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Forward And Backward Error Analysis

Truncation Error
Absolute Error

by introducing more precise citations. (February 2012) (Learn how and when to remove this template message) In the mathematical subfield of numerical analysis, numerical stability is backward error analysis example a generally desirable property of numerical algorithms. The precise definition of stability backward stable depends on the context. One is numerical linear algebra and the other is algorithms for solving ordinary and relative condition number partial differential equations by discrete approximation. In numerical linear algebra the principal concern is instabilities caused by proximity to singularities of various kinds, such as very small or nearly colliding numerical error eigenvalues. On the other hand, in numerical algorithms for differential equations the concern is the growth of round-off errors and/or initially small fluctuations in initial data which might cause a large deviation of final answer from the exact solution. Some numerical algorithms may damp out the small fluctuations (errors) in the input data; others might magnify such errors. Calculations that can

Truncation Error

be proven not to magnify approximation errors are called numerically stable. One of the common tasks of numerical analysis is to try to select algorithms which are robust– that is to say, do not produce a wildly different result for very small change in the input data. An opposite phenomenon is instability. Typically, an algorithm involves an approximate method, and in some cases one could prove that the algorithm would approach the right solution in some limit. Even in this case, there is no guarantee that it would converge to the correct solution, because the floating-point round-off or truncation errors can be magnified, instead of damped, causing the deviation from the exact solution to grow exponentially.[1] Contents 1 Stability in numerical linear algebra 2 Stability in numerical differential equations 3 See also 4 References Stability in numerical linear algebra[edit] There are different ways to formalize the concept of stability. The following definitions of forward, backward, and mixed stability are often used in numerical linear algebra. Diagram showing the forward error Δy and the backward error Δx, and their relati

the forward https://en.wikipedia.org/wiki/Numerical_stability and backward error analysis. Backward error propagation: How much error in input would be required to explain http://www.physics.arizona.edu/~restrepo/475A/Notes/sourcea-/node13.html all output error? Assumes that approximate solution to problem is good IF IT IS THE exact solution to a ``nearby'' problem. Example Want to approximate . We evaluate its accuracy at . Backward Error: 1) Find such that The following are different and cannot be compared: Juan Restrepo 2003-04-12

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