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2.5. Vector Norms 29 Exercises 1. Relative Conditioning of Multiplication. Let x , y , ˜ x , ˜ y be nonzero scalars. Show: v v v v xy − ˜ x ˜ y xy v v v v ( 2 + ǫ)ǫ , where ǫ = max bv v v v x − ˜ x x v v v v , v v v v y − ˜ y y v v v v B , and if ǫ 1, then v v v v xy − ˜ x ˜ y xy v v v v 3 ǫ . Therefore, if the relative error in the inputs is not too large, then the condition number of multiplication is at most 3. We can conclude that multiplication is well-conditioned in the relative sense, provided the inputs have small relative perturbations. 2. Relative Conditioning of Division. Let x , y , ˜ x , ˜ y be nonzero scalars, and let ǫ = max bv v v v x − ˜ x x v v v v , v v v v y − ˜ y y v v v v B . Show: If ǫ < 1, then v v v v x/y − ˜ x/ ˜ y x/y v v v v 2 ǫ 1 ǫ , and if ǫ < 1 / 2, then v v v v x/y − ˜ ˜ y x/y v v v v 4 ǫ . Therefore, if the relative error in the operands is not too large, then the condition number of division is at most 4. We can conclude that division is well-conditioned in the relative sense, provided the inputs have small relative perturbations. 2.5 Vector Norms In the context of linear system solution, the error in the solution constitutes a vector. If we do not want to pay attention to individual components of the error, perhaps because there are too many components, then we can combine all errors into a single number. This is akin to a grade point average which combines all grades into a single number. Mathematically, this “combining” is accomplished by norms. We start with vector norms, which measure the length of a vector. Defnition 2.6. A vector norm b·b is a function from C n to R with three properties: N1: b x b ≥ 0 for all x C n , and b x b = 0 if and only if x = 0 . Downloaded 01/17/14 to 143.215.200.123. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php
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30 2. Sensitivity, Errors, and Norms N2: b x + y b ≤ b x b+b y b for all x , y C n (triangle inequality). N3: b α x b = | α |b x b for all α C , x C n . The vector p -norms below are useful for computational purposes, as well as analysis. Fact 2.7 (Vector p -Norms). Let x C n with elements x = ( x 1 ... x n ) T . The p-norm b x b p = n s j = 1 | x j | p 1 /p , p 1, is a vector norm. Example. If e j is a canonical vector, then b e j b p = 1 for p 1. If e = ( 1 1 ··· 1 ) T R n , then b e b 1 = n , b e b = 1, b e b p = n 1 /p , 1 < p < . The three p -norms below are the most popular, because they are easy to compute. One norm: b x b 1 = n j = 1 | x j | . Two (or Euclidean) norm: b x b 2 = r n j = 1 | x j | 2 = x x . InFnity (or maximum) norm: b x b = max 1 j n | x j | . Example. If x = ( 1 2 n ) T R n , then b x b 1 = 1 2 n(n + 1 ) , b x b 2 = R 1 6 n(n + 1 )( 2 n + 1 ) , b x b = n . The inequalities below bound inner products in terms of norms.
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