SHOGUN  v3.0.0
 All Classes Namespaces Files Functions Variables Typedefs Enumerations Enumerator Friends Macros Groups Pages
RationalApproximation.h File Reference

Go to the source code of this file.


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...

SHOGUN Machine Learning Toolbox - Documentation