MATH 343:001 (11814) Applied Linear Algebra Fall 2009 Homework 4 Due Friday, November 6 Turn in answers to the following problems. Section 5.2: 41, 42 Section 5.3: 8, 10, 12 Section 5.4: 9, 38 MATLAB PROBLEMS. (1) This problem refers to the matrix appearing in Section 1.7 (page 59) associated to the second order diﬀerential equation-d 2 u dx 2 = f ( x ), 0 ≤ x ≤ 1 (this is the one-dimensional Poisson equation). Generate the matrix for an arbitrary n as follows A = 2 * diag ( ones ( n, 1))-diag ( ones ( n-1 , 1) , 1)-diag ( ones ( n-1 , 1) ,-1) Look at what you get with n = 5 to conﬁrm you get the same matrix as in page 60. Now generate A with n = 100. Compute the eigenvalues and eigenvectors using [ V,D ] = eig ( A ); and plot the eigenvectors corresponding to the three smallest eigenvalues (the command diag (D) will list just the diagonal entries of the matrix D ). What pattern do you see? Turn in a plot and your comments. (I prefer a single ﬁgure with all three plots if you can do it.) (2) Gerschgorin Circles. With each
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