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\mathbf{z}_{n})\mathbf{u}-\mathbf{f}_{t}\left( t_{n},\mathbf{z}_{n}\right)
(s-t_{n})-\mathbf{f}\left( t_{n},\mathbf{z}_{n}\right) +\mathbf{f}_{\mathbf{x
}}(t_{n},\mathbf{z}_{n})\mathbf{z}_{n}</math>.
=== Local Linear discretization ===
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\propto h^{\alpha }).</math> The HOLL discretization (4) [[Rate of convergence|converges]] with order to the solution of nonlinear ODEs, but it match the solution of the linear ODEs.
HOLL discretizations can be derived in two ways: 1) by approximating the integral representation (2) of r; and 2) by using a numerical integrator for the
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<small><math>\mathbf{q}(t_{n},\mathbf{z}_{n};s\mathbf{,\xi })=\mathbf{f}(s,\mathbf{z}_{n}+
\mathbf{\phi }\left( t_{n},\mathbf{z}_{n};s-t_{n}\right) +\mathbf{\xi })-
\mathbf{f}_{\mathbf{x}}(t_{n},\mathbf{z}_{n})\mathbf{\phi }\left( t_{n},
\mathbf{z}_{n};s-t_{n}\right) -\mathbf{f}_{t}\left( t_{n},\mathbf{z}
_{n}\right) (s-t_{n})-\mathbf{f}\left( t_{n},\mathbf{z}_{n}\right) .</math></small>
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The resulting approximation is often called Locally Linearized discretization.
Known HOLL discretizations are the following
* ''Locally Linearized Runge Kutta discretization''▼
▲Locally Linearized Runge Kutta discretization
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<math>\mathbf{z}_{n+1}=\mathbf{z}_{n}+\mathbf{\phi }(t_{n},\mathbf{z}
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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''▼
▲Local Linear Taylor discretization
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<small><math display="block">\mathbf{z}_{n+1}=\mathbf{z}_{n}+\mathbf{\phi }(t_{n},\mathbf{z}
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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''▼
▲''[[Exponential integrator|Exponential Rosembrock discretization]]'' (poner link) is obtained by approximating the integral (2) by [[Numerical integration|aquadrature rule]].
▲''Linealized Exponential Adams discretization''
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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
=== 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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==== Computing integrals involving matrix exponential ====
Among a number of algorithms to compute the integrals <math>\phi _{j}</math>, those based on rational Padé and Krylov subspaces approximations for exponential matrix are preferred. For this, a central role is playing by the expression
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If <math>\mathbf{P}_{p,q}(2^{-k}\mathbf{H}h)
</math> denotes the (p; q)-[[Padé approximant|Padé approximation]] of <math>e^{2^{-k}\mathbf{H}h}
</math> and ''k'' is the smallest integer number such that <math>
</math>
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