MIT OpenCourseWare http://ocw.mit.edu 18.085 Computational Science and Engineering IFall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.� 18.085 Quiz 3 December 7, 2007 Professor Strang Your PRINTED name is: Grading 1 2 3 ** NOTE AT NOON A BIG CHEMISTRY CLASS IS COMING !!! 1) (30 pts.) (a) Solve by a Fourier sine series u(x) = bk sin kx: ⎧ ⎨ 1 0 < x < π −u ��+ 4 u(x) = f(x) = ⎩ with u(−π) = u(π) = 0 . −1 −π < x < 0 That right side f(x) is the square wave SW(x) on page 318. (b) What is the decay rate of the coefficients bk ? What is the smoothness of u(x) — which derivative jumps ? 11 3 1 xn−1 + xn + x = y < n < 5 5 5 n+1 n − ∞ ∞� 2) (30 pts.) This problem is about the equation (a) Suppose the vector x = (. . . , x−1, x0, x1, . . .) is known. The equation is a non-cyclic convolution a ∗ x = y. What is the infinite vector a ? Transform the equation into the frequency domain using X(ω) = xn einω and Y (ω) and A(ω). What is A(ω) in this problem ? (b) Suppose the vector y is known but the vector x is not known. We want to find x. Take two steps: 1. Give a formula for X(ω) using known things like Y (ω) and 15 , 53 , 51 , or A. 2. Does your formula involve any division by zero or is it safe ? The last step in this deconvolution would recove r the Fourier coeffi-cients xn from your function X(ω) but this is not on the exam ! 23) (40 pts.) This circulant equation Cd = b is a cyclic convolution: ⎤⎡⎤⎡⎤⎡ d0 1 Cd = ⎢⎢⎢⎢⎢⎢⎣ 4 −1 −1 −1 −1 4 −1 −1 −1 −1 4 −1 ⎢⎢⎢⎢⎢⎢⎣ ⎥⎥⎥⎥⎥⎥⎦ d1 d2 ⎥⎥⎥⎥⎥⎥⎦ = ⎢⎢⎢⎢⎢⎢⎣ 0 0 ⎥⎥⎥⎥⎥⎥⎦ = b is c � d = b −1 −1 −1 4 d3 0 (a) The eigenvectors of that matrix C are the four columns e0, e1, e2, e3 of the Fourier matrix F (this F is on page 347). Multiply F times the e’s to find the four eigenvalues. Check that their sum is c orrect. (b) Write the right side b = (1, 0, 0, 0) as a combination of those four eigenvectors (columns of F ). Using the eigenvalues, the solution d is what combination of the four eigenvectors ? Find the vector d . (c) A direct way to solve c � d = b would be to take the 4-point discrete transform of both sides. What are the transforms of b and c in this problem ? What is the transform of the solution d ? Isn’t this just the same method in different words (yes or no). Thank you for taking 18.085 !
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