and Mohamed, J.L. 1985, Computational Methods for Integral Equations (Cambridge, U.K.: Cambridge University Press). [1]Linz, P. 1985, Analytical and Numerical Methods for Volterra Equations (Philadelphia: S.I.A.M.).[2]Atkinson, K.E. 1976, A Survey of Numerical Methods for the Solution of Fredholm IntegralEquations of the Second Kind (Philadelphia: S.I.A.M.). [3]Brunner, H. 1988, in Numerical Analysis 1987, Pitman Research Notes in Mathematics vol.
170,D.F. Griffiths and G.A. Watson, eds. (Harlow, Essex, U.K.: Longman Scientific and Technical), pp. 18–38. [4]Smithies, F. 1958, Integral Equations (Cambridge, U.K.: Cambridge University Press).Kanwal, R.P. 1971, Linear Integral Equations (New York: Academic Press).Green, C.D. 1969, Integral Equation Methods (New York: Barnes & Noble).18.1 Fredholm Equations of the Second KindWe desire a numerical solution for f(t) in the equationZbK(t, s)f(s) ds + g(t)f(t) = λ(18.1.1)aThe method we describe, a very basic one, is called the Nystrom method. It requiresthe choice of some approximate quadrature rule:Zby(s) ds =aNXwj y(sj )(18.1.2)j=1Here the set {wj } are the weights of the quadrature rule, while the N points {sj }are the abscissas.What quadrature rule should we use? It is certainly possible to solve integralequations with low-order quadrature rules like the repeated trapezoidal or Simpson’sSample 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).special quadrature rules, but they are also sometimes blessings in disguise, since theycan spoil a kernel’s smoothing and make problems well-conditioned.In §§18.4–18.7 we face up to the issues of inverse problems. §18.4 is anintroduction to this large subject.We should note here that wavelet transforms, already discussed in §13.10, areapplicable not only to data compression and signal processing, but can also be usedto transform some classes of integral equations into sparse linear problems that allowfast solution.
You may wish to review §13.10 as part of reading this chapter.Some subjects, such as integro-differential equations, we must simply declareto be beyond our scope. For a review of methods for integro-differential equations,see Brunner [4].It should go without saying that this one short chapter can only barely touch ona few of the most basic methods involved in this complicated subject.792Chapter 18.Integral Equations and Inverse Theoryf(t) = λNXwj K(t, sj )f(sj ) + g(t)(18.1.3)j=1Evaluate equation (18.1.3) at the quadrature points:f(ti ) = λNXwj K(ti , sj )f(sj ) + g(ti )(18.1.4)j=1Let fi be the vector f(ti ), gi the vector g(ti ), Kij the matrix K(ti , sj ), and definee ij = Kij wjK(18.1.5)Then in matrix notation equation (18.1.4) becomese ·f=g(1 − λK)(18.1.6)This is a set of N linear algebraic equations in N unknowns that can be solvedby standard triangular decomposition techniques (§2.3) — that is where the O(N 3 )operations count comes in.
The solution is usually well-conditioned, unless λ isvery close to an eigenvalue.Having obtained the solution at the quadrature points {ti }, how do you get thesolution at some other point t? You do not simply use polynomial interpolation.This destroys all the accuracy you have worked so hard to achieve. Nystrom’s keyobservation was that you should use equation (18.1.3) as an interpolatory formula,maintaining the accuracy of the solution.We here give two routines for use with linear Fredholm equations of the secondkind. The routine fred2 sets up equation (18.1.6) and then solves it by LUdecomposition with calls to the routines ludcmp and lubksb.
The Gauss-Legendrequadrature is implemented by first getting the weights and abscissas with a call togauleg. Routine fred2 requires that you provide an external function that returnsg(t) and another that returns λKij . It then returns the solution f at the quadraturepoints. It also returns the quadrature points and weights. These are used by thesecond routine fredin to carry out the Nystrom interpolation of equation (18.1.3)and return the value of f at any point in the interval [a, b].Sample 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).rules. We will see, however, that the solution method involves O(N 3 ) operations,and so the most efficient methods tend to use high-order quadrature rules to keepN as small as possible. For smooth, nonsingular problems, nothing beats Gaussianquadrature (e.g., Gauss-Legendre quadrature, §4.5).
(For non-smooth or singularkernels, see §18.3.)Delves and Mohamed [1] investigated methods more complicated than theNystrom method. For straightforward Fredholm equations of the second kind, theyconcluded “. . . the clear winner of this contest has been the Nystrom routine . . . withthe N -point Gauss-Legendre rule. This routine is extremely simple. . . . Such resultsare enough to make a numerical analyst weep.”If we apply the quadrature rule (18.1.2) to equation (18.1.1), we get18.1 Fredholm Equations of the Second Kind793#include "nrutil.h"indx=ivector(1,n);omk=matrix(1,n,1,n);gauleg(a,b,t,w,n);Replace gauleg with another routine if not usingfor (i=1;i<=n;i++) {Gauss-Legendre quadrature.efor (j=1;j<=n;j++)Form 1 − λK.omk[i][j]=(float)(i == j)-(*ak)(t[i],t[j])*w[j];f[i]=(*g)(t[i]);}ludcmp(omk,n,indx,&d);Solve linear equations.lubksb(omk,n,indx,f);free_matrix(omk,1,n,1,n);free_ivector(indx,1,n);}float fredin(float x, int n, float a, float b, float t[], float f[],float w[], float (*g)(float), float (*ak)(float, float))Given arrays t[1..n] and w[1..n] containing the abscissas and weights of the Gaussianquadrature, and given the solution array f[1..n] from fred2, this function returns the value off at x using the Nystrom interpolation formula.
On input, a and b are the limits of integration,and n is the number of points used in the Gaussian quadrature. g and ak are user-suppliedexternal functions that respectively return g(t) and λK(t, s).{int i;float sum=0.0;for (i=1;i<=n;i++) sum += (*ak)(x,t[i])*w[i]*f[i];return (*g)(x)+sum;}One disadvantage of a method based on Gaussian quadrature is that there is nosimple way to obtain an estimate of the error in the result. The best practical methodis to increase N by 50%, say, and treat the difference between the two estimates as aconservative estimate of the error in the result obtained with the larger value of N .Turn now to solutions of the homogeneous equation.
If we set λ = 1/σ andg = 0, then equation (18.1.6) becomes a standard eigenvalue equatione · f = σfK(18.1.7)which we can solve with any convenient matrix eigenvalue routine (see Chapter11). Note that if our original problem had a symmetric kernel, then the matrix KSample 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).void fred2(int n, float a, float b, float t[], float f[], float w[],float (*g)(float), float (*ak)(float, float))Solves a linear Fredholm equation of the second kind.
Характеристики
Тип файла PDF
PDF-формат наиболее широко используется для просмотра любого типа файлов на любом устройстве. В него можно сохранить документ, таблицы, презентацию, текст, чертежи, вычисления, графики и всё остальное, что можно показать на экране любого устройства. Именно его лучше всего использовать для печати.
Например, если Вам нужно распечатать чертёж из автокада, Вы сохраните чертёж на флешку, но будет ли автокад в пункте печати? А если будет, то нужная версия с нужными библиотеками? Именно для этого и нужен формат PDF - в нём точно будет показано верно вне зависимости от того, в какой программе создали PDF-файл и есть ли нужная программа для его просмотра.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
