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RationalApproximation.h File Reference

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singleton  SGVector< T >
 shogun vector More...
singleton  CLinearOperator< RetType, OperandType >
 Abstract template base class that represents a linear operator, e.g. a matrix. More...
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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