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Because of this approximation the error probability derived is not exact, but a lower bound, since for low SNRs error events may correspond to paths that do not have the minimum distance.
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However, the accuracy of the assumption and the limitation of the analytical solutions have not been examined and because of the approximation, the analytical solutions may lead to significant errors for the buckling loads of shallow parabolic arches in some cases.
For the case of CDE-SD, the difference is slightly bigger because of the approximation in (55).
Because of these approximation errors, the following problems may exist: The pre-image vector may consist of non-integer components.
Note that, because of the approximation done in (12), (18) can be seen as a lower bound of the algorithm exact error probability.
Then, the proposed algorithm reverts to SUD, and its performance is even worse than SUD in Fig. 8 because of the approximation in the proposed detection algorithm.
The CCM still adds another source of error in the data analysis, because of the approximation taken in the calculation of dC/da, which is performed by taking the derivative of the C = f(a) curve after fitting cubic polynomials [24].
We note that, conceptually, the 2-dimensional approach should cover all BPs identifiable by the 1-dimensional approach, but it might not be always so because of the approximation procedure in the multiple test corrections.
Unfortunately, because of the approximations involved, this argument does not prove the theorem about the area of a circle.
Because of simultaneous engineering this approximation should take place in an early planning stage with uncertain information.
Because of its advantage, this approximation procedure has received great attention, and many people have been devoted to the study of the Carathéodory scheme for SDEs.
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