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Unformatted text preview: MIT 18.335, Fall 2005: Midterm, Solutions November 10, 2005 Name: • Do all of the 8 problems • Justify your answers • Exam time 80 minutes Grading 1 / 10 2 / 10 3 / 10 4 / 10 5 / 15 6 / 15 7 / 15 8 / 15 /100 1 1. (10 points) (a) Show that ρ ( A ) ≤ bardbl A bardbl for any induced matrix norm, where ρ ( A ) is the spectral radius of A (the largest absolute eigenvalue). (b) Show that the condition number κ ( A ) ≥ 1, where κ is computed with any induced matrix norm. Solution: (a) From the definitions of induced matrix norm and eigenvalues: bardbl A bardbl = sup x negationslash =0 bardbl Ax bardbl bardbl x bardbl ≥ bardbl ρ ( A ) x 1 bardbl bardbl x 1 bardbl = ρ ( A ) (b) From the definition of condition number and the bound on induced matrix norms on matrix products: κ ( A ) = bardbl A − 1 bardblbardbl A bardbl ≥ bardbl A − 1 A bardbl = bardbl I bardbl = 1 2 2. (10 points) Consider the following algorithm applied to an m × n matrix A : for j = 1 to n v j = A : ,j R 1: j − 1 ,j = Q ∗ : , 1: j − 1 A : ,j v j = v j − R 1: j − 1 ,j Q : , 1: j − 1 R jj = bardbl v j bardbl 2 Q : ,j = v j /R jj (a) What does the algorithm compute, and which method is used? (b) Calculate the operation count (leading term). Show your work. Correction: Line 4 should be: v j = v j − Q : , 1: j − 1 R 1: j − 1 ,j Solution: (a) The algorithm computes the reduced QR factorization of A , using the classical GramSchmidt algorithm. It does not use the modified GramSchmidt, since A : ,j is used instead of v j at Line 3. (b) Most work done in lines 3 and 4, with about 4 mj operations per iteration. The total is then: n summationdisplay j =1 4 mj ∼ 2 mn 2 3 3. (10 points) Suppose you solve a linear system Ax = b by a backward stable algorithm on a computer with ǫ machine = 10...
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 Spring '10
 Johnson
 Singular value decomposition, ℓk e∗, Ax ρ

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