B-spline environment. More...

#include <bspline.h>

Public Member Functions
	Bspline (int order, Real th, rArrayView cknots1, rArrayView rknots, rArrayView cknots2)
	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 (Real r) const
	Apply the ECS transformation. More...

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

Complex	clamp (Complex z, Real a, Real b) const
	Restrict value into a given range. 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...

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

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

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

cArrayView	t () const
	B-spline knot sequence. 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...

int	iR1 () const
	Index of knot between leading complex grid and real grid. More...

int	iR2 () const
	Index of knot between real grid and trailing complex grid. More...

Real	R1 () const
	Beginning of the real grid. More...

Real	R2 () const
	End of the real grid. More...

Real	Rmin () const
	Beginning of the grid (unrotated). More...

Real	Rmax () const
	End of the grid (unrotated). More...

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

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

rArray const &	cknots1 () const
	complex knots More...

rArray const &	cknots2 () const

std::size_t	hash () const
	Return (almost) unique identification for this B-spline object. More...

Static Public Member Functions
static cArray	zip (Bspline const &bx, Bspline const &by, const cArrayView coeff, const rArrayView xgrid, const rArrayView ygrid, Complex(Bspline::evalXBSpline)(int, int, int, Complex) const =&Bspline::bspline, Complex(Bspline::evalYBSpline)(int, int, int, Complex) const =&Bspline::bspline)

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::Bspline	(	int	order,
		Real	th,
		rArrayView	cknots1,
		rArrayView	rknots,
		rArrayView	cknots2
	)

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

Parameters

order	B-spline order.
th	ECS angle in radians.
cknots1	Leading complex-rotated knots.
rknots	Real knot array.
cknots2	Trailing complex-rotated knots.

Member Function Documentation

◆ B()

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

Evaluates single B-spline at several points at once. This results in increased speed, because some data need to be computed only once. Also, some operation have been reorganized to a form that allows SIMD auto-vectorization.

Note that all evaluation points must be in the interval bounded by 'iknot'-th and 'iknot+1'-th knot.

For parameters description see bspline.

◆ 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).

◆ cknots1()

rArray const& Bspline::cknots1 ( ) const

inline

◆ cknots2()

rArray const& Bspline::cknots2 ( ) const

inline

◆ clamp()

Complex Bspline::clamp	(	Complex	z,
		Real	a,
		Real	b
	)		const

Returns the value unchanged if it fits into the given range, or one of the bounds (the nearer one) if it doesn't. The bounds a and b are (unrotated) coordinates along the contour.

◆ dB()

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

For parameters description see dspline.

This routine simply calls dspline for every evaluation point.

◆ dspline()

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).

◆ ECStheta()

Real Bspline::ECStheta ( ) const

inline

◆ eval() [1/2]

Complex Bspline::eval	(	const cArrayView	coeff,
		Real	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.

◆ eval() [2/2]

Complex Bspline::eval	(	const cArrayView	coeff,
		Real	x,
		Real	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.

◆ hash()

std::size_t Bspline::hash ( ) const

The identifier is calculated as a simple hash of the data contained in the object.

◆ iR1()

int Bspline::iR1 ( ) const

inline

This is also the index of the first possibly purely real B-spline.

◆ iR2()

int Bspline::iR2 ( ) const

inline

This is the index of the trailing knot of the last possibly purely real B-spline. The index of that B-spline is iR2 - order - 1.

◆ knot()

int Bspline::knot ( Complex x ) const

Finds knot (interval \( \langle t_i,t_{i+1} \rangle \)) for 'x'.

Parameters

x	Complex coordinate.

◆ Nknot()

int Bspline::Nknot ( ) const

inline

◆ Nreknot()

int Bspline::Nreknot ( ) const

inline

◆ Nspline()

int Bspline::Nspline ( ) const

inline

◆ order()

int Bspline::order ( ) const

inline

◆ R1()

Real Bspline::R1 ( ) const

inline

◆ R2()

Real Bspline::R2 ( ) const

inline

◆ rknots()

rArray const& Bspline::rknots ( ) const

inline

◆ Rmax()

Real Bspline::Rmax ( ) const

inline

◆ Rmin()

Real Bspline::Rmin ( ) const

inline

◆ rotate()

Complex Bspline::rotate ( Real r ) const

inline

Parameters

r	Real coordinate.

◆ t() [1/2]

cArrayView Bspline::t ( ) const

inline

◆ t() [2/2]

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

inline

◆ unrotate()

Real Bspline::unrotate ( Complex z ) const

inline

Parameters

z	Complex coordinate.

◆ zip() [1/3]

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.

◆ zip() [2/3]

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.

◆ zip() [3/3]

cArray Bspline::zip	(	Bspline const &	bx,
		Bspline const &	by,
		const cArrayView	coeff,
		const rArrayView	xgrid,
		const rArrayView	ygrid,
		Complex(Bspline::*)(int, int, int, Complex) const	evalXBSpline = `&Bspline::bspline`,
		Complex(Bspline::*)(int, int, int, Complex) const	evalYBSpline = `&Bspline::bspline`
	)

static

Zip 2D expansion.

Evaluate 2D function given as a B-spline expansion over a carthesian product of two 1D grids. The user may supply the B-spline evaluation functions for the individual axes, thus allowing evaluation of partial derivatives besides the plain B-spline expansion.

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

src/bspline.h
src/bspline.cpp

Public Member Functions

Static Public Member Functions

Detailed Description

Constructor & Destructor Documentation

◆ Bspline()

Member Function Documentation

◆ B()

◆ bspline()

◆ cknots1()

◆ cknots2()

◆ clamp()

◆ dB()

◆ dspline()

◆ ECStheta()

◆ eval() [1/2]

◆ eval() [2/2]

◆ hash()

◆ iR1()

◆ iR2()

◆ knot()

◆ Nknot()

◆ Nreknot()

◆ Nspline()

◆ order()

◆ R1()

◆ R2()

◆ rknots()

◆ Rmax()

◆ Rmin()

◆ rotate()

◆ t() [1/2]

◆ t() [2/2]

◆ unrotate()

◆ zip() [1/3]

◆ zip() [2/3]

◆ zip() [3/3]