Error Propagation System Of Equations
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Error Propagation Formulas
vote favorite If I have a system of equations, $Ax=B$ where the elements of $B$ have been experimentally determined and as such each element has some uncertainty, how would I propagate this to the elements of $x$? $$ \left[\begin{matrix} a_{11} & a_{12}\\ a_{21} & a_{22}\\ \end{matrix}\right] \left[\begin{matrix} x_{11}\\x_{21}\end{matrix}\right]= \left[\begin{matrix} b_{11}\pm\sigma_{b_{11}}\\b_{21}\pm\sigma_{b_{21}}\end{matrix}\right] $$ For instance, in a system like the one above, how do I account for the error in $B$ when solving for $x$? I am trying to find $\sigma_{x_{11}}$ and $\sigma_{x_{12}}$. measurement-error error-propagation share|improve this question edited Apr 28 '13 at 23:49 sashkello 1,35111124 asked Apr 28 '13 at 23:01 johnish 1235 add a comment| 1 Answer 1 active oldest votes up vote 5 down vote accepted Let me translate into statistician. So $B$ is a random variable where $B = \beta + \varepsilon$, with $\text{Var}(\varepsilon)$ = $\Sigma_B$, for $\Sigma_B$ known. An observation is taken, and the observed value of $B$ is $b$. Assuming $A$ is invertible, the solution of $Ax
Order Error Analysis of a Linear System of Equations by use of Error Propagation Matrices connected to the Pseudo Inverse SolutionArticle · January 2003 with 187 ReadsSource: CiteSeer1st Per- Ake Wedin2nd Gunilla WikstromAbstractThe singular value decomposition (SVD) of a matrix A
Error Propagation Calculus
= U#V is a useful tool for analyzing the e#ect of errors in A error propagation example of the pseudoinverse solution to Ax = b. Let E be the error propagation matrix such that the first order error error propagation division propagation result dx = E A dA(:) is satisfied. Then the SVD of E is directly available from the SVD of A. It is shown how to calculate E A in di#erent cases: well-, over- http://stats.stackexchange.com/questions/57532/propagation-of-uncertainty-through-a-linear-system-of-equations and underdetermined as well as rank-deficient. Illustrative small examples are analyzed as is the connection between the singular values of E and condition numbers. A few steps are also taken towards the analysis of a regularized solution of Ax = b.Do you want to read the rest of this article?Request full-text CitationsCitations1ReferencesReferences0Interpretation and Practical Use of Error Propagation Matrices[Show abstract] [Hide abstract] ABSTRACT: Di#erent kinds of linear systems of equations, Ax https://www.researchgate.net/publication/2866284_First_Order_Error_Analysis_of_a_Linear_System_of_Equations_by_use_of_Error_Propagation_Matrices_connected_to_the_Pseudo_Inverse_Solution = b where A , often occur when solving real-world problems. The singular value decomposition of A can then be used to construct error propagation matrices and by use of these it is possible to investigate how changes in both the matrix A and vector b a#ect the solution x. Theoretical error bounds based on condition numbers indicate the worst case but the use of experimental error analysis makes it possible to also have information about the e#ect of a more limited amount of perturbations and are in that sense more realistic. In this paper it is shown how the e#ect of perturbations can be analyzed by a semiexperimental analysis for the case m = n and m > n. The analysis combines the theory of the error propagation matrices with an experimental error analysis based on randomly generated perturbations that takes the structure of A into account. Keywords : pseudoinverse solution, perturbation theory, singular value decomposition, experimental error analysis Paper III Contents 1Article · Jan 2003 Gunilla WikstromPer- Ake WedinReadPeople who read this publication also readLinear algebra software for large-scale accelerated multicore computing Full-text · Article · May 2016 A. AbdelfattahH. AnztJ. Dongarra+6 more authors ...A. YarKhanRead full-text2.5D forward modeling and inversion of frequency-domain airborne electromagnetic data Full-text · Ar
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