ex33.hpp File Reference

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Functions
void	RationalApproximation_AAA (const Vector &val, const Vector &pt, Array< double > &z, Array< double > &f, Vector &w, double tol, int max_order)
void	ComputePolesAndZeros (const Vector &z, const Vector &f, const Vector &w, Array< double > &poles, Array< double > &zeros, double &scale)
void	PartialFractionExpansion (double scale, Array< double > &poles, Array< double > &zeros, Array< double > &coeffs)
void	ComputePartialFractionApproximation (double &alpha, Array< double > &coeffs, Array< double > &poles, double lmax=1000., double tol=1e-10, int npoints=1000, int max_order=100)

Function Documentation

ComputePartialFractionApproximation()

void ComputePartialFractionApproximation	(	double &	alpha,
		Array< double > &	coeffs,
		Array< double > &	poles,
		double	lmax = 1000.,
		double	tol = 1e-10,
		int	npoints = 1000,
		int	max_order = 100
	)

ComputePartialFractionApproximation: compute a rational approximation (RA) in partial fraction form, e.g., f(z) ≈ Σ_i c_i / (z - p_i), from sampled values of the function f(z) = z^{-a}, 0 < a < 1.

Parameters

[in]	alpha	Exponent a in f(z) = z^-a
[in]	lmax,npoints	f(z) is uniformly sampled npoints times in the interval [ 0, lmax ]
[in]	tol	Relative tolerance
[in]	max_order	Maximum number of terms (order) of the RA
[out]	coeffs	Coefficients c_i
[out]	poles	Poles p_i

NOTES: When MFEM is not built with LAPACK support, only alpha = 0.33, 0.5, and 0.99 are possible. In this case, if alpha != 0.33 and alpha != 0.99, then alpha = 0.5 is used by default.

See pg. A1501 of Nakatsukasa et al. [1].

Definition at line 295 of file ex33.hpp.

ComputePolesAndZeros()

void ComputePolesAndZeros	(	const Vector &	z,
		const Vector &	f,
		const Vector &	w,
		Array< double > &	poles,
		Array< double > &	zeros,
		double &	scale
	)

ComputePolesAndZeros: compute the poles [out] and zeros [out] of the rational function f(z) = C p(z)/q(z) from its ration barycentric form.

Parameters

[in]	z	Support points in rational barycentric form
[in]	f	Data values at support points z
[in]	w	Weights in rational barycentric form
[out]	poles	Array of poles (roots of p(z))
[out]	zeros	Array of zeros (roots of q(z))
[out]	scale	Scaling constant in f(z) = C p(z)/q(z)

See pg. A1501 of Nakatsukasa et al. [1].

Definition at line 175 of file ex33.hpp.

PartialFractionExpansion()

void PartialFractionExpansion	(	double	scale,
		Array< double > &	poles,
		Array< double > &	zeros,
		Array< double > &	coeffs
	)

PartialFractionExpansion: compute the partial fraction expansion of the rational function f(z) = Σ_i c_i / (z - p_i) from its poles [in] and zeros [in].

Parameters

[in]	poles	Array of poles (same as p_i above)
[in]	zeros	Array of zeros
[in]	scale	Scaling constant
[out]	coeffs	Coefficients c_i

Definition at line 245 of file ex33.hpp.

RationalApproximation_AAA()

void RationalApproximation_AAA	(	const Vector &	val,
		const Vector &	pt,
		Array< double > &	z,
		Array< double > &	f,
		Vector &	w,
		double	tol,
		int	max_order
	)

RationalApproximation_AAA: compute the rational approximation (RA) of data val [in] at the set of points pt [in].

Parameters

[in]	val	Vector of data values
[in]	pt	Vector of sample points
[in]	tol	Relative tolerance
[in]	max_order	Maximum number of terms (order) of the RA
[out]	z	Support points of the RA in rational barycentric form
[out]	f	Data values at support points z
[out]	w	Weights of the RA in rational barycentric form

See pg. A1501 of Nakatsukasa et al. [1].

Definition at line 52 of file ex33.hpp.