Local linearization method: Difference between revisions

Revision as of 17:52, 26 June 2017

Revision as of 18:13, 26 June 2017

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<math>\phi _{j}(\mathbf{A},h)=\int\limits_{0}^{h}e^{(h-s)\mathbf{A}}s^{j-1}ds,\qquad j=1,2..., </math>

where A is an ''d <math>\times</math>� d'' matrix. Every numerical implementation <math>\mathbf{y}_{n}</math> of a Local Linear discretization <math>\mathbf{z}_{n}</math> of any �orderorder is generically called Local Linearization scheme.

==== Computing integrals involving matrix exponential ====

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\in \mathbb{R}^{(d+lr)\times (d+lr)},

</math>

\mathbf{L}=[\mathbf{I}$ $\mathbf{0}_{d\times l)}]$, $\mathbf{r}=[\mathbf{0}%

_{1\times (d+l-1)}$ $1]^{\intercal }$, and $\mathbf{v}_{i}=\mathbf{a}%

<math>\mathbf{L}=[\mathbf{I}$ $\mathbf{0}_{d\times l)}]$, $\mathbf{r}=[\mathbf{0}%

_{1\times (d+l-1)}$ $1\quad1]^{\intercal }$\quad, and $\quad \mathbf{v}_{i}=\mathbf{a}%

_{i}(i-1)!

</math>

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>\left\Vert 2^{-k}\mathbf{H}h\right\Vert \leq \frac{1}{2},

</math>