B-spline environment. More...

#include <bspline.h>

Public Member Functions
	Bspline (int order, rArrayView const &rknots, double th, rArrayView const &cknots)
	Constructor. More...

Complex	bspline (int i, int iknot, int k, Complex r) const
	Evaluate B-spline. More...

Complex	dspline (int i, int iknot, int k, Complex r) const
	Evaluate derivative of a B-spline. More...

void	B (int i, int iknot, int n, const Complex x, Complex y) const
	B-spline. More...

void	dB (int i, int iknot, int n, const Complex x, Complex y) const
	Derivative of a B-spline. More...

Complex	rotate (double r) const
	Apply the ECS transformation. More...

double	unrotate (Complex z) const
	Apply the inverse ECS transformation. More...

cArray	zip (const cArrayView coeff, const rArrayView grid) const
	Zip 1D expansion. More...

cArray	zip (const cArrayView coeff, const rArrayView xgrid, const rArrayView ygrid) const
	Zip 2D expansion. More...

void	writeVTK (std::ofstream &out, const cArrayView coeff, const rArrayView xgrid, const rArrayView ygrid) const
	Zip 2D expansion to VTK. More...

int	knot (Complex x) const
	Get knot index for coordinate. More...

Complex	eval (const cArrayView coeff, double x) const
	Evaluate 1D B-spline expansion. More...

Complex	eval (const cArrayView coeff, double x, double y) const
	Evaluate 2D B-spline expansion. More...

Complex const &	t (int i) const
	B-spline knot sequence. More...

int	Nspline () const
	Number of B-splines. More...

int	Nknot () const
	Number of knots. More...

int	Nreknot () const
	Number of real knots. More...

int	order () const
	B-spline order. More...

double	R0 () const
	End of real grid. More...

double	Rmax () const
	End of complex grid (real, unrotated). More...

double	ECStheta () const
	ECS rotation angle. More...

rArray const &	rknots () const
	real knots More...

rArray const &	cknots () const
	complex knots More...

Detailed Description

This class is used to manage B-spline computations within the context of the exterior complex scaling. That is, it allows (upon construction) specification of the real and complex knot sequences

\[ t_1^{R}, t_2^{R}, ..., t_{\mathrm{Nreknot}}^{R} \ , \]

\[ t_{\mathrm{Nreknot}+1}^{C}, ..., t_{\mathrm{Nknot}}^{C} \ , \]

which then serve as the knot data of the B-spline set. See bspline and dspline on the definition formulae.

Constructor & Destructor Documentation

Bspline::Bspline	(	int	order,
		rArrayView const &	rknots,
		double	th,
		rArrayView const &	cknots
	)

Setup the knot sequence, which will consist of two parts.

Parameters

order	B-spline order.
rknots	Real knot array (usually including R₀).
th	ECS angle in radians.
cknots	To-be-complex knot array (including R₀ and Rmax).

Member Function Documentation

void Bspline::B	(	int	i,
		int	iknot,
		int	n,
		const Complex *	x,
		Complex *	y
	)		const

Vectorized interface to the bspline function. For parameters description see bspline.

Complex Bspline::bspline	(	int	i,
		int	iknot,
		int	k,
		Complex	r
	)		const

Function evaluates a B-spline in one point. The recursive Cox de Boor formula is used:

\[ B_i^{(0)}(x) = \cases { 1 & t_i \le x \le t_{i+1} \cr 0 & \mbox{otherwise} }\ , \]

\[ B_i^{(k)}(x) = \frac{x - t_i}{t_{i+k-1} - t_i} B_i^{(k-1)}(x) + \frac{t_{i+k} - x}{t_{i+k} - t_{i+1}} B_{i+1}^{(k-1)}(x) \ . \]

Parameters

i	Index of the B-spline.
iknot	Index of the left knot.
k	B-spline order.
r	Coordinate (independent variable).

rArray const& Bspline::cknots ( ) const

inline

void Bspline::dB	(	int	i,
		int	iknot,
		int	n,
		const Complex *	x,
		Complex *	y
	)		const

Vectorized interface to the dspline function. For parameters description see bspline.

Complex Bspline::dspline	(	int	i,
		int	iknot,
		int	k,
		Complex	r
	)		const

This function evaluated the derivative of a B-spline, using the formula

Parameters

i	Index of the B-spline.
iknot	Index of the left knot.
k	B-spline order.
r	Coordinate (independent variable).

double Bspline::ECStheta ( ) const

inline

Complex Bspline::eval	(	const cArrayView	coeff,
		double	x
	)		const

Evaluates a B-spline expansion in a single point x. This function is faster than calling

* Complex z = zip (coeff, rArray({x}));

*

though the results are the same.

Complex Bspline::eval	(	const cArrayView	coeff,
		double	x,
		double	y
	)		const

Evaluates a double B-spline expansion in a single point (x,y). This function is faster than calling

* Complex z = zip (coeff, rArray({x}), rArray({y}));

*

though the results are the same.

int Bspline::knot ( Complex x ) const

Finds knot (interval \( \left< t_i,t_{i+1} \right> \)) for 'x'.

Parameters

x	Complex coordinate.

int Bspline::Nknot ( ) const

inline

int Bspline::Nreknot ( ) const

inline

int Bspline::Nspline ( ) const

inline

int Bspline::order ( ) const

inline

double Bspline::R0 ( ) const

inline

rArray const& Bspline::rknots ( ) const

inline

double Bspline::Rmax ( ) const

inline

Complex Bspline::rotate ( double r ) const

inline

Parameters

r	Real coordinate.

Complex const& Bspline::t ( int i ) const

inline

double Bspline::unrotate ( Complex z ) const

inline

Parameters

z	Complex coordinate.

void Bspline::writeVTK	(	std::ofstream &	out,
		const cArrayView	coeff,
		const rArrayView	xgrid,
		const rArrayView	ygrid
	)		const

cArray Bspline::zip	(	const cArrayView	coeff,
		const rArrayView	grid
	)		const

Evaluate 1D function given as a B-spline expansion over a grid.

Parameters

coeff	Expansion coefficients.
grid	Evaluation points (unrotated). They are assumed to be sorted. The length of `coeff` must be at least equal to the spline count and it is these first `Nspline` coefficients that are used in evaluation.

cArray Bspline::zip	(	const cArrayView	coeff,
		const rArrayView	xgrid,
		const rArrayView	ygrid
	)		const

Evaluate 2D function given as a B-spline expansion over a carthesian product of two 1D grids.

Parameters

coeff	Expansion coefficients.
xgrid	Evaluation points at x-axis (unrotated). They are assumed to be sorted.
ygrid	Evaluation points at y-axis (unrotated). They are assumed to be sorted. The length of `coeff` must be at least equal to the spline count squared and it is these first `Nspline**2` coefficients that are used in evaluation.

The documentation for this class was generated from the following files:

src/bspline.h
src/bspline.cpp

Public Member Functions

Detailed Description

Constructor & Destructor Documentation

Member Function Documentation