First assume trial values for a and b call these a

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First, assume trial values for A and B ; call these A 0 and B 0 . You should pick values that are close to the correct ones. In our example you wouldn’t need to do this for B , but it’s easier to treat all coefficients identically. These trial values produce predicted values y 0 ,m : sin( A 0 t m ) + B 0 t m = y 0 ,m . (7.2) Subtract equation 7.2 from 7.1, and express the differences as derivatives. Letting δA = A A 0 and δB = B B 0 , this gives δA [ t m cos( A 0 t m )] + δBt m = y m y 0 ,m . (7.3) This is linear in ( δA, δB ) so you can solve for them using standard least squares. Increment the original guessed values to calculate A 0 ,new = A 0 + δA and B 0 ,new = B 0 + δB , These won’t be exact because higher derivatives (including cross derivatives) come into play, so you need to use these new values to repeat the process. This is an iterative procedure and you keep going until the changes become “small”. The generalization to an arbitrarily large number of unknown coefficients is obvious.
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– 21 – We now offer some cautionary and practical remarks. (0) In linear least squares, the curvature and covariance matrices are set by the values of the independent variable, which here is denoted by t , and are independent of the datapoint values. Here, the matrix elements change from one iteration to the next because they depend on the guessed parameters, and sometimes they even depend on the datapoint values. (1) Multiple minima: Nonlinear problems often have multiple minima in σ 2 . A classical case is fitting multiple Gaussians to a spectral line profile. Gaussians are most definitely not orthogonal functions and in some cases several solutions may give almost comparably good values of σ 2 , each one being a local minimum. For example, for the case of two blended Gaussians, one can often fit two narrow Gaussians or the combination of a wide and narrow Gaussian, the two fits giving almost equal σ 2 . The lower of these is the real minimum but, given the existence of systematic errors and such, not necessarily the best solution. The best solution is often determined by physical considerations; in this case, for example, you might have physical reasons to fit a broad plus narrow Gaussian, so you’d choose this one even if its σ 2 weren’t the true minimum. (2) The Initial Guess: When there are multiple minima, the one to which the solution converges is influenced by your initial guess. To fully understand the range of possible solutions, you should try different initial guesses and see what happens. If the solutions always converge to the same answer, then you can have some confidence (but not full confidence) that the solution is unique. (3) Iterative stability: If your initial guess is too far from the true solution, then the existence of higher derivatives means that the computed corrections can be too large and drive the iterative solution into instability. It is often a good idea to multiply the derived correction factors ( δA and δB above) by a factor F less than unity, for example F = 0 . 5 or 0.75. This increases the
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