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m Moving Category:Numerical integration (quadrature) to Category:Numerical integration per Wikipedia:Categories for discussion/Speedy |
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which results from the approximation of <math>\mathbf{g}_{n}</math> in (4.2) by its order-''p'' truncated [[Taylor series|Taylor expansion]].
* ''Multistep-type exponential propagation discretization''
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which results from the interpolation of <math>\mathbf{g}_{n}</math> in (4.2) by a polynomial of degree ''p'' on <math>t_{n},\ldots, t_{n-p+1}</math>, where <math>\nabla ^{j}\mathbf{g}_{n}(t_{m},\mathbf{z}_{m})</math> denotes the ''j''-th [[backward difference]] of <math>\mathbf{g}_{n}(t_{m},\mathbf{z}_{m})</math>.
* ''Runge Kutta type Exponential Propagation discretization'' <ref name=":17">Tokman M. (2006). "Efficient integration of large stiff systems of ODEs with exponential propagation iterative (EPI) methods". J. Comput. Physics. 213 (2): 748–776. [[doi:10.1016/j.jcp.2005.08.032]].</ref>
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which results from the interpolation of <math>\mathbf{g}_{n}</math> in (4.2) by a polynomial of degree ''p'' on <math>t_{n},\ldots, t_{n}+(p-1)h/p</math>,
* ''Linealized exponential Adams discretization''<ref name=":7">M. Hochbruck.; A. Ostermann. (2011). "Exponential multistep methods of Adams-type". BIT Numer. Math. 51 (4): 889–908. [[doi:10.1007/s10543-011-0332-6]].</ref>
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which results from the interpolation of <math>\mathbf{g}_{n}</math> in (4.2) by a [[Hermite polynomials|Hermite polynomial]] of degree ''p'' on <math>t_{n},\ldots, t_{n-p+1}</math>.
=== Local linearization schemes ===
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and <math>\widetilde{\mathbf{z}}^i:\left[ t_n-\tau_i,t_n\right] \longrightarrow \mathbb{R}^d</math> is a suitable approximation to <math>\mathbf{x}(t)</math> for all <math>t\in \lbrack t_n-\tau_i,t_n]</math> such that <math>\widetilde{\mathbf{z}}^i(t_n)=\mathbf{z}_n.</math> Here,<div style="text-align: center;">
<math>\mathbf{A}_n=\mathbf{f}_x(t_n,\mathbf{z}_n,\widetilde{\mathbf{z}}_{t_n}^1(-\tau_1),\ldots,\widetilde{\mathbf{z}}_{t_n}^m(-\tau_d)),
\text{ }\mathbf{B}_n^i=\mathbf{f}_{x_t(-\tau_i)}(t_n,\mathbf{z}_n,\widetilde{\mathbf{z}}_{t_n}^1(-\tau_1),\ldots,\widetilde{\mathbf{z}}_{t_n}^m(-\tau_d)) </math>
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[[Category:Numerical analysis]]
[[Category:Numerical integration
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