2-norm Of The Error
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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 l2 norm error eigenvectors of a matrix. This section provides measures for errors in these quantities,
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which we need in order to express error bounds. First consider scalars. Let the scalar be an approximation of the energy norm error true answer . We can measure the difference between and either by the absolute error , or, if is nonzero, by the relative error . Alternatively, it is sometimes more convenient to use instead
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of the standard expression for relative error (see section4.2.1). If the relative error of is, say 10-5, then 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 infinity norm error 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 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
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Home MATLAB Answers File Exchange Cody Blogs Newsreader Link Exchange ThingSpeak Anniversary Home Post A New Message Advanced 2 norm of a function Search Help Trial software Norm of the error Subject: Norm of the error From: Richard Date: 30 Mar, 2006 09:08:00 Message: 1 of 4 Reply to this message Add http://www.netlib.org/lapack/lug/node75.html author to My Watch List View original format Flag as spam Good day All! I need to write a function to evaluate "the norm of the error" between measured and predicted outputs. This function is to be used in MATLAB's fminsearch to find various response parameters. Question: Does anyone know how to find the norm of the error using these https://www.mathworks.com/matlabcentral/newsreader/view_thread/120815 two datasets? I'm not looking for a MATLAB function, rather the actual procedure itself. I've googled on "norm of error" etc, and it is mentioned often but I can't find any actual step-by-step guides for how to find the norm of the error. Any ideas welcome! Cheers4now Richard Subject: Norm of the error From: vijitnair@gmail.com Date: 30 Mar, 2006 06:22:05 Message: 2 of 4 Reply to this message Add author to My Watch List View original format Flag as spam I guess you are looking for an objective function to be used in fminsearch. there are more than a few choices mean square error sum of squared errors mean absolute error normalized error chi-square error The choice is dictated by the kind of data set you have. Sum of squared residuals/least squares is the most popular but it is also very susceptible to noise. This might help http://www.itl.nist.gov/div898/handbook/pmd/section1/pmd14.htm Subject: Norm of the error From: joa Date: 30 Mar, 2006 18:16:21 Message: 3 of 4 Reply to this message Add author to My Watch List View origi
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 http://www.math.pitt.edu/~sussmanm/2071Spring09/lab05/ the ``sizes'' of vectors and matrices, which we do by using norms. We then need to consider whether we can bound the size of the product of a matrix and vector, given that we know the ``size'' of the two factors. In order for this 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 norm error error and backward error in a linear algebra 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 l2 norm error page itself. Vector Norms A vector norm assigns a size to a 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) e