The Probability That a Numerical Analysis Problem is Difficult
The Probability That a Numerical Analysis Problem is Difficult
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2) Prob( K (A) 3= *) = "^, ' + 0( — x" Remark: The upper bound in (5. 1) exceeds the asymptotic value in (5. 2) by a factor of only about e 2 n A /(n 2 -l) for sufficiently large x. However, even for n = 10, x must exceed about 5300 for the upper bound to drop below 1. For n = 100, x must exceed 2. 210 7 for the upper bound to drop below 1.
Applying estimate (4. 20) to equation (3. 2) yields Theorem 5. 2: Let A be a random real n by n matrix distributed in such a way that A/||A|| f is uniformly... distributed on the unit sphere. Let k(A) = ||A|| F -||A _1 ||. Then C ^- y*)" 2 " 1 ^ Prob(K(A) * x) * i *k)0 is a constant proportional to the volume of the variety of singular matrices inside the unit ball.
Remark: When n = 10 x must exceed 4900 for the upper bound in (5. 3) to be less than 1. More generally, for large n x must exceed about 4. 93n 3 for the upper bound to be less than 1. One can prove this by noting that the upper bound may also be written as 2[(l+2n/x)" : - 1].
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To quickly assess the difficulty of the text, read a short excerpt:
2) Prob( K (A) 3= *) = "^, ' + 0( — x" Remark: The upper bound in (5. 1) exceeds the asymptotic value in (5. 2) by a factor of only about e 2 n A /(n 2 -l) for sufficiently large x. However, even for n = 10, x must exceed about 5300 for the upper bound to drop below 1. For n = 100, x must exceed 2. 210 7 for the upper bound to drop below 1.
Applying estimate (4. 20) to equation (3. 2) yields Theorem 5. 2: Let A be a random real n by n matrix distributed in such a way that A/||A|| f is uniformly... distributed on the unit sphere. Let k(A) = ||A|| F -||A _1 ||. Then C ^- y*)" 2 " 1 ^ Prob(K(A) * x) * i *k)
Remark: When n = 10 x must exceed 4900 for the upper bound in (5. 3) to be less than 1. More generally, for large n x must exceed about 4. 93n 3 for the upper bound to be less than 1. One can prove this by noting that the upper bound may also be written as 2[(l+2n/x)" : - 1].
What to read after The Probability That a Numerical Analysis Problem is Difficult?
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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