Norm Of The Error Vector

an eigenvalue of a matrix, Vector, such as the solution x of a linear system Ax=b, Matrix, such as a matrix inverse A-1, and Subspace, such as the space spanned by one or more eigenvectors of a matrix. This section provides measures for norm error matlab errors in these quantities, which we need in order to express error bounds. First consider l2 error norm scalars. Let the scalar be an approximation of the true answer . We can measure the difference between and either by the absolute

Relative Norms Definition

error , or, if is nonzero, by the relative error . Alternatively, it is sometimes more convenient to use instead of the standard expression for relative error (see section4.2.1). If the relative error of is, say 10-5, then

L2 Relative Error Norm

we say that is accurate to 5 decimal digits. In order to measure the error in vectors, we need to measure the size or norm of a vector x. A popular norm is the magnitude of the largest component, , which we denote . This is read the infinity norm of x. See Table4.2 for a summary of norms. Table 4.2: Vector and matrix norms Vector Matrix one-norm two-norm Frobenius norm |x|F = |x|2 infinity-norm l2 norm error matlab If is an approximation to the exact vector x, we will refer to as the absolute error in (where p is one of the values in Table4.2), and refer to as the relative error in (assuming ). As with scalars, we will sometimes use for the relative error. As above, if the relative error of is, say 10-5, then we say that is accurate to 5 decimal digits. The following example illustrates these ideas: Thus, we would say that approximates x to 2 decimal digits. Errors in matrices may also be measured with norms. The most obvious generalization of to matrices would appear to be , but this does not have certain important mathematical properties that make deriving error bounds convenient (see section4.2.1). Instead, we will use , where A is an m-by-n matrix, or ; see Table4.2 for other matrix norms. As before is the absolute error in , is the relative error in , and a relative error in of 10-5 means is accurate to 5 decimal digits. The following example illustrates these ideas: so is accurate to 1 decimal digit. Here is some related notation we will use in our error bounds. The condition number of a matrix A is defined as , where A is square and invertible, and p is or one of the other possibilities in Table4.2. The condition nu

Exercise 7 Introduction The objects we work with in linear systems are vectors and matrices. In order to make statements about the size of these objects, and the errors we make in solutions, we want to be able to describe the ``sizes'' of vectors

Relative Error Between Two Vectors

and matrices, which we do by using norms. We then need to consider whether we can bound 2 norm condition number of a matrix the size of the product of a matrix and vector, given that we know the ``size'' of the two factors. In order for this matrix norm example to happen, we will need to use matrix and vector norms that are compatible. These kinds of bounds will become very important in error analysis. We will then consider the notions of forward error and backward error in a linear algebra http://www.netlib.org/lapack/lug/node75.html computation. From the definitions of norms and errors, we can define the condition number of a matrix, which will give us an objective way of measuring how ``bad" a matrix is, and how many digits of accuracy we can expect when solving a particular linear system. This lab will take two sessions. You may find it convenient to print the pdf version of this lab rather than the web page itself. Vector Norms A vector norm assigns a size to a http://www.math.pitt.edu/~sussmanm/2071Spring09/lab05/ vector, in such a way that scalar multiples do what we expect, and the triangle inequality is satisfied. There are three common vector norms in dimensions: The vector norm The (or ``Euclidean'') vector norm The vector norm To compute the norm of a vector in Matlab: norm(x,1); norm(x,2)= norm(x); norm(x,inf) (Recall that inf is the Matlab symbol corresponding to .) Exercise 1: For each of the following vectors: x1 = [ 1; 2; 3 ] x2 = [ 1; 0; 0 ] x3 = [ 1; 1; 1 ] compute the vector norms, using the appropriate Matlab commands. Be sure your answers are reasonable. L1 L2 L Infinity x1 __________ __________ __________ x2 __________ __________ __________ x3 __________ __________ __________ Matrix Norms A matrix norm assigns a size to a matrix, again, in such a way that scalar multiples do what we expect, and the triangle inequality is satisfied. However, what's more important is that we want to be able to mix matrix and vector norms in various computations. So we are going to be very interested in whether a matrix norm is compatible with a particular vector norm, that is, when it is safe to say: There are five common matrix norms: The or ``max column sum'' matrix norm: The matrix norm: where is a (necessarily real) eigenvalue of or where is a singular value of ; The or ``max row sum'' matrix norm: The ``Frobenius'' matrix norm: The ``spectral'' matrix nor

Random Entry New in MathWorld MathWorld Classroom About MathWorld Contribute to MathWorld Send a Message to the Team MathWorld Book Wolfram Web Resources» 13,594 entries Last updated: Wed Oct 19 2016 Created, developed, and nurturedbyEricWeisstein at WolframResearch Calculus and Analysis>Norms> http://mathworld.wolfram.com/L2-Norm.html History and Terminology>Notation> Interactive Entries>Interactive Demonstrations> L^2-Norm The -norm (also written "-norm") is a vector norm defined for a complex vector (1) by (2) where on the right denotes the complex modulus. The -norm is the vector norm that is commonly encountered in vector algebra and vector operations (such as the dot product), where it is commonly denoted . However, if desired, a more explicit (but more norm error cumbersome) notation can be used to emphasize the distinction between the vector norm and complex modulus together with the fact that the -norm is just one of several possible types of norms. For real vectors, the absolute value sign indicating that a complex modulus is being taken on the right of equation (2) may be dropped. So, for example, the -norm of the vector is given norm error matlab by (3) The -norm is also known as the Euclidean norm. However, this terminology is not recommended since it may cause confusion with the Frobenius norm (a matrix norm) is also sometimes called the Euclidean norm. The -norm of a vector is implemented in the Wolfram Language as Norm[m, 2], or more simply as Norm[m]. The "-norm" (denoted with an uppercase ) is reserved for application with a function , (4) with denoting an angle bracket. SEE ALSO: Angle Bracket, Complete Set of Functions, L1-Norm, L2-Space, L-infty-Norm, Parallelogram Law, Vector Norm REFERENCES: Gradshteyn, I.S. and Ryzhik, I.M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, pp.1114-1125, 2000. Horn, R.A. and Johnson, C.R. "Norms for Vectors and Matrices." Ch.5 in Matrix Analysis. Cambridge, England: Cambridge University Press, 1990. CITE THIS AS: Weisstein, Eric W. "L^2-Norm." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/L2-Norm.html Wolfram Web Resources Mathematica» The #1 tool for creating Demonstrations and anything technical. Wolfram|Alpha» Explore anything with the first computational knowledge engine. Wolfram Demonstrations Project» Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more. Computerbasedmath.org» Join the initiative for modernizing math education. Online Integral Calculator

be down. Please try the request again. Your cache administrator is webmaster. Generated Fri, 21 Oct 2016 22:28:50 GMT by s_wx1011 (squid/3.5.20)