Differential Equations Solutions 67

# Differential Equations Solutions 67 - 77 −5 −5 x 10 2...

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Unformatted text preview: 77 −5 −5 x 10 2 change in w2 change in x2 1 0.5 0 −0.5 0.5 x 10 0 −0.5 0.5 −0.05 0 0.05 change in x1 0.1 0 change in w1 0.5 1 −7 x 10 1 0 −1 −1 0.2 0 −0.05 −1 −3 x 10 0.05 −0.1 −0.1 0 −2 −1.5 1 change in w2 0.1 0 change in x1 1 −2 −1 −0.5 −3 x 10 2 1 −5 1 −1 −1 change in x2 0 change in x1 change in w2 change in x2 −1 −1 −0.5 −3 x 10 2 x 10 −0.5 0 change in w1 0.5 1 −5 x 10 0.1 0 −0.1 −0.2 −1 −0.5 0 change in w1 0.5 1 −3 x 10 Figure 13.2. Challenge 1, with α = −[0.30, 0.31] and η = 10−6 (top row), η = 10 (middle row), and η = 10−2 (bottom row). On the left, we plot the two components of x − xtrue and on the right w − wtrue . −4 Note that results will vary with the particular sequence of random errors generated. CHALLENGE 13.4. We solved this problem using MATLAB’s lsqnonlin and the two parameters α using several initial guesses: [−1, −2], [−5, −6], [−2, −6], [0, −6], and [−1, −3]. All runs except the fourth produced values α = [−1.6016, −2.6963] and a residual of 0.0024011. The fourth run produced a residual of .49631. The residuals for the ﬁve runs are shown in Figure 13.6. The four “good” residuals look like white noise of size about 10−4 , giving some conﬁdence in the ﬁt. We tested the sensitivity of the residual norm to changes in the parameters by creating a contour plot in the neighborhood of the optimal values computed above, ...
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