k where a maxa 0 you might nd the matrix

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Unformatted text preview: test set (using the metric found using the data set above). 5.16 Polynomial approximation of inverse using eigenvalue information. We seek a polynomial of degree k , p(a) = c0 + c1 a + c2 a2 + · · · + ck ak , for which p(A) = c0 I + c1 A + c2 A2 · · · + ck Ak is an approximate inverse of the nonsingular matrix A, for all A ∈ A ⊂ Rn×n . When x = p(A)b ˆ is used as an approximate solution of the linear equation Ax = b, the associated residual norm is A(p(A)b) − b 2 . We will judge our polynomial (i.e., the coefficients c0 , . . . , ck ) by the worst case residual over A ∈ A and b in the unit ball: Rwc = sup A∈A, b 2 ≤1 A(p(A)b) − b 2 . The set of matrices we take is A = {A ∈ Sn | σ (A) ⊆ Ω}, where σ (A) is the set of eigenvalues of A (i.e., its spectrum), and Ω ⊂ R is a union of a set of intervals (that do not contain 0). (a) Explain how to find coefficients c⋆ , . . . , c⋆ that minimize Rwc . Your solution can involve ex0 k pressions that involve the supremum of a polynomial (with scalar argument) over an interval. (b) Carry out your metho...
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This note was uploaded on 09/10/2013 for the course C 231 taught by Professor F.borrelli during the Fall '13 term at Berkeley.

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