Changes between Version 3 and Version 4 of u/erica/norms

Timestamp:

08/22/13 15:35:21 (11 years ago)

Author:

Erica Kaminski

Comment:

—

u/erica/norms

@@ -29 +29 @@
 = Error of vectors = 
 
-This is different than scalars in that now we must think of ''norms'' of error vectors (scalar measures of their "lengths"). There are 3 different ways to measure their "lengths", namely the 1-norm, 2-norm, and infinity-norm (aka max-norm).
+Now we are considering how to compute the error of an approximate vector of s quantities at a given position (say from solving a system of s equations at a given point). This is different than scalars in that now we must think of ''norms'' of error vectors (scalar measures of their "lengths"). There are 3 different ways to measure their "lengths", namely the 1-norm, 2-norm, and infinity-norm (aka max-norm).
 
-Thus, if we define the error vector as e=
+If we define the error vector as [[latex($\vec{e}=\vec{\hat{z}} - \vec{z}$)]], then the different norms are given by,
+
+1-norm:
+
+[[latex($||e||_1 = \sum_{i=1} ^{s} |e_i| $)]], 
+
+2-norm:
+
+[[latex($||e||_2 = \sum_{i=1} ^{s}(|e_i|^2)^{1/2} $)]], and
+
+infinity-norm:
+
+[[latex($||e||_{\infty} = max_{(1\leq i\leq s)} |e_i| $)]]
+
+Note the bound on the infinity-norm that we don't get from the others, this may be helpful in determining the error of some schemes. 
+
+These norms are **equivalent** so long as the dimension of z is fixed as h goes to 0. 
+
+** This is NOT true when approximating functions!!