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Unformatted text preview: MIT OpenCourseWare http://ocw.mit.edu 18.085 Computational Science and Engineering I
Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.085 Quiz 1 October 5, 2007 Professor Strang
Grading 1 2 3 Your PRINTED name is: SOLUTIONS 1) (39 pts.) With h = 1 3 there are 4 meshp oints 0, 1 , 2 , 1 and displacements u0 , u1 , u2 , u3 . 33 a) Write down the matrices A0 , A1 , A2 with three rows that produce the ﬁrst diﬀerences ui − ui−1 : A0 has 0 boundary conditions on u A1 has 1 boundary condition u0 = 0 (left end ﬁxed) A2 has 2 boundary conditions u0 = u3 = 0. b) Write down all three matrices AT A0 , AT A1 , AT A2 . 0 1 2 CROSS OUT IF FALSE / GIVE REASON BASED ON COLUMNS OF A ! K0 = AT A0 is (singular) (invertible) (p ositive deﬁnite) 0 Reason: K1 = AT A1 is (singular) (invertible) (p ositive deﬁnite) 1 Reason: c) Find all solutions w = (w1 , w2 , w3 ) to each of these equations: AT w = 0 0 AT w = 0 1 AT w = 0 2 1 Solution. a) � � A0 = � � b) � 1 −1 ⎡ � ⎡ � −1 1 −1 1 −1 1 ⎢ ⎢ ⎢ ⎣ ⎡ � � A1 = � −1 � � 1 1 −1 1 ⎢ ⎢ ⎢ ⎣ ⎡ � ⎢ � ⎢ A2 = � −1 ⎢ 1 � ⎣ 0 −1 � 1 0 ⎡ � ⎢ � ⎢ � −1 ⎢ 2 −1 T ⎢ A0 A0 = � � ⎢ � −1 2 −1 ⎢ � ⎣ −1 1 � singular c) � � AT A1 = � −1 1 � 2 −1 ⎢ ⎢ 2 −1 ⎢ ⎣ −1 1 AT A2 = � 2 � 2 −1 −1 2 ⎡ ⎣ � (invertible) (positive deﬁnite) �⎡ � ⎢w � ⎢� 1⎢ � 1 −1 ⎢� ⎢ T ⎢�w ⎢ A0 w = � 2 � ⎢ � 1 −1 ⎢ � ⎣ � ⎣ w3 1 AT w = 0 1 −� � −1 ⎡ −� w=0 w=0 �⎡ � ⎡ w1 �⎡ ⎢ 1 −1 0� ⎢ 0 � ⎣ � w2 ⎢ = � ⎣ AT w = � 2 0 1 −1 � ⎣ 0 w3 −� �⎡ 1 �⎢ �⎢ w = c�1⎢ �⎣ 1 2 2) (33 pts.) a) Find the eigenvalues �1 , �2 , �3 and unit eigenvectors y1 , y2 , y3 of B . � Hint: one eigenvector is (1, 0, −1)/ 2. � ⎡ 1 −1 0 ⎢ � � ⎢ B = � −1 2 −1 ⎢ . � ⎣ 0 −1 1 b) Factor B into Q�QT with Q−1 = QT . Draw a graph of the energy function f (u1 , u2 , u3 ) =
1T u Bu. 2 This is a surface in 4dimensional u1 , u2 , u3 , f space so your graph may not be perfect—OK to describe it in 1 sentence. c) What diﬀerential equation with what boundary conditions on y (x) at x = 0 and 1 is the continuous analog of B y = �y ? What are the eigenfunctions y (x) and eigenvalues � in this diﬀerential equation ? At which x’s would you sample the ﬁrst three eigenfunctions to get the three eigenvectors in part (a) ? 3 Solution. �⎡ 1 �⎢ �⎢ a) y1 = � 1 ⎢ has B y1 = 0 so �1 = 0 (check: trace =4) �⎣ 1 � 3 � ⎡ 1 � ⎢ � ⎢ given vector y2 = � 0 ⎢ has B y2 = y2 so �2 = 1 � ⎣ −1 � 2 � ⎡ 1 � ⎢ ⎢ � y3 is orthogonal to y1 , y2 y3 = � −2 ⎢ with �3 = 3 � ⎣ 1 � 6 b) The orthonormal eigenvectors are the columns of Q (orthonormal gives Q T Q = I ). � ⎡ � ⎡ 1 1 1 � ⎢ 0 � ⎢ ⎢ � �1 ⎢ 0 −2 ⎢ � Then B = Q�QT with Q = � �=� 1 ⎢ ⎣ ⎣ � 1 −1 1 3 � � � 3 2 6 1 The graph of f = 2 uT Bu has a valley along the line of eigenvectors u = (c, c, c). The surface goes up the orthogonal directions. f �⎡ 1 �⎢ �⎢ �1⎢ �⎣ 1 4 c) B is freefree so the equation is −y �� = �y with y � = 0 at x = 0, 1. yk = cos k�x , k = 0, 1, 2, . . . ⎤1 , 3, 5 666 ⎥ to get (a multiple of) Sample at the points x = the discrete eigenvectors y1 , y2 , y3 . 5 3) (28 pts.) The ﬁxedﬁxed ﬁgure shows n = 2 masses and m = 4 springs. Displacements u1 , u2 . c1 m1 c2 m2 c4 c3 a) Write down the stretchingdisplacement matrix A in e = Au. b) What is the stiﬀness matrix K = AT CA for this system ? c) Theory question about any AT CA. C is symmetric positive deﬁnite. What condition on A assures that uT AT CAu > 0 for every → vector u = 0 ? Explain why this is greater than zero and where you use your condition on A. 6 Solution. a) � e1 �⎢� ⎢� ⎡ �⎢� ⎢ � e2 ⎢ � −1 1 ⎢ u1 ⎢ � ⎣ = Au � ⎢=� �⎢� ⎢ � e3 ⎢ � 0 1 ⎢ u2 �⎣� ⎣ 0 −1 e4 b) � c1 c2 c3 ⎡� 1 0 ⎡ ⎡ � 1 0 ⎡ AT CA = � � 1 −1 0 1 ⎡� � 0 0� ⎣� � 1 −1 � � ⎢� ⎢� ⎡ ⎢� ⎢ ⎢ � −1 c + c2 −c2 1⎢ ⎢� ⎢=� 1 ⎣ ⎢� ⎢ ⎢� 0 1⎢ −c2 c2 + c 3 + c 4 ⎣� ⎣ c4 0 −1 c) Write uT AT CAu = (Au)T C (Au) = eT Ce Since C is p ositive deﬁnite, this is positive unless e = 0. Condition on A: Independent columns. Then e = Au is zero only if u = 0. So AT CA is positive deﬁnite. 7 ...
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