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[2]Delves, L.M., and Mohamed, J.L. 1985, Computational Methods for Integral Equations (Cambridge, U.K.: Cambridge University Press). [3]Atkinson, K.E. 1976, A Survey of Numerical Methods for the Solution of Fredholm IntegralEquations of the Second Kind (Philadelphia: S.I.A.M.). [4]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).f (x).5804Chapter 18.Integral Equations and Inverse Theory18.4 Inverse Problems and the Use of A PrioriInformationminimize:A[u]orminimize:B[u](18.4.1)(Of course these will generally give different answers for u.) As another possibility,now suppose that we want to minimize A[u] subject to the constraint that B[u] havesome particular value, say b. The method of Lagrange multipliers gives the variationδδ{A[u] + λ1 (B[u] − b)} =(A[u] + λ1 B[u]) = 0δuδu(18.4.2)where λ1 is a Lagrange multiplier.
Notice that b is absent in the second equality,since it doesn’t depend on u.Next, suppose that we change our minds and decide to minimize B[u] subjectto the constraint that A[u] have a particular value, a. Instead of equation (18.4.2)we haveδδ{B[u] + λ2 (A[u] − a)} =(B[u] + λ2 A[u]) = 0δuδu(18.4.3)with, this time, λ2 the Lagrange multiplier. Multiplying equation (18.4.3) by theconstant 1/λ2 , and identifying 1/λ2 with λ1 , we see that the actual variations areexactly the same in the two cases. Both cases will yield the same one-parameterfamily of solutions, say, u(λ1 ).
As λ1 varies from 0 to ∞, the solution u(λ1 )varies along a so-called trade-off curve between the problem of minimizing A andthe problem of minimizing B. Any solution along this curve can equally wellbe thought of as either (i) a minimization of A for some constrained value of B,or (ii) a minimization of B for some constrained value of A, or (iii) a weightedminimization of the sum A + λ1 B.The second preliminary point has to do with degenerate minimization principles.In the example above, now suppose that A[u] has the particular formA[u] = |A · u − c|2(18.4.4)for some matrix A and vector c.
If A has fewer rows than columns, or if A is squarebut degenerate (has a nontrivial nullspace, see §2.6, especially Figure 2.6.1), thenminimizing A[u] will not give a unique solution for u. (To see why, review §15.4,and note that for a “design matrix” A with fewer rows than columns, the matrixAT · A in the normal equations 15.4.10 is degenerate.) However, if we add anymultiple λ times a nondegenerate quadratic form B[u], for example u · H · u with Ha positive definite matrix, then minimization of A[u] + λB[u] will lead to a uniquesolution for u.
(The sum of two quadratic forms is itself a quadratic form, with thesecond piece guaranteeing nondegeneracy.)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).Later discussion will be facilitated by some preliminary mention of a coupleof mathematical points. Suppose that u is an “unknown” vector that we plan todetermine by some minimization principle. Let A[u] > 0 and B[u] > 0 be twopositive functionals of u, so that we can try to determine u by either.
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