Name: Entry #: Group#: Department of Computer Science and Engineering CS 210: Numerical and Scientific Computing Circle the correct answer for each of the following questions. 1. Which of the following is not an appropriate use for interpolation? (a) To read between the lines of a table (b) To replace a complicated function by a simple one (c) To smooth out error in a noisy function (d) To integrate a tabular function Choices (a), (b) and (d) are legitimate uses of interpolation. However, since by definition it fits the data points exactly, interpolation does not smooth out a noisy function. Correct answer is (C). 2. Which of the following types of polynomial interpolants costs (n 2 ) arithmetic operations to determine from n given data points? (a) Monomial basis (b) Lagrange (c) Newton (d) All of the above Correct answer is (C). 3. For which of the following reasons is the characteristic polynomial of a matrix not useful, in general, for computing the eigenvalues of the matrix? (a) Its coefficients may not be well determined numerically. (b) Its roots may be sensitive to perturbations in the coefficients. (c) Its roots may be difficult to compute. (d) All of the above. The characteristics polynomial may exhibit all three of these drawbacks. Correct answer is (d) 4. Which of the following properties does not necessarily imply that an n x n real Matrix A is diagonalizable (i.e., is similar to a diagonal matrix)? (a) A has n distinct eigenvalues. (b) A has only real eigenvalues. (c) A is equal to its transpose. (d) A commutes with its transpose. Having all real eigenvalues is not enough to ensure that a matrix is 1 1 diagonalizable. For example, the matrix is not similar to any diagonal 0 1 matrix. On the other hand, if a matrix has distinct eigenvalues (a), is symmetric (C), or is normal (d), then it is always diagonalizable. Correct answer is (b). 5. For which of the following purposes is Inverse Iteration useful for a given matrix? (a) Computing the eigenvalue of smallest magnitude (b) Computing the eigenvalue closest to a given value (c) Computing the eigenvector corresponding to a given approximate eigenvalue (d) All of the above Inverse iteration converges to the eigenvector corresponding to the eigenvalue of smallest magnitude of a given matrix. Thus, it is useful for (a), with no shift, or (b), with shift equal to the given value, or (C), with shift equal to the given approximate eigenvalue. So correct answer is (d).
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