Fact 27 vector p norms let x cn with elements x x1 p

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Unformatted text preview: x for all α ∈ C, x ∈ Cn . The vector p-norms below are useful for computational purposes, as well as analysis. Fact 2.7 (Vector p -Norms). Let x ∈ Cn with elements x = x1 p-norm 1/p ... xn T . The n x p = is a vector norm. j =1 |xj |p p ≥ 1, , Example. • If ej is a canonical vector, then ej • If e = 1 e 1 1 = n, T ∈ Rn , then e ··· 1 ∞ = 1, p = 1 for p ≥ 1. e p = n1/p , 1 < p < ∞. The three p-norms below are the most popular, because they are easy to compute. • One norm: x 1 n j =1 |xj |. = • Two (or Euclidean) norm: x 2 = • Infinity (or maximum) norm: x Example. If x = 1 x 1 2 n ··· 1 = n(n + 1), 2 x T 2 = ∞ n 2 j =1 |xj | √ = x∗x. = max1≤j ≤n |xj |. ∈ Rn , then 1 n(n + 1)(2n + 1), 6 x ∞ = n. The inequalities below bound inner products in terms of norms. Fact 2.8. Let x , y ∈ Cn . Then Hölder inequality: |x ∗ y | ≤ x 1 y ∞ Cauchy–Schwarz inequality: |x ∗ y | ≤ x Moreover, |x ∗ y | = x 2 y 2 2 y 2. if and only if x and y are multiples of each other. Example. Let x ∈ Cn with elements x = x1 · · · xn and Cauchy–Schwarz inequality imply, respectively, n n i =1 xi ≤ n max |xi |, 1≤i ≤n i =1 xi ≤ T √ . The Hölder inequality n x 2. Downloaded 01/17/14 to 143.215.200.123. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 2.5. Vector Norms 31 Definition 2.9. A nonzero vector x ∈ Cn is called unit-norm vector in the · norm if x = 1. The vector x/ x has unit norm. Example. Let e be the n × 1 vector of all ones. Then 1= e ∞ = 1 e n 1 = √e n 1 . 2 The canonical vectors ei have unit norm in any p-norm. Normwise Errors. We determine how much information the norm of an error gives about individual, componentwise errors. Definition 2.10. If x is an approximation to a vector x ∈ Cn , then x − x is a ˜ ˜ x −x ˜ x −x ˜ normwise absolute error. If x = 0 or x = 0, then x and x are normwise ˜ ˜ relative errors. How much do we lose when we replace componentwise errors by normwise errors? For vectors x , x ∈ Cn , the infinity norm is equal...
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