19-Norms-and-Conditioning-4UP

19-Norms-and-Conditioning-4UP - Vector norms Norms...

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Unformatted text preview: Vector norms Norms & errors Conditioning Norms and Conditioning Dhavide Aruliah UOIT MATH 2070U c D. Aruliah (UOIT) Norms and Conditioning MATH 2070U 1 / 19 Vector norms Norms & errors Conditioning Norms and Conditioning 1 Vector norms 2 Norms, errors and linear equations 3 Conditioning of linear equations c D. Aruliah (UOIT) Norms and Conditioning MATH 2070U 2 / 19 Vector norms Norms & errors Conditioning Motivation for vector norms Real numbers are ordered : given a , b R , either a < b , a > b , or a = b Vectors in R n are not ordered,e.g., expressions like 1- 1 1 > - 1 1- 1 or - 1 2- 1 < 1- 1 3 do not make sense (at least as scalars) Norms provide a way to order vectors, measure distance c D. Aruliah (UOIT) Norms and Conditioning MATH 2070U 4 / 19 Vector norms Norms & errors Conditioning Vector norms Definition (Vector norm) Given a vector space V , a norm is a function kk : V [ 0, ) satisfying three postulates: 1 k v k > 0 if v 6 = for every v V 2 k v k = | |k v k for every C , v V 3 k u + v k k u k + k v k for every u , v V (triangle inequality) k x k provides notion of length of vector x k x- y k provides notion of distance between vectors x , y c D. Aruliah (UOIT) Norms and Conditioning MATH 2070U 5 / 19 Vector norms Norms & errors Conditioning The ` 2-norm The ` 2-norm k x k 2 : = " n k = 1 | x k | 2 # 1 2 x R n Also called Euclidean norm or 2-norm Given x in workspace, use norm(x,2) in MATLAB 3,- 4, 0, 3 2 T 2 = s ( 3 ) 2 + (-...
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19-Norms-and-Conditioning-4UP - Vector norms Norms...

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