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 4, 2004 Professor Strang Your name is: Grading 1. 2. 3. 4.OPEN BOOK EXAM Write solutions onto these pages ! Circles around short answers please !! 1) (32 pts.) This problem is about the symmetric matrix ⎤⎡ 2 − 1 0 H = ⎢⎢⎢⎣ ⎥⎥⎥⎦ − 1 2 − 1 0 − 1 1 (a) By elimination find the triangular L and diagonal D in H = LDLT . H(b) What is the smallest number q that could replace the corner entry 33 = 1 and still leave H positive semi definite ? q = (c) H comes from the 3-step framework for a hanging line of springs: A C AT displacements −→ elongations −→ spring forces −→ external force f What are the specific matrices A and C in H = ATCA ? (d) What are the requirements on any m by n matrix A and any symmetric matrix C for ATCA to be positive definite ?x 2 2) (24 pts.) Suppose we make three measurements b1,b2,b3 at times t1,t2,t3.They would fit exactly on a straight line b = C + Dt if we could solve C + Dt1 = b1 C + Dt2 = b2 C + Dt3 = b3 . (a) If b1,b2,b3 are equally reliable what are the equations for the best values C and D ? Don’t solve the equations—write them in terms of t’s and b’s (not just some letter A). (b) Suppose the errors in b1,b2,b3 are independent with variances σ12,σ22,σ2 3 (covariances = 0 because independent). Find the new equations for C and D. Use t’s, b’s and ci =1/σi 2—by method  or the best  1 21 Remember how the covariance matrix Σ (called V in the book) enters the equations 2 Divide the three equations above by σ1,σ2,σ3.Then do ordinary least squares because the rescaled errors have variances = 1. (c) Suppose σ1 =1,σ2 = 1, but σ3 →∞so the third measurement is (exactly reliable )(totally unreliable)CROSS OUTONE. In this case the best straight line goes through which points ? 3xx 43) (20 pts.) (a) Suppose du is approximated by a centered difference:dx du u(x +∆x) − u(x − ∆x)≈ dx 2∆x 1 2 3 4 5With equally spaced points x = h, 2h, 3h, 4h, 5h = 6, 6, 6, 6, and zero 6 boundary conditions u0 = u6 =0, write down the 3 by 3 centered first difference matrix ∆: ui+1 − ui−1(∆u)i = . 2h Show that this matrix ∆ is singular (because 3 is odd) by solving ∆u =0. (b) Removing the last row and column of a positive definite matrix K always leaves a positive definite matrix L. Why ? Explain using one of the tests for positive definiteness. 5xxx 64) (24 pts.) The equation to solve is −u + u = δ(x −12) with a unit point load at x = 12 and zero boundary conditions u(0) = u(1) = 0. −u(a) Solve −u − u = 0 starting from x =0 with u(0) = 0. There will be one arbitrary constant A. Replace x by 1 − x in your answer, to solve  − u = 0 ending at u(1) = 0 with arbitrary constant B. (b) Use the “jump conditions” at x = 12to find A and B. 7xxxx