Padé Approximants
- is the “best” approximation of a function near a specific point by a rational function of a given order
Definition
Given a function 𝑓 and two integers 𝑚≥0 and 𝑛≥1, the Padé approximant of order [𝑚/𝑛] is the rational function:
which agrees with 𝑓(𝑥) to the highest possible order, which amounts to
- \begin{align} f(0) &= R(0), \\ f'(0) &= R'(0), \\ f''(0) &= R''(0), \\ &\mathrel{\;\vdots} \\ f^{(m+n)}(0) &= R^{(m+n)}(0). \end{align}