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Diagonal Padé approximant to the exponential

Definition

For an integer $m \geq 0$ , put $a_{m, j} : = \frac{(2 m - j)! m!}{(2 m)! j! (m - j)!} (0 \leq j \leq m), p_{m} (z) : = \sum_{j = 0}^{m} a_{m, j} z^{j}, q_{m} (z) : = p_{m} (- z) .$ The diagonal Padé rational function of order $m$ for the scalar exponential The complex exponential by its power series is $r_{m} (z) : = \frac{p_{m} (z)}{q_{m} (z)}$ at every $z \in C$ for which $q_{m} (z) \neq 0$ . The pair is normalized by $p_{m} (0) = q_{m} (0) = 1$ . This definition does not assign a value at a zero of the denominator.

Depends on

The complex exponential by its power series

Used by

Dependency tree · two levels

9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Jan Zur, Matrix Functions, TU Berlin full lecture notes (standard reference, not scraped)
Awad H. Al-Mohy and Nicholas J. Higham, A New Scaling and Squaring Algorithm for the Matrix Exponential (standard reference, not scraped)