SHOGUN  v3.0.0
RationalApproximation.h File Reference

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## Classes

class  CRationalApproximation
Abstract base class of the rational approximation of a function of a linear operator (A) times vector (v) using Cauchy's integral formula -

$f(\text{A})\text{v}=\oint_{\Gamma}f(z)(z\text{I}-\text{A})^{-1} \text{v}dz$

Computes eigenvalues of linear operator and uses Jacobi elliptic functions and conformal maps [2] for quadrature rule for discretizing the contour integral and computes complex shifts, weights and constant multiplier of the rational approximation of the above expression as

$f(\text{A})\text{v}\approx \eta\text{A}\Im-\left(\sum_{l=1}^{N}\alpha_{l} (\text{A}-\sigma_{l}\text{I})^{-1}\text{v}\right)$

where $$\alpha_{l},\sigma_{l}\in\mathbb{C}$$ are respectively the shifts and weights of the linear systems generated from the rational approximation, and $$\eta\in\mathbb{R}$$ is the constant multiplier, equals to $$\frac{-8K(\lambda_{m}\lambda_{M})^{\frac{1}{4}}}{k\pi N}$$. More...

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