Padé Table - Notation

Notation

A function f(z) is represented by a formal power series:


f(z) = c_0 + c_1z + c_2z^2 + \cdots = \sum_{n=0}^\infty c_nz^n,

where c0 ≠ 0, by convention. The (m, n)th entry Rm, n in the Padé table for f(z) is then given by


R_{m,n}(z) = \frac{P_m(z)}{Q_n(z)} =
\frac{a_0 + a_1z + a_2z^2 + \cdots + a_mz^m}{b_0 + b_1z + b_2z^2 + \cdots + b_nz^n}

where Pm(z) and Qn(z) are polynomials of degrees not more than m and n, respectively. The coefficients {ai} and {bi} can always be found by considering the expression


Q_n(z) \left(c_0 + c_1z + c_2z^2 + \cdots + c_{m+n}z^{m+n}\right) = P_m(z)

and equating coefficients of like powers of z up through m + n. For the coefficients of powers m + 1 to m + n, the right hand side is 0 and the resulting system of linear equations contains a homogeneous system of n equations in the n + 1 unknowns bi, and so admits of infinitely many solutions each of which determines a possible Qn. Pm is then easily found by equating the first m coefficients of the equation above. However, it can be shown that, due to cancellation, the generated rational functions Rm, n are all the same, so that the (m, n)th entry in the Padé table is unique. Alternatively, we may require that b0 = 1, thus putting the table in a standard form.

Although the entries in the Padé table can always be generated by solving this system of equations, that approach is computationally expensive. More efficient methods have been devised, including the epsilon algorithm.

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