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Unformatted text preview: x for all α ∈ C, x ∈ Cn . The vector pnorms below are useful for computational purposes, as well as
analysis.
Fact 2.7 (Vector p Norms). Let x ∈ Cn with elements x = x1
pnorm 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 pnorms 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 unitnorm 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 pnorm.
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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 Spring '07
 McClain
 Multiplication, Scalar

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