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 1 October 2, 2006 Professor Strang Your PRINTED name is: Grading 1 2 3 1) (36 pts.) (a) Suppose u(x) is linear on each side of x = 0, with slopes u �(x) = A on ⎡⎥⎤⎣⎥the left and u �(x) = B on the right: Ax Bx for x � 0 u(x) = for x � 0 What is the second derivative u ��(x) ? Give the answer at every x. (b) Take discrete values U n at all the whole numbers x = n: U n = ⎡⎥⎤⎣⎥An for n � 0 Bn for n � 0 For each n, what is the second difference �2U n ? Using coefficients 1, −2, 1 (notice signs !) give the answer �2U n at every n. (c) Solve the differential equation −u ��(x) = �(x) from x = −2 to x = 3 with boundary values u(−2) = 0 and u(3) = 0. (d) Approximate problem (c) by a difference equation with h = �x = 1. What is the matrix in the equation K U = F ? What is the solution U ? 1x 22) (24 pts.) A symmetric matrix K is “positive definite” if u TKu > 0 for every nonzero vector u. (a) Suppose K is positive definite and u is a (nonzero) eigenvector, so Ku = �u. From du the definition above show that � > 0. What solution u(t) to = Ku comes from dt knowing this eigenvector and eigenvalue ? (b) Our second-difference matrix K4 has the form ATA: � ⎢� 1 ⎢� ⎢ �1 −1 ⎧��−1 1 ⎧⎧�2 −1 ⎧K4 = �1 −1 ⎧�−1 1 ⎧= �−1 2 −1 ⎧�1 −1 ⎧� ⎧ �−1 2 −1 ⎧� ⎧�−1 1 ⎧ � ⎧� 1 −1 ⎨�� ⎧⎨� −1 2 ⎨ −1 Convince me how K4 = ATA proves that uTK4u = uTATAu > 0 for every nonzero vector u. (Show me why u TATAu � 0 and why > 0.) (c) This matrix is positive definite for which b ? Semidefinite for which b ? What are its pivots ?? ⎩ 2 b ⎦ S = b 4 3xx 43) (40 pts.) (a) Suppose I measure (with possible error) u1 = b1 and u2 − u1 = b2 and u3 − u2 = b3 and finally u3 = b4. What matrix equation would I solve to find the best least squares estimate u⎪1, u⎪2, u⎪3 ? Tell me the matrix and the right side in K u⎪= f. (b) What 3 by 2 matrix A gives the spring stretching e = A u from the displacements u1, u2 of the masses ? (c) Find the stiffness matrix K = ATCA. Assuming positive c1, c2, c3 show u1 that K is invertible and positive definite. c3 u2 c1 m1 � c2 m2 � 5xxx
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