L2 Norm Error

consists of computing, at each node: the difference the relative difference the percentage where is the exact solution and is the calculated solution, and printing, for each degree of freedom: the error 2 norm of a vector norm the relative error norm the relative error norm the maximum error by indicating

Euclidean Norm Of A Matrix

in addition the number and coordinates of the node where the maximum occurs. Preprocessor NORMXX compares the calculated solution with the l2 norm matlab exact solution for those cases where the solution to a problem is known analytically. It calls module NORME: SUBROUTINE NORME (M,XM,DM,NFMAIL,NIMAIL,NFCOOR,NICOOR,NFB,NIB, + NFBS,NIBS,INDICB,NSM,FONINT,SOLEX,DSOLEX) C ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ C AIM : IPRINT THE EXACT SOLUTION, THE CALCULATED SOLUTION, l1 norm vs l2 norm C --- THE ABSOLUTE AND RELATIVE DIFFERENCES BETWEEN THEM, C THE L1,L2 ERRORS AND MAX C ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ where M, XM and DM designates the super array, NFMAIL, NIMAIL are the file number and level of structure MAIL, NFCOOR, NICOOR are the file number and level of structure COOR, NFB, NIB are the file number and level of structure B, NFBS, NIBS are the file number and level of structure

L2 Norm Numpy

B on exit, INDICB is the save option: 1 : the error is stored in NFBS (used in this case), 0 if not, NSM is the number of the load case to consider (between 1 and NDSM), FONINT is a logical set to .TRUE. if functions SOLEX or DSOLEX are input as interpreted functions, and set to .FALSE. if they are input in the classical manner, SOLEX, DSOLEX are the functions used to input the exact solution (in single or double precision). Depending on the value of FONINT, functions SOLEX or DSOLEX must be written using the following format: FUNCTION SOLEX(I,X,Y,Z) DOUBLE PRECISION FUNCTION DSOLEX(I,X,Y,Z) where I is the degree of freedom number of the node with coordinates X, Y and Z. 2.10.2 Norm corresponding to D.S. TAE Preprocessor NORMXX compares the calculated stresses with the exact solution for those cases where the stresses of elasticity problem is known analytically, using the same method of computation as for a D.S. B as seen above. It calls module NORTAE: SUBROUTINE NORTAE(M,XM,DM,NFTAE,NITAE,NFTAES,NITAES,INDICB, + FONINT,SOLEX,DSOLEX,NSM,NC1) C ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ C AIM : PRINT THE EXACT SOLUTION THE CALCULATED SOLUTION, C --- THE ABSOLUTE AND RELATIVE DIFFERENCES BETWEEN THEM, C THE L1,L2 ERRORS AND MAX FOR A SOLUTION GIVEN BY TAES C ................................

Exercise 7 Introduction The objects we work with in linear systems are vectors and matrices. In order to make statements about the size of euclidean-norm example these objects, and the errors we make in solutions, we want to norm of a vector example be able to describe the ``sizes'' of vectors and matrices, which we do by using norms. We then need

L2 Norm Regularization

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 https://www.rocq.inria.fr/modulef/Doc/GB/Guide6-10/node21.html 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 computation. From the definitions of norms and errors, we can define the condition number of a matrix, which will give http://www.math.pitt.edu/~sussmanm/2071Spring09/lab05/ 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 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 __________ __________ __________ Matri

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 http://www.netlib.org/lapack/lug/node75.html eigenvectors of a matrix. This section provides measures for errors in these quantities, which we need in order to express error bounds. First consider scalars. Let the scalar be an approximation of the 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 2 norm instead 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 norm of a 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 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

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