15.5 Nonlinear Models 681 Lawson,C.L.,and Hanson,R.1974,So/ving Least Squares Problems (Englewood Cliffs,NJ: Prentice-Hall). Forsythe,G.E.,Malcolm,M.A.,and Moler,C.B.1977,Computer Methods for Mathematical Computations (Englewood Cliffs,NJ:Prentice-Hall),Chapter 9. 15.5 Nonlinear Models We now consider fitting when the model depends nonlinearly on the set of M unknown parameters ak,k=1,2,...,M.We use the same approach as in previous sections,namely to define a x-merit function and determine best-fit parameters by its minimization.With nonlinear dependences,however,the minimization must 、 proceed iteratively.Given trial values for the parameters,we develop a procedure that improves the trial solution.The procedure is then repeated until x2 stops(or effectively stops)decreasing. How is this problem different from the general nonlinear function minimization problem already dealt with in Chapter 10?Superficially,not at all:Sufficiently a 9 close to the minimum,we expect the x2 function to be well approximated by a quadratic form,which we can write as 1 X2(a)≈y-da+2aD.a (15.5.1) 星 9 where d is an M-vector and D is an M x M matrix.(Compare equation 10.6.1.) If the approximation is a good one,we know how to jump from the current trial parameters acur to the minimizing ones amin in a single leap,namely amin acur +D-1.[-Vx2(acur)] (15.5.2) (Compare equation 10.7.4.) On the other hand,(15.5.1)might be a poor local approximation to the shape 10.621 of the function that we are trying to minimize at acur.In that case,about all we Numerica can do is take a step down the gradient,as in the steepest descent method(810.6). 431 In other words. Recipes anext acur -constant x Vx2(acur) (15.5.3) North where the constant is small enough not to exhaust the downhill direction. To use (15.5.2)or (15.5.3),we must be able to compute the gradient of the x2 function at any set of parameters a.To use(15.5.2)we also need the matrix D,which is the second derivative matrix(Hessian matrix)of the x2 merit function,at any a. Now,this is the crucial difference from Chapter 10:There,we had no way of directly evaluating the Hessian matrix.We were given only the ability to evaluate the function to be minimized and(in some cases)its gradient.Therefore,we had to resort to iterative methods not just because our function was nonlinear,but also in order to build up information about the Hessian matrix.Sections 10.7 and 10.6 concerned themselves with two different techniques for building up this information.15.5 Nonlinear Models 681 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). Lawson, C.L., and Hanson, R. 1974, Solving Least Squares Problems (Englewood Cliffs, NJ: Prentice-Hall). Forsythe, G.E., Malcolm, M.A., and Moler, C.B. 1977, Computer Methods for Mathematical Computations (Englewood Cliffs, NJ: Prentice-Hall), Chapter 9. 15.5 Nonlinear Models We now consider fitting when the model depends nonlinearly on the set of M unknown parameters ak, k = 1, 2,...,M. We use the same approach as in previous sections, namely to define a χ2 merit function and determine best-fit parameters by its minimization. With nonlinear dependences, however, the minimization must proceed iteratively. Given trial values for the parameters, we develop a procedure that improves the trial solution. The procedure is then repeated until χ2 stops (or effectively stops) decreasing. How is this problem different from the general nonlinear function minimization problem already dealt with in Chapter 10? Superficially, not at all: Sufficiently close to the minimum, we expect the χ2 function to be well approximated by a quadratic form, which we can write as χ2(a) ≈ γ − d · a + 1 2 a · D · a (15.5.1) where d is an M-vector and D is an M × M matrix. (Compare equation 10.6.1.) If the approximation is a good one, we know how to jump from the current trial parameters acur to the minimizing ones amin in a single leap, namely amin = acur + D−1 · −∇χ2(acur)  (15.5.2) (Compare equation 10.7.4.) On the other hand, (15.5.1) might be a poor local approximation to the shape of the function that we are trying to minimize at a cur. In that case, about all we can do is take a step down the gradient, as in the steepest descent method (§10.6). In other words, anext = acur − constant × ∇χ2(acur) (15.5.3) where the constant is small enough not to exhaust the downhill direction. To use (15.5.2) or (15.5.3), we must be able to compute the gradient of the χ 2 function at any set of parameters a. To use (15.5.2) we also need the matrix D, which is the second derivative matrix (Hessian matrix) of the χ2 merit function, at any a. Now, this is the crucial difference from Chapter 10: There, we had no way of directly evaluating the Hessian matrix. We were given only the ability to evaluate the function to be minimized and (in some cases) its gradient. Therefore, we had to resort to iterative methods not just because our function was nonlinear, but also in order to build up information about the Hessian matrix. Sections 10.7 and 10.6 concerned themselves with two different techniques for building up this information