tut2 - / k A k and K ( A ) is the condition number of the...

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cep 2008/09 MT3802 - Numerical Analysis Tutorial Sheet 2 1. For a sub-ordinate matrix norm and an invertible matrix A , where Ax = b , show that ( i ) k A s k ≤ k A k s s N ( ii ) 1 k A k k x k k b k ≤ k A - 1 k 2. Assume that ( A + δ A ) - 1 is computed as an approximation to A - 1 . Show that ( A + δ A ) - 1 - A - 1 = - ( A + δ A ) - 1 A . A - 1 = - ( I + A - 1 δ A ) - 1 . A - 1 δ A . A - 1 . Provided that γK ( A ) < 1, show that k ( A + δ A ) - 1 - A - 1 k k A - 1 k γK ( A ) 1 - γK ( A ) where γ = k δ A k
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Unformatted text preview: / k A k and K ( A ) is the condition number of the matrix A . Hint: Use the identity from the rst part of the question and remember (from Section 1.7.1) that k ( I + B )-1 k 1 / (1- k B k ) if k B k &amp;lt; 1 and 1 1- k A-1 A k 1 1- k A-1 kk A k ....
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