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{{User sandbox|Lilian|Local Linearization Method}}
Consider the d-dimensional [[Ordinary differential equation|Ordinary Differential Equation]] (ODE).
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}h_{n}\sum_{j=1}^{i-1}a_{ij}\mathbf{k}_{j}), </math>
which is obtained by solving (5) via a s-stage [[Runge–Kutta methods|RK scheme]] with coefficients <math>\mathbf{c}=\left[ c_{i}\right] , \mathbf{A}=\left[ a_{ij}\right] \quad and \quad \mathbf{b}=\left[ b_{j}\right]</math>
Local Linear Taylor discretization
<small><math display="inline">\mathbf{z}_{n+1}=\mathbf{z}_{n}+\mathbf{\phi }(t_{n},\mathbf{z}%
_{n};h_{n})+\int_{0}^{h_{n}}e^{\left( h_{n}-s\right) \mathbf{f}_{\mathbf{x}%
}\left( t_{n},\mathbf{z}_{n}\right) }\sum_{j=2}^{p}\frac{\mathbf{c}_{n,j}}{j!%
}s^{j}ds,\text{ with }\mathbf{c}_{n,j}=\left( \frac{d^{j+1}\mathbf{x}\left(
t\right) }{dt^{j+1}}-\mathbf{f}_{\mathbf{x}}\left( t_{n},\mathbf{z}%
_{n}\right) \frac{d^{j}\mathbf{x}\left( t\right) }{dt^{j}}\right) \mid _{t=%
\mathbf{z}_{n}}, </math></small>
which results from the approximation of <math>\mathbf{g}_{n}</math>in (2) by its order-''p'' truncated [[Taylor series|Taylor expansion]].
''[[Exponential integrator|Exponential Rosembrock discretization]]'' (poner link) is obtained by approximating the integral (2) by [[Numerical integration|aquadrature rule]].
''Linealized Exponential Adams discretization''
<small><math display="inline">\mathbf{z}_{n+1}=\mathbf{z}_{n}+\mathbf{\phi }(t_{n},\mathbf{z}
_{n};h_{n})+h_{n}\sum_{j=1}^{p}\sum_{l=1}^{j}\frac{\gamma _{j+1}}{l}\nabla
^{l}\mathbf{g}_{n}(t_{n},\mathbf{z}_{n}),\quad with \quad \gamma
_{j+1}=(-1)^{j+1}\int\limits_{0}^{1}e^{(1-\theta )h_{n}\mathbf{f}_{\mathbf{x}
}\left( t_{n},\mathbf{z}_{n}\right) }\theta \left(
\begin{array}{c}
-\theta \\
j%
\end{array}%
\right) d\theta , </math></small>
which results from the interpolation of <math>\mathbf{g}_{n}</math>in (2) by a [[Hermite polynomials|Hermite polynomial]] of degree ''p'', where <math>\nabla ^{l}\mathbf{g}%
_{n}(t_{m},\mathbf{z}_{m})</math> denotes the ''l''-th backward di§erence of <math>\mathbf{g}_{n}(t_{m},\mathbf{z}_{m})</math>.
=== Local Linearization schemes ===
All numerical implementation <math>\mathbf{y}_{n}</math> of the LL (or of a HOLL) discretization <math>\mathbf{z}_{n}</math> involves approximations <math>\widetilde{\phi }_{j}</math> to
integrals �<math>\phi _{j}</math> of the form
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