Gaussian quadrature accuracy

The (n + 1)-dimensional Krylov space constructed in Eq. (C.5) spans locally over p(t) and the subsequent n actions of A(t). As an orthogonal but incomplete basis set, the Gaussian quadrature accuracy of order would be expected for the n-dimensional Krylov space approximation. It thus allows the time-local evolution, p t + St) exp[A(t)St]p t) be evaluated accurately with a fairly large St, The project-out error can be estimated similarly as that of the short-iterative-Lanczos Hilbert-space propagator [51]. 

As has already been mentioned, Eq. (3.36) is not a Gaussian quadrature approximation its degree of accuracy is not known a priori (and strongly depends on the choice of moments on which the formula is constructed) and the algorithms for its derivation, from the moments of the NDF, are not well known (unlike for the univariate case). 

The DVR is related to, but distinct from pseudo-spectral and collocation methods of solving differential equations. For the DVR there is an orthogonal transformation which defines die relation of die DVR to the finite basis representation (FBR). > Thus, for example, the Hermidan character of operators remains obvious in the DVR. Both pseudo-spect and collocation methods, however, use a "mixed" representation operators and, as such, do not display the Hermitian character of operators such as H. Thus the advantages of the DVR are that the accuracy is that of a Gaussian quadrature and it is a true representation, while the collocation methods permit more freedom in the choice of points, a distinct advantage in some multidimensional problems. 

The integrals over the particle surface are usually computed by using appropriate quadrature formulas. For particles with piecewise smooth surfaces, the numerical stability and accuracy of the T-matrix calculations can be improved by using separate Gaussian quadratures on each smooth section [8,170]. 

Since the EKF is based on the first-order Taylor series expansion, the accuracy and stability of the EKF may not be sufficient for many applications with large uncertainties. Many quadrature-based Gaussian approximation filters can be used in the same filtering framework to improve the performance of the EKF. 

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