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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) | 15.5 NonlinearModels 681 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. S 15.5 Nonlinear Models i I I f GO _ O 3 X-X Q Q O 3- V V Zt -4. We now consider fitting when the model depends nonlinearly on the set of M I 2-1 g unknown parameters ak k 1 2 . . M. We use the same approach as in previous S. 5 sections namely to define a 2 merit function and determine best-fit parameters 5 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 2 stops or 5 effectively stops decreasing. How is this problem different from the general nonlinear function minimization g 3 3 problem already dealt with in Chapter 10 Superficially not at all Sufficiently close to the minimum we expect the 2 function to be well approximated by a 5 quadratic form which we can write as W O 1 2 a y - d a -a D a 15.5.1 8 I p where d is an M-vector and D is an M x M matrix. Compare equation 10.6.1. 4-111. 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 3 9 ju CO amin acur D 1 -VX2 acur 15.5.2 0 m Compare equation 10.7.4. 8 On the other hand 15.5.1 might be a poor local approximation to the shape jj of the function that we are trying to minimize at acur. In that case about all we can do is take a step down the gradient as in the steepest descent method 10.6 . 3 g a In other words c - Q Q a v anext acur - constant xV 2 aCur 15.5.3 P a SB 8 S 20 . 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 2 g functionat any set of parameters a. To use 15.5.2 we also need
