Lau1DCubicSpline Class Reference

Class for defining a 1D cubic spline based on a set of knots. More...

#include <Lau1DCubicSpline.hh>

Public Types
enum	LauSplineType { StandardSpline, AkimaSpline, LinearInterpolation }
	Define the allowed interpolation types. More...
enum	LauSplineBoundaryType { Clamped, Natural, NotAKnot }
	Define the allowed boundary condition types. More...

Public Member Functions
	Lau1DCubicSpline (const std::vector< Double_t > &xs, const std::vector< Double_t > &ys, LauSplineType type=Lau1DCubicSpline::StandardSpline, LauSplineBoundaryType leftBound=Lau1DCubicSpline::NotAKnot, LauSplineBoundaryType rightBound=Lau1DCubicSpline::NotAKnot, Double_t dydx0=0.0, Double_t dydxn=0.0)
	Constructor. More...
virtual	~Lau1DCubicSpline ()
	Destructor. More...
Double_t	evaluate (Double_t x) const
	Evaluate the function at given point. More...
void	updateYValues (const std::vector< Double_t > &ys)
	Update the y-values of the knots. More...
void	updateType (LauSplineType type)
	Update the type of interpolation to perform. More...
void	updateBoundaryConditions (LauSplineBoundaryType leftBound, LauSplineBoundaryType rightBound, Double_t dydx0=0.0, Double_t dydxn=0.0)
	Update the boundary conditions for the spline. More...

Private Member Functions
	Lau1DCubicSpline (const Lau1DCubicSpline &rhs)
	Copy constructor - not implemented. More...
Lau1DCubicSpline &	operator= (const Lau1DCubicSpline &rhs)
	Copy assignment operator - not implemented. More...
void	init ()
	Initialise the class. More...
void	calcDerivatives ()
	Calculate the first derivative at each knot. More...
void	calcDerivativesStandard ()
	Calculate the first derivatives according to the standard method. More...
void	calcDerivativesAkima ()
	Calculate the first derivatives according to the Akima method. More...
	ClassDef (Lau1DCubicSpline, 0)

Private Attributes
const UInt_t	nKnots_
	The number of knots in the spline. More...
std::vector< Double_t >	x_
	The x-value at each knot. More...
std::vector< Double_t >	y_
	The y-value at each knot. More...
std::vector< Double_t >	dydx_
	The first derivative at each knot. More...
std::vector< Double_t >	a_
	The 'a' coefficients used to determine the derivatives. More...
std::vector< Double_t >	b_
	The 'b' coefficients used to determine the derivatives. More...
std::vector< Double_t >	c_
	The 'c' coefficients used to determine the derivatives. More...
std::vector< Double_t >	d_
	The 'd' coefficients used to determine the derivatives. More...
LauSplineType	type_
	The type of interpolation to be performed. More...
LauSplineBoundaryType	leftBound_
	The left-hand boundary condition on the spline. More...
LauSplineBoundaryType	rightBound_
	The right-hand boundary condition on the spline. More...
Double_t	dydx0_
	The gradient at the left boundary for a clamped spline. More...
Double_t	dydxn_
	The gradient at the right boundary for a clamped spline. More...

Detailed Description

Class for defining a 1D cubic spline based on a set of knots.

Class for defining a 1D cubic spline based on a set of knots. Interpolation between the knots is performed either by one of two types of cubic spline (standard or Akima) or by linear interpolation. The splines are defined by a piecewise cubic function which, between knots i and i+1, has the form

f_i(x) = (1 - t)*y_i + t*y_i+1 + t*(1 - t)(a*(1 - t) + b*t)

where t = (x - x_i)/(x_i+1 - x_i), a = k_i *(x_i+1 - x_i) - (y_i+1 - y_i), b = -k_i+1*(x_i+1 - x_i) + (y_i+1 - y_i)

and k_i is (by construction) the first derivative at knot i. f(x) and f'(x) are continuous at the internal knots by construction.

For the standard splines, f''(x) is required to be continuous at all internal knots placing n-2 constraints on the n parameters, k_i. The final two constraints are set by the boundary conditions. At each boundary, the function may be:

(i) Clamped : f'(x) = C at the last knot (ii) Natural : f''(x) = 0 at the last knot (iii) Not a knot : f'''(x) continuous at the second last knot

The algorithms used in these splines can be found on: http://en.wikipedia.org/wiki/Spline_interpolation#Algorithm_to_find_the_interpolating_cubic_spline http://en.wikipedia.org/wiki/Tridiagonal_matrix_algorithm

For the Akima splines, the values of k_i are determined from the slopes of the four nearest segments (a_i-1, a_i, a_i+1 and a_i+2) as

k_i = ( | a_i+2 - a_i+1 | a_i + | a_i - a_i-1 | a_i+1 ) / ( | a_i+2 - a_i+1 | + | a_i - a_i-1 | )

and as

k_i = ( a_i + a_i+1 ) / 2

in the special case a_i-1 == a_i != a_i+1 == a_i+2.

Boundary conditions are specified by the relations a_2 - a_1 = a_1 - a_0 = a_0 - a_-1 and a_n-1 - a_n-2 = a_n - a_n-1 = a_n+1 - a_n

The algorithms used in these splines can be found in: J.ACM vol. 17 no. 4 pp 589-602

Definition at line 84 of file Lau1DCubicSpline.hh.

Member Enumeration Documentation

enum Lau1DCubicSpline::LauSplineBoundaryType

Define the allowed boundary condition types.

These are only supported by standard splines

Enumerator
Clamped	clamped boundary - f'(x) = C
Natural	natural boundary - f''(x) = 0
NotAKnot	'not a knot' boundary - f'''(x) continuous at second last knot

Definition at line 98 of file Lau1DCubicSpline.hh.

enum Lau1DCubicSpline::LauSplineType

Define the allowed interpolation types.

Enumerator
StandardSpline	standard cubic splines with f''(x) continuous at all internal knots
AkimaSpline	Akima cubic splines with f'(x) at each knot defined locally by the positions of only five knots
LinearInterpolation	Linear interpolation between each pair of knots

Definition at line 88 of file Lau1DCubicSpline.hh.

Constructor & Destructor Documentation

Lau1DCubicSpline::Lau1DCubicSpline	(	const std::vector< Double_t > &	xs,
		const std::vector< Double_t > &	ys,
		LauSplineType	type = Lau1DCubicSpline::StandardSpline,
		LauSplineBoundaryType	leftBound = Lau1DCubicSpline::NotAKnot,
		LauSplineBoundaryType	rightBound = Lau1DCubicSpline::NotAKnot,
		Double_t	dydx0 = 0.0,
		Double_t	dydxn = 0.0
	)

Constructor.

/param [in] xs the x-values of the knots /param [in] ys the y-values of the knots /param [in] leftBound the left-hand boundary condition /param [in] rightBound the right-hand boundary condition /param [in] dydx0 the gradient at the left-hand boundary of a clamped spline /param [in] dydxn the gradient at the right-hand boundary of a clamped spline

Definition at line 41 of file Lau1DCubicSpline.cc.

Lau1DCubicSpline::~Lau1DCubicSpline ( )

virtual

Destructor.

Definition at line 58 of file Lau1DCubicSpline.cc.

Lau1DCubicSpline::Lau1DCubicSpline ( const Lau1DCubicSpline &  rhs )

private

Copy constructor - not implemented.

Member Function Documentation

void Lau1DCubicSpline::calcDerivatives ( )

private

Calculate the first derivative at each knot.

Definition at line 164 of file Lau1DCubicSpline.cc.

void Lau1DCubicSpline::calcDerivativesAkima ( )

private

Calculate the first derivatives according to the Akima method.

Definition at line 284 of file Lau1DCubicSpline.cc.

void Lau1DCubicSpline::calcDerivativesStandard ( )

private

Calculate the first derivatives according to the standard method.

Definition at line 179 of file Lau1DCubicSpline.cc.

Lau1DCubicSpline::ClassDef ( Lau1DCubicSpline  , 0    )

private

Double_t Lau1DCubicSpline::evaluate

(

Double_t

x

)

const

Evaluate the function at given point.

Parameters

[in]

x

the x-coordinate

Returns

the value of the spline at x

Definition at line 62 of file Lau1DCubicSpline.cc.

void Lau1DCubicSpline::init ( )

private

Initialise the class.

Definition at line 146 of file Lau1DCubicSpline.cc.

Lau1DCubicSpline& Lau1DCubicSpline::operator= ( const Lau1DCubicSpline &  rhs )

private

Copy assignment operator - not implemented.

void Lau1DCubicSpline::updateBoundaryConditions	(	LauSplineBoundaryType	leftBound,
		LauSplineBoundaryType	rightBound,
		Double_t	dydx0 = 0.0,
		Double_t	dydxn = 0.0
	)

Update the boundary conditions for the spline.

/param [in] leftBound the left-hand boundary condition /param [in] rightBound the right-hand boundary condition /param [in] dydx0 the gradient at the left-hand boundary of a clamped spline /param [in] dydxn the gradient at the right-hand boundary of a clamped spline

Definition at line 119 of file Lau1DCubicSpline.cc.

void Lau1DCubicSpline::updateType

(

LauSplineType

type

)

Update the type of interpolation to perform.

Parameters

[in]

type

the type of interpolation

Definition at line 111 of file Lau1DCubicSpline.cc.

void Lau1DCubicSpline::updateYValues

(

const std::vector< Double_t > &

ys

)

Update the y-values of the knots.

Parameters

[in]

ys

the y-values of the knots

Definition at line 105 of file Lau1DCubicSpline.cc.

Member Data Documentation

std::vector<Double_t> Lau1DCubicSpline::a_

private

The 'a' coefficients used to determine the derivatives.

Definition at line 182 of file Lau1DCubicSpline.hh.

std::vector<Double_t> Lau1DCubicSpline::b_

private

The 'b' coefficients used to determine the derivatives.

Definition at line 184 of file Lau1DCubicSpline.hh.

std::vector<Double_t> Lau1DCubicSpline::c_

private

The 'c' coefficients used to determine the derivatives.

Definition at line 186 of file Lau1DCubicSpline.hh.

std::vector<Double_t> Lau1DCubicSpline::d_

private

The 'd' coefficients used to determine the derivatives.

Definition at line 188 of file Lau1DCubicSpline.hh.

Double_t Lau1DCubicSpline::dydx0_

private

The gradient at the left boundary for a clamped spline.

Definition at line 199 of file Lau1DCubicSpline.hh.

std::vector<Double_t> Lau1DCubicSpline::dydx_

private

The first derivative at each knot.

Definition at line 179 of file Lau1DCubicSpline.hh.

Double_t Lau1DCubicSpline::dydxn_

private

The gradient at the right boundary for a clamped spline.

Definition at line 201 of file Lau1DCubicSpline.hh.

LauSplineBoundaryType Lau1DCubicSpline::leftBound_

private

The left-hand boundary condition on the spline.

Definition at line 194 of file Lau1DCubicSpline.hh.

const UInt_t Lau1DCubicSpline::nKnots_

private

The number of knots in the spline.

Definition at line 172 of file Lau1DCubicSpline.hh.

LauSplineBoundaryType Lau1DCubicSpline::rightBound_

private

The right-hand boundary condition on the spline.

Definition at line 196 of file Lau1DCubicSpline.hh.

LauSplineType Lau1DCubicSpline::type_

private

The type of interpolation to be performed.

Definition at line 191 of file Lau1DCubicSpline.hh.

std::vector<Double_t> Lau1DCubicSpline::x_

private

The x-value at each knot.

Definition at line 175 of file Lau1DCubicSpline.hh.

std::vector<Double_t> Lau1DCubicSpline::y_

private

The y-value at each knot.

Definition at line 177 of file Lau1DCubicSpline.hh.

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

• inc/Lau1DCubicSpline.hh
• src/Lau1DCubicSpline.cc