Differential Equations Solutions 65

Differential Equations Solutions 65 - α values produce...

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Chapter 13 Solutions: Case Study: Fitting Exponentials: An Interest in Rates CHALLENGE 13.1. Sample MATLAB programs to solve this problem (and the others in this chapter) are available on the website. The results are shown in Figures 13.1 and 13.2. Note that the shape of the w clusters are rather circular; the sensitivity in the two components is approximately equal. This is not true of the x clusters; they are elongated in the direction corresponding to the eigenvector of the smallest singular value, since small changes in the data in this direction cause large changes in the solution. The length of the x cluster (and thus the sensitivity of the solution) is greater in Figure 13.2 because the condition number is larger. CHALLENGE 13.2. The results are shown in Figure 13.3. One thing to note is that the sensitivity is not caused by the conditioning of the linear parameters; as t final is varied, the condition number κ ( A ) varies from 62 to 146, which is quite small. But the plots dramatically illustrate the fact that a wide range of
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Unformatted text preview: α values produce small residuals for this problem. This is an inherent limitation in the problem and we cannot change it. It means, though, that we need to be very careful in computing and reporting results of exponential ±tting. One important requirement on the data is that there be a sufficiently large number of points in the range where each of the exponential terms is large. CHALLENGE 13.3. When the true α = [ − . 3 , − . 4], the computations with 4 parameters produced unreliable results: [ − . 343125 , − 2 . 527345] for the ±rst guess and [ − . 335057 , − . 661983] for the second. The results for 2 parameters were some-what better but still unreliable: [ − . 328577 , − . 503422] for the ±rst guess and [ − . 327283 , − . 488988] for the second. Note that all of the runs produced one signi±cant ±gure for the larger of the rate constants but had more trouble with the smaller. 75...
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