Differential Equations Solutions 68

# Differential Equations Solutions 68 - and the log = -2.36...

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78 Chapter 13. Solutions: Case Study: Fitting Exponentials: An Interest in Rates -0.8 -0.6 -0.4 -0.2 -0.8 -0.6 -0.4 -0.2 α 1 α 2 t final = 1 -0.8 -0.6 -0.4 -0.2 -0.8 -0.6 -0.4 -0.2 α 1 t final = 2 -0.8 -0.6 -0.4 -0.2 -0.8 -0.6 -0.4 -0.2 α 1 t final = 3 -0.8 -0.6 -0.4 -0.2 -0.8 -0.6 -0.4 -0.2 α 1 t final = 4 -0.8 -0.6 -0.4 -0.2 -0.8 -0.6 -0.4 -0.2 α 1 t final = 5 -0.8 -0.6 -0.4 -0.2 -0.8 -0.6 -0.4 -0.2 α 1 t final = 6 Figure 13.3. Challenge 2. Contour plots of the residual norm as a function of the estimates of α for various values of t final . The contours marked are 10 2 , 10 6 ,and 10 10 . shown in Figure 13.7. If the contours were square, then reporting the uncertainty in α as ± some value would be appropriate, but as we can see, this is far from the case. The log=-2.6 contour outlines a set of α values that changes the residual norm by less than 1%, the log = -2.5 contour denotes a change of less than 5%,
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Unformatted text preview: and the log = -2.36 contour corresponds to a 10% change. The best value found was = [ 1 . 601660 , 2 . 696310], with residual norm 0.002401137 = 10 2 . 6196 . Our uncertainty in the rate constants is rather large. The true solution, the value used to generate the data, was = [ 1 . 6 , 2 . 7] with x 1 = x 2 = 0 . 75, and the standard deviation of the white noise was 10 4 . Variants of Pronys method [1] provide alternate approaches to exponential tting. Exponential tting is a very dicult problem, even when the number of terms n is known. It becomes even easier to be fooled when determining n is part of the problem!...
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## This note was uploaded on 01/21/2012 for the course MAP 3302 taught by Professor Dr.robin during the Fall '11 term at University of Florida.

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