Numerical Analysis I - Final Exam Matrikelnummer:

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1 Dr. Behrens Center for Mathematical Sciences Technische Universität München Winter Term 2005/2006 Name: Numerical Analysis I - Final Exam Matrikelnummer: I agree to the publication of the results of this examination on the course home page. Only the Matrikelnummer will be associated to the result (no names or other personal data). Signature Hints: Make sure that your copy of the exam is complete and that the copy is readable. This exam should consist of 9 (nine) pages. Make sure you filled the Name and Matrikelnummer fields. If appropriate, also sign the agreement to publish the results. You have 90 Minutes time to solve the assignments. Please read the assignments carefully, before you start solving them. In places where you find a box of page width, full answers are expected. If the space provided is not sufficient, use the page s back side or the last page for notes. In places where you find small boxes, you may mark the box(es) corresponding to the right solution(s). Assisting material is allowed, but limited to one DIN A4 sheet of paper with your own notes and reminders (cheating sheet). Except for this sheet, no other utilities are permitted. achievable points 1st corrector Assignment 1 10 Assignment 2 4 Assignment 3 10 Mult. choice Signature 1

2 Assignment 1: 10 Point(s) Gaussian elimination for matrix with special structure: Let A be a sparse matrix: A = (a ij ) 1 i,j n with a ij 0 for (i = 1, j = 1 : n), (j = 1, i = 1 : n), and (i = j, j = 1 : n); a ij = 0 otherwise. (a) Sketch the sparsity pattern of such a matrix, i.e. the pattern of those entries that are different from zero. (b) Show that the first step of Gaussian Elimination without pivoting changes all zero entries to non-zero values (so-called fill-in). 2

3 (c) Find permutation matrices P 1 and P 2 such that for Gaussian elimination without pivoting applied to the matrix à = P 1AP 2 no fill-in occurs. Hint: the fill-in occurs due to non-zero entries in the first row of the matrix. (d) Write down the algorithm that implements Gaussian elimination as discussed in (c) using only three vectors to store A. (e) What is the leading order of the computational cost of the optimal algorithm in (d)? (flop). (a) O(n), (b) O(n 2 ), (c) O(n 3 )

4 Assignment 2: 4 Point(s) Finite differences: Let f C 2 in a neighborhood of x, i.e. f is two times continuously differentiable. Verify that the error corresponding to the backward finite difference approximation of f fulfills the following formula (δ f)( x) = f( x) f( x h) h f ( x) (δ f)( x) = h 2 f (η), with η [ x h, x]. 4

5 Assignment 3: 10 Point(s) QR-algorithm for symmetric tri-diagonal matrix: Let T R n n be a symmetric tri-diagonal matrix (i.e. T i,j 0 if j {i 1, i, i + 1} for i = 1 : n). Let QR = T be the corresponding QR-factorization. a) Show that T + = RQ again is a tri-diagonal matrix. b) Let s R an eigenvalue and let T unreduced (i.e. T has no zero entries on the sub-diagonal). The shifted QR-factorization is given by T si = QR. Show that the last row of T +,s = RQ + si is given by se n, where I is the n n unit matrix, and e n the n-th unit vector. 5

6 c) Design an efficient algorithm for the QR-factorization of a tri-diagonal matrix T. d) If you could not solve c), you might still have had an idea of the structure of the algorithm. So, what kind of unitary transformation Q would you have used and why (only keywords)? 6

7 Assignment 4: Increasing the length of the mantissa of a floating point number increases the range of values that can be represented true false Assignment 5: The Cholesky factorization works for all symmetric matrices true Assignment 6: false All backward stable algorithms are forward stable true false Assignment 7: An algorithm is forward stable, if the error in the result is not larger than the number of operations times the machine precision amplified by the condition of the problem true false Assignment 8: For the condition number of a matrix A R n n we have that κ(a) = κ(αa) with α R \ {0}, i.e. it is scaling invariant under multiplication by a scalar true false Assignment 9: n The Lebesgue constant Λ n = max x [a,b] i=0 L i(x) (L i (x) the Lagrangian polynomials) is independent of the node distribution {x 0,..., x n } true false Assignment 10: The Fast Fourier Transform (FFT) for a sample length n = p, p a prime number, can be performed in O(n 2 ) flop true false Assignment 11: The Sampling Theorem (Shannon/Nyquist) tells us that a frequency k has to be sampled with a rate of at most n 2 k true false Assignment 12: Periodic cubic spline interpolation uses the following two additional conditions to close the interpolation problem: s 3(x 0 ) = f (x 0 ) and s 3(x n ) = f (x n ), for f a given function, s 3 the piecewise cubic interpolating polynomial, and x 0, x n the start and end node, respectively true false Assignment 13: 2 Point(s) The normal equation A T Ax = A T b corresponding to a least squares problem (LSP) can always be solved by a backward stable algorithm true false The normal equation corresponding to a LSP is badly conditioned true false 7

8 Assignment 14: To compute the QR-factorization A = QR of a sparse matrix A R n n, Householder reflections are efficient true false Assignment 15: In an interpolatory quadrature rule Î(f) = (b a) k λ kf(x k ), we require k λ k = 1, because this guarantees that Î(f) is a positive form true false Assignment 16: The Clenshaw-Curtis/Féjer quadrature scheme is based on an interpolatory quadrature with equidistant nodes true false Assignment 17: Let A = V DV 1 be the eigenvalue decomposition of A C n n, with D = diag(λ 1,..., λ n ). The condition of the eigenvalue problem is given by κ(d) true false Assignment 18: If A is a normal matrix, then A is diagonalizable with unitary Q, i.e. Q H AQ = D with D a diagonal matrix true false Assignment 19: Vector-iteration converges extremely rapidly for computing one single eigenvalue. true Assignment 20: The QR-algorithm for eigenvalue problems of the form Ax = λx, A normal, converges locally quadratically true false false 8

9 Take your notes here: 9