Local linearization method: Difference between revisions

Revision as of 18:31, 4 July 2017

Revision as of 18:54, 4 July 2017

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\mathbf{r}\right\vert \varpropto h^{p+q+1}.</math>

If <math>\mathbf{\mathbf{k}}_{m,k}^{p,q}(h,\mathbf{H},\mathbf{r})</math> � denotes the ''(m; p; q; k)'' [[Krylov subspace|Krylov-Padé approximation]] of <math>e^{h\mathbf{H}}\mathbf{r}</math>

<math>\left\vert \sum\nolimits_{i=1}^{l}\phi _{i}(\mathbf{A},h)\mathbf{a}_{i}-

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<math>\mathbf{L}=\left[

\begin{array}{ll}

\mathbf{I}_{d} & \mathbf{0}_{d\times 2}%

\end{array}%

\right]</math>and <math>\mathbf{r}^{\intercal }=\left[

\begin{array}{ll}

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==== Order 1.5 SLL schemes ====

<small><math>\mathbf{y}_{n+1} =\mathbf{y}_{n}+\mathbf{L}(\mathbf{P}_{p,q}(2^{-k_{n}}

\mathbf{M}_{n}h_{n}))^{2^{k_{n}}}\mathbf{r}

+\sum\limits_{i=1}^{m}\left( \mathbf{g}_{i}(t_{n})\Delta \mathbf{w}

_{n}^{i}+\mathbf{f}_{\mathbf{x}}(t_{n},\widetilde{\mathbf{y}}_{n})\mathbf{g}

_{i}(t_{n})\Delta \mathbf{z}_{n}^{i}+\frac{d\mathbf{g}_{i}(t_{n})}{dt}

(\Delta \mathbf{w}_{n}^{i}h_{n}-\Delta \mathbf{z}_{n}^{i})\right) , (16)</math></small>