Approximaths

Nuttall’s Padé approximant

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Let f(z) = \sum_{k=0}^\infty C_k z^k be a power series. The denominator Q_{n-1}(z) of the Padé approximant P(n, n-1) is given by Nuttall’s compact form:

\displaystyle Q_{n-1}(z) = \frac{ \begin{vmatrix} C_0 & C_1 & C_2 & \cdots & C_{n-1} & C_n \\ C_1 & C_2 & C_3 & \cdots & C_n & C_{n+1} \\ C_2 & C_3 & C_4 & \cdots & C_{n+1} & C_{n+2} \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ C_{n-2} & C_{n-1} & C_n & \cdots & C_{2n-3} & C_{2n-2} \\ 1 & z & z^2 & \cdots & z^{n-2} & z^{n-1} \end{vmatrix} }{ \begin{vmatrix} C_0 & C_1 & C_2 & \cdots & C_{n-1} \\ C_1 & C_2 & C_3 & \cdots & C_n \\ C_2 & C_3 & C_4 & \cdots & C_{n+1} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ C_{n-1} & C_n & C_{n+1} & \cdots & C_{2n-1} \end{vmatrix} }

The numerator P_n(z) is obtained by satisfying the Padé approximation condition: f(z) Q_{n-1}(z) - P_n(z) = O(z^{2n}).

The compact form of Nuttall’s Padé approximant P(n, n-1) is particularly valuable in numerical analysis and theoretical physics for its efficiency in computing Padé approximants without explicitly solving large linear systems.

By expressing the denominator Q_{n-1}(z) as a ratio of determinants, it provides a direct and elegant method to capture the approximant’s poles, which is crucial for analyzing singularities of functions, especially in Stieltjes series or meromorphic functions.

This formulation simplifies calculations, facilitates the study of convergence properties, and connects Padé approximants to orthogonal polynomials, enabling applications in areas like quantum field theory and asymptotic analysis where rapid computation and singularity detection are essential.

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