Content deleted Content added
Lilianmm87 (talk | contribs) |
Lilianmm87 (talk | contribs) agregar nuevas fórmulas |
||
Line 155:
<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 �order is generically called Local Linearization scheme.
==== 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
<math>\sum\nolimits_{i=1}^{l}\phi _{i}(\mathbf{A},h)\mathbf{a}_{i}=\mathbf{L}e^{h
\mathbf{H}}\mathbf{r,} </math>
where <math>\mathbf{a}_{i}</math> are d-dimensional vectors,
<math>\mathbf{H}= \begin{bmatrix}
\mathbf{A} & \mathbf{v}_{l} & \mathbf{v}_{l-1} & \cdots & \mathbf{v}_{1} \\
\mathbf{0} & \mathbf{0} & 1 & \cdots & 0 \\
\mathbf{0} & \mathbf{0} & 0 & \ddots & 0 \\
\vdots & \vdots & \vdots & \ddots & 1 \\
\mathbf{0} & \mathbf{0} & 0 & \cdots & 0%
\end{bmatrix}
\in \mathbb{R}^{(d+lr)\times (d+lr)},
\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}%
_{i}(i-1)!
</math>
| |||