On Condition Numbers And the Distance to the Nearest Ill Posed Problem
On Condition Numbers And the Distance to the Nearest Ill Posed Problem
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To quickly assess the difficulty of the text, read a short excerpt:
LU^JIk. P"-^) Proof: From Lemma 3 we know that if i|«I|=l then ID I ^ A = iig-'-' W -Jf'P'W l IP'MP = lliph" - j^'pVII b'WP - M'-'p - ^'3. "p\\ Ip'WP Ip'WP where |||! is the 2-norm of a matrix. The matrix ii"^ - i'x'^ is an n + 1 by n + 1 matrix whose norm we may bound simply by ilil"^ - i'i'^l ^ lUII • lli"^ll + Ill'IP ^n'max (LlxP-^) . This implies IT I- "^axfLUF-^llbll Now consider the function y{s) — y\p, '{x, )\. We have just shown 12 j-y(s)^n^max(l, \x, ^''-')\\pM-y\s) so that by Lemma 2 ...y(s) remains finite for '^'' 2n'- max {\\pM, \\pM kP""^) as claimed. Q. E. D.
At first glance it would seem hard to apply Theorem 10 since sq is defined in terms of itself. In practice, however, one would apply the theorem when it is possible to make x a multiple zero by only a small perturbation in p. Thus, x, should not vary much from x nor should \\pj\\ vary much from \\p\\. In such cases an approximate lower bo'ind is thus provided by the expression Ip'MP 2n^ ■ |lp||-max(l, !i|2''--''-) 6.
What to read after On Condition Numbers And the Distance to the Nearest Ill Posed Problem?
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To quickly assess the difficulty of the text, read a short excerpt:
LU^JIk. P"-^) Proof: From Lemma 3 we know that if i|«I|=l then ID I ^ A = iig-'-' W -Jf'P'W l IP'MP = lliph" - j^'pVII b'WP - M'-'p - ^'3. "p\\ Ip'WP Ip'WP where |||! is the 2-norm of a matrix. The matrix ii"^ - i'x'^ is an n + 1 by n + 1 matrix whose norm we may bound simply by ilil"^ - i'i'^l ^ lUII • lli"^ll + Ill'IP ^n'max (LlxP-^) . This implies IT I- "^axfLUF-^llbll Now consider the function y{s) — y\p, '{x, )\. We have just shown 12 j-y(s)^n^max(l, \x, ^''-')\\pM-y\s) so that by Lemma 2 ...y(s) remains finite for '^'' 2n'- max {\\pM, \\pM kP""^) as claimed. Q. E. D.
At first glance it would seem hard to apply Theorem 10 since sq is defined in terms of itself. In practice, however, one would apply the theorem when it is possible to make x a multiple zero by only a small perturbation in p. Thus, x, should not vary much from x nor should \\pj\\ vary much from \\p\\. In such cases an approximate lower bo'ind is thus provided by the expression Ip'MP 2n^ ■ |lp||-max(l, !i|2''--''-) 6.
What to read after On Condition Numbers And the Distance to the Nearest Ill Posed Problem?
You can find similar books in the "Read Also" column, or choose other free books by James Demmel to read onlineMoreLess
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