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 12, 2005 Professor Strang 1) (40 pts.) Here is a network with 9 numbered nodes and then separately the 12 edges. 7 8 9 5 8 12 4 6 7 11 1 2 3 3 45 69101 2 (a) What is the shape of the incidence matrix A ? What is its 4th row ? (b) What is the 5th column of A ? What is the (5, 5) entry in ATA ? Write the whole 5th row of ATA. (c) How many independent solutions to Kirchhoff’s Law ATw = 0 ? Find one of them. (d) Without writing down this matrix A, explain why ATA is or is not symmetric positive definite. 1 Your PRINTED name is: Grading 1 2 3x 22) (30 pts.) (a) For a hanging elastic bar, with u(0) = 0 at the top and u �(0) = 0 at the bottom and elastic constant c(x) = 1, what is the displacement u(x) when a unit point load f (x) = �(x − a) acts at the point x = a ? Draw a graph of u(x). (b) What is the limit of u(x) as the unit load moves to the bottom (a � 1) ? Suppose it moves to the top (a � 0) ? Draw graphs of u(x) in those two cases. (c) Choose a matrix equation that approximates the differential equation in part (a). (Describe the matrix—OK to put the load at a meshpoint.) If the load moves to the lowest meshpoint (number N ), what displace-ments correspond to your answer in part (b) ? 3xx 43) (30 pts.) Suppose you measure your initial position u1 = b1, and then you measure the step lengths u2 − u1 = b2 and u3 − u2 = b3. At the end you make a last measurement u3 = b4. (a) Under what conditions on b1, b2, b3, b4 will these four equations have an exact solution ? Create (don’t solve) a set of equations for the best estimates � u2, �u1, � u3. (b) Draw a picture of masses, springs, and forces (write in all constants) that would lead to the same equations for the displacements. (c) Suppose the variances for errors in the measurements are �12, �22, �32, �42 . What equations should you solve (DON’T DO IT) for the statistically best estimate �u ? If �4 � � so that b4 becomes completely unreliable, what answer do you expect for the best �u ? 5xxx
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