Spline for two sampling points. More...

#include <dumux/common/spline.hh>

Inheritance diagram for Dumux::Spline< Scalar, 2 >:

Description

template<class Scalar>
class Dumux::Spline< Scalar, 2 >

Spline for two sampling points.

For this type of spline there is no natural spline.

Public Member Functions
	Spline ()

template<class ScalarArrayX , class ScalarArrayY >
	Spline (const ScalarArrayX &x, const ScalarArrayY &y, Scalar m0, Scalar m1)
	Convenience constructor for a full spline. More...

template<class PointArray >
	Spline (const PointArray &points, Scalar m0, Scalar m1)
	Convenience constructor for a full spline. More...

	Spline (Scalar x0, Scalar x1, Scalar y0, Scalar y1, Scalar m0, Scalar m1)
	Convenience constructor for a full spline. More...

int	numSamples () const
	Returns the number of sampling points. More...

void	set (Scalar x0, Scalar x1, Scalar y0, Scalar y1, Scalar m0, Scalar m1)
	Set the sampling points and the boundary slopes of the spline. More...

template<class ScalarContainer >
void	setXYArrays (int nSamples, const ScalarContainer &x, const ScalarContainer &y, Scalar m0, Scalar m1)
	Set the sampling points and the boundary slopes of the spline. More...

template<class ScalarContainerX , class ScalarContainerY >
void	setXYContainers (const ScalarContainerX &x, const ScalarContainerY &y, Scalar m0, Scalar m1)
	Set the sampling points and the boundary slopes of the spline. More...

template<class PointArray >
void	setArrayOfPoints (int nSamples, const PointArray &points, Scalar m0, Scalar m1)
	Set the sampling points and the boundary slopes of the spline. More...

template<class PointContainer >
void	setContainerOfPoints (const PointContainer &points, Scalar m0, Scalar m1)
	Set the sampling points and the boundary slopes from an STL-like container of points. More...

template<class TupleContainer >
void	setContainerOfTuples (const TupleContainer &tuples, Scalar m0, Scalar m1)
	Set the sampling points and the boundary slopes from an STL-like container of tuples. More...

bool	applies (Scalar x) const
	Return true if the given x is in range [x1, xn]. More...

Scalar	xMin () const
	Return the x value of the leftmost sampling point. More...

Scalar	xMax () const
	Return the x value of the rightmost sampling point. More...

void	printCSV (Scalar xi0, Scalar xi1, int k) const
	Prints k tuples of the format (x, y, dx/dy, isMonotonic) to stdout. More...

Scalar	eval (Scalar x, bool extrapolate=false) const
	Evaluate the spline at a given position. More...

Scalar	evalDerivative (Scalar x, bool extrapolate=false) const
	Evaluate the spline's derivative at a given position. More...

Scalar	intersect (Scalar a, Scalar b, Scalar c, Scalar d) const
	Find the intersections of the spline with a cubic polynomial in the whole intervall, throws Dune::MathError exception if there is more or less than one solution. More...

Scalar	intersectInterval (Scalar x0, Scalar x1, Scalar a, Scalar b, Scalar c, Scalar d) const
	Find the intersections of the spline with a cubic polynomial in a sub-intervall of the spline, throws Dune::MathError exception if there is more or less than one solution. More...

int	monotonic (Scalar x0, Scalar x1) const
	Returns 1 if the spline is monotonically increasing, -1 if the spline is mononously decreasing and 0 if the spline is not monotonous in the interval (x0, x1). More...

int	monotonic () const
	Same as monotonic(x0, x1), but with the entire range of the spline as interval. More...

Protected Member Functions
void	assignXY_ (Scalar x0, Scalar x1, Scalar y0, Scalar y1)

Scalar	x_ (int i) const
	Returns the x coordinate of the i-th sampling point. More...

Scalar	y_ (int i) const
	Returns the y coordinate of the i-th sampling point. More...

Scalar	moment_ (int i) const
	Returns the moment (i.e. second derivative) of the spline at the i-th sampling point. More...

void	assignSamplingPoints_ (DestVector &destX, DestVector &destY, const SourceVector &srcX, const SourceVector &srcY, int numSamples)
	Set the sampling point vectors. More...

void	assignFromArrayList_ (DestVector &destX, DestVector &destY, const ListIterator &srcBegin, const ListIterator &srcEnd, int numSamples)

void	assignFromTupleList_ (DestVector &destX, DestVector &destY, ListIterator srcBegin, ListIterator srcEnd, int numSamples)
	Set the sampling points. More...

void	makePeriodicSystem_ (Matrix &M, Vector &d)
	Make the linear system of equations Mx = d which results in the moments of the periodic spline. More...

void	makeFullSystem_ (Matrix &M, Vector &d, Scalar m0, Scalar m1)
	Make the linear system of equations Mx = d which results in the moments of the full spline. More...

void	makeNaturalSystem_ (Matrix &M, Vector &d)
	Make the linear system of equations Mx = d which results in the moments of the natural spline. Stoer 2005: Numerische Mathematik 1, p. 111 [64]. More...

Scalar	eval_ (Scalar x, int i) const

Scalar	evalDerivative_ (Scalar x, int i) const

int	monotonic_ (int i, Scalar x0, Scalar x1) const

int	intersectSegment_ (Scalar *sol, int segIdx, Scalar a, Scalar b, Scalar c, Scalar d, Scalar x0=-1e100, Scalar x1=1e100) const
	Find all the intersections of a segment of the spline with a cubic polynomial within a specified interval. More...

int	segmentIdx_ (Scalar x) const

Scalar	h_ (int i) const
	Returns x[i] - x[i - 1]. More...

Scalar	a_ (int i) const

Scalar	b_ (int i) const

Scalar	c_ (int i) const

Scalar	d_ (int i) const

int	numSamples_ () const
	Returns the number of sampling points. More...

Protected Attributes
Vector	xPos_

Vector	yPos_

Vector	m_

Constructor & Destructor Documentation

◆ Spline() [1/4]

template<class Scalar >

Dumux::Spline< Scalar, 2 >::Spline ( )

inline

◆ Spline() [2/4]

template<class Scalar >

template<class ScalarArrayX , class ScalarArrayY >

Dumux::Spline< Scalar, 2 >::Spline	(	const ScalarArrayX &	x,
		const ScalarArrayY &	y,
		Scalar	m0,
		Scalar	m1
	)

inline

Convenience constructor for a full spline.

Parameters

x	An array containing the \(x\) values of the spline's sampling points
y	An array containing the \(y\) values of the spline's sampling points
m0	The slope of the spline at \(x_0\)
m1	The slope of the spline at \(x_n\)

◆ Spline() [3/4]

template<class Scalar >

template<class PointArray >

Dumux::Spline< Scalar, 2 >::Spline	(	const PointArray &	points,
		Scalar	m0,
		Scalar	m1
	)

inline

Convenience constructor for a full spline.

Parameters

points	An array of \((x,y)\) tuples of the spline's sampling points
m0	The slope of the spline at \(x_0\)
m1	The slope of the spline at \(x_n\)

◆ Spline() [4/4]

template<class Scalar >

Dumux::Spline< Scalar, 2 >::Spline	(	Scalar	x0,
		Scalar	x1,
		Scalar	y0,
		Scalar	y1,
		Scalar	m0,
		Scalar	m1
	)

inline

Convenience constructor for a full spline.

Parameters

x0	The \(x\) value of the first sampling point
x1	The \(x\) value of the second sampling point
y0	The \(y\) value of the first sampling point
y1	The \(y\) value of the second sampling point
m0	The slope of the spline at \(x_0\)
m1	The slope of the spline at \(x_n\)

Member Function Documentation

◆ a_()

Scalar Dumux::SplineCommon_< Scalar , Spline< Scalar, 2 > >::a_ ( int i ) const

inlineprotectedinherited

◆ applies()

bool Dumux::SplineCommon_< Scalar , Spline< Scalar, 2 > >::applies ( Scalar x ) const

inlineinherited

Return true if the given x is in range [x1, xn].

◆ assignFromArrayList_()

void Dumux::SplineCommon_< Scalar , Spline< Scalar, 2 > >::assignFromArrayList_	(	DestVector &	destX,
		DestVector &	destY,
		const ListIterator &	srcBegin,
		const ListIterator &	srcEnd,
		int	numSamples
	)

inlineprotectedinherited

◆ assignFromTupleList_()

void Dumux::SplineCommon_< Scalar , Spline< Scalar, 2 > >::assignFromTupleList_	(	DestVector &	destX,
		DestVector &	destY,
		ListIterator	srcBegin,
		ListIterator	srcEnd,
		int	numSamples
	)

inlineprotectedinherited

Set the sampling points.

Here we assume that the elements of the source vector have an [] operator where v[0] is the x value and v[1] is the y value if the sampling point.

◆ assignSamplingPoints_()

void Dumux::SplineCommon_< Scalar , Spline< Scalar, 2 > >::assignSamplingPoints_	(	DestVector &	destX,
		DestVector &	destY,
		const SourceVector &	srcX,
		const SourceVector &	srcY,
		int	numSamples
	)

inlineprotectedinherited

Set the sampling point vectors.

This takes care that the order of the x-values is ascending, although the input must be ordered already!

◆ assignXY_()

template<class Scalar >

void Dumux::Spline< Scalar, 2 >::assignXY_	(	Scalar	x0,
		Scalar	x1,
		Scalar	y0,
		Scalar	y1
	)

inlineprotected

◆ b_()

Scalar Dumux::SplineCommon_< Scalar , Spline< Scalar, 2 > >::b_ ( int i ) const

inlineprotectedinherited

◆ c_()

Scalar Dumux::SplineCommon_< Scalar , Spline< Scalar, 2 > >::c_ ( int i ) const

inlineprotectedinherited

◆ d_()

Scalar Dumux::SplineCommon_< Scalar , Spline< Scalar, 2 > >::d_ ( int i ) const

inlineprotectedinherited

◆ eval()

Scalar Dumux::SplineCommon_< Scalar , Spline< Scalar, 2 > >::eval	(	Scalar	x,
		bool	extrapolate = `false`
	)		const

inlineinherited

Evaluate the spline at a given position.

Parameters

x	The value on the abscissa where the spline ought to be evaluated
extrapolate	If this parameter is set to true, the spline will be extended beyond its range by straight lines, if false calling extrapolate for \( x \not [x_{min}, x_{max}]\) will cause a failed assertation.

◆ eval_()

Scalar Dumux::SplineCommon_< Scalar , Spline< Scalar, 2 > >::eval_	(	Scalar	x,
		int	i
	)		const

inlineprotectedinherited

◆ evalDerivative()

Scalar Dumux::SplineCommon_< Scalar , Spline< Scalar, 2 > >::evalDerivative	(	Scalar	x,
		bool	extrapolate = `false`
	)		const

inlineinherited

Evaluate the spline's derivative at a given position.

Parameters

x	The value on the abscissa where the spline's derivative ought to be evaluated
extrapolate	If this parameter is set to true, the spline will be extended beyond its range by straight lines, if false calling extrapolate for \( x \not [x_{min}, x_{max}]\) will cause a failed assertation.

◆ evalDerivative_()

Scalar Dumux::SplineCommon_< Scalar , Spline< Scalar, 2 > >::evalDerivative_	(	Scalar	x,
		int	i
	)		const

inlineprotectedinherited

◆ h_()

Scalar Dumux::SplineCommon_< Scalar , Spline< Scalar, 2 > >::h_ ( int i ) const

inlineprotectedinherited

Returns x[i] - x[i - 1].

◆ intersect()

Scalar Dumux::SplineCommon_< Scalar , Spline< Scalar, 2 > >::intersect	(	Scalar	a,
		Scalar	b,
		Scalar	c,
		Scalar	d
	)		const

inlineinherited

Find the intersections of the spline with a cubic polynomial in the whole intervall, throws Dune::MathError exception if there is more or less than one solution.

◆ intersectInterval()

Scalar Dumux::SplineCommon_< Scalar , Spline< Scalar, 2 > >::intersectInterval	(	Scalar	x0,
		Scalar	x1,
		Scalar	a,
		Scalar	b,
		Scalar	c,
		Scalar	d
	)		const

inlineinherited

Find the intersections of the spline with a cubic polynomial in a sub-intervall of the spline, throws Dune::MathError exception if there is more or less than one solution.

◆ intersectSegment_()

int Dumux::SplineCommon_< Scalar , Spline< Scalar, 2 > >::intersectSegment_	(	Scalar *	sol,
		int	segIdx,
		Scalar	a,
		Scalar	b,
		Scalar	c,
		Scalar	d,
		Scalar	x0 = `-1e100`,
		Scalar	x1 = `1e100`
	)		const

inlineprotectedinherited

Find all the intersections of a segment of the spline with a cubic polynomial within a specified interval.

◆ makeFullSystem_()

void Dumux::SplineCommon_< Scalar , Spline< Scalar, 2 > >::makeFullSystem_	(	Matrix &	M,
		Vector &	d,
		Scalar	m0,
		Scalar	m1
	)

inlineprotectedinherited

Make the linear system of equations Mx = d which results in the moments of the full spline.

◆ makeNaturalSystem_()

void Dumux::SplineCommon_< Scalar , Spline< Scalar, 2 > >::makeNaturalSystem_	(	Matrix &	M,
		Vector &	d
	)

inlineprotectedinherited

Make the linear system of equations Mx = d which results in the moments of the natural spline. Stoer 2005: Numerische Mathematik 1, p. 111 [64].

◆ makePeriodicSystem_()

void Dumux::SplineCommon_< Scalar , Spline< Scalar, 2 > >::makePeriodicSystem_	(	Matrix &	M,
		Vector &	d
	)

inlineprotectedinherited

Make the linear system of equations Mx = d which results in the moments of the periodic spline.

When solving Mx = d, it should be noted that x[0] is trash and needs to be set to x[n-1]

◆ moment_()

template<class Scalar >

Scalar Dumux::Spline< Scalar, 2 >::moment_ ( int i ) const

inlineprotected

Returns the moment (i.e. second derivative) of the spline at the i-th sampling point.

◆ monotonic() [1/2]

int Dumux::SplineCommon_< Scalar , Spline< Scalar, 2 > >::monotonic ( ) const

inlineinherited

Same as monotonic(x0, x1), but with the entire range of the spline as interval.

◆ monotonic() [2/2]

int Dumux::SplineCommon_< Scalar , Spline< Scalar, 2 > >::monotonic	(	Scalar	x0,
		Scalar	x1
	)		const

inlineinherited

Returns 1 if the spline is monotonically increasing, -1 if the spline is mononously decreasing and 0 if the spline is not monotonous in the interval (x0, x1).

In the corner case where the whole spline is flat, it returns 2.

◆ monotonic_()

int Dumux::SplineCommon_< Scalar , Spline< Scalar, 2 > >::monotonic_	(	int	i,
		Scalar	x0,
		Scalar	x1
	)		const

inlineprotectedinherited

◆ numSamples()

template<class Scalar >

int Dumux::Spline< Scalar, 2 >::numSamples ( ) const

inline

Returns the number of sampling points.

◆ numSamples_()

int Dumux::SplineCommon_< Scalar , Spline< Scalar, 2 > >::numSamples_ ( ) const

inlineprotectedinherited

Returns the number of sampling points.

◆ printCSV()

void Dumux::SplineCommon_< Scalar , Spline< Scalar, 2 > >::printCSV	(	Scalar	xi0,
		Scalar	xi1,
		int	k
	)		const

inlineinherited

Prints k tuples of the format (x, y, dx/dy, isMonotonic) to stdout.

If the spline does not apply for parts of [x0, x1] it is extrapolated using a straight line. The result can be inspected using the following commands:

--------— snip --------— ./yourProgramm > spline.csv gnuplot

gnuplot> plot "spline.csv" using 1:2 w l ti "Curve", \ "spline.csv" using 1:3 w l ti "Derivative", \ "spline.csv" using 1:4 w p ti "Monotonic" --------— snap --------—

◆ segmentIdx_()

int Dumux::SplineCommon_< Scalar , Spline< Scalar, 2 > >::segmentIdx_ ( Scalar x ) const

inlineprotectedinherited

◆ set()

template<class Scalar >

void Dumux::Spline< Scalar, 2 >::set	(	Scalar	x0,
		Scalar	x1,
		Scalar	y0,
		Scalar	y1,
		Scalar	m0,
		Scalar	m1
	)

inline

Set the sampling points and the boundary slopes of the spline.

Parameters

x0	The \(x\) value of the first sampling point
x1	The \(x\) value of the second sampling point
y0	The \(y\) value of the first sampling point
y1	The \(y\) value of the second sampling point
m0	The slope of the spline at \(x_0\)
m1	The slope of the spline at \(x_1\)

◆ setArrayOfPoints()

template<class Scalar >

template<class PointArray >

void Dumux::Spline< Scalar, 2 >::setArrayOfPoints	(	int	nSamples,
		const PointArray &	points,
		Scalar	m0,
		Scalar	m1
	)

inline

Set the sampling points and the boundary slopes of the spline.

Parameters

nSamples	The number of sampling points (must be >= 2)
points	An array of \((x,y)\) tuples of the spline's sampling points
m0	The slope of the spline at \(x_0\)
m1	The slope of the spline at \(x_1\)

◆ setContainerOfPoints()

template<class Scalar >

template<class PointContainer >

void Dumux::Spline< Scalar, 2 >::setContainerOfPoints	(	const PointContainer &	points,
		Scalar	m0,
		Scalar	m1
	)

inline

Set the sampling points and the boundary slopes from an STL-like container of points.

Parameters

points	An array of \((x,y)\) tuples of the spline's sampling points
m0	The slope of the spline at \(x_0\)
m1	The slope of the spline at \(x_1\)

◆ setContainerOfTuples()

template<class Scalar >

template<class TupleContainer >

void Dumux::Spline< Scalar, 2 >::setContainerOfTuples	(	const TupleContainer &	tuples,
		Scalar	m0,
		Scalar	m1
	)

inline

Set the sampling points and the boundary slopes from an STL-like container of tuples.

Parameters

tuples	An array of \((x,y)\) tuples of the spline's sampling points
m0	The slope of the spline at \(x_0\)
m1	The slope of the spline at \(x_1\)

◆ setXYArrays()

template<class Scalar >

template<class ScalarContainer >

void Dumux::Spline< Scalar, 2 >::setXYArrays	(	int	nSamples,
		const ScalarContainer &	x,
		const ScalarContainer &	y,
		Scalar	m0,
		Scalar	m1
	)

inline

Set the sampling points and the boundary slopes of the spline.

Parameters

nSamples	The number of sampling points (must be >= 2)
x	An array containing the \(x\) values of the sampling points
y	An array containing the \(y\) values of the sampling points
m0	The slope of the spline at \(x_0\)
m1	The slope of the spline at \(x_1\)

◆ setXYContainers()

template<class Scalar >

template<class ScalarContainerX , class ScalarContainerY >

void Dumux::Spline< Scalar, 2 >::setXYContainers	(	const ScalarContainerX &	x,
		const ScalarContainerY &	y,
		Scalar	m0,
		Scalar	m1
	)

inline

Set the sampling points and the boundary slopes of the spline.

Parameters

x	An array containing the \(x\) values of the sampling points
y	An array containing the \(y\) values of the sampling points
m0	The slope of the spline at \(x_0\)
m1	The slope of the spline at \(x_1\)

◆ x_()

template<class Scalar >

Scalar Dumux::Spline< Scalar, 2 >::x_ ( int i ) const

inlineprotected

Returns the x coordinate of the i-th sampling point.

◆ xMax()

Scalar Dumux::SplineCommon_< Scalar , Spline< Scalar, 2 > >::xMax ( ) const

inlineinherited

Return the x value of the rightmost sampling point.

◆ xMin()

Scalar Dumux::SplineCommon_< Scalar , Spline< Scalar, 2 > >::xMin ( ) const

inlineinherited

Return the x value of the leftmost sampling point.

◆ y_()

template<class Scalar >

Scalar Dumux::Spline< Scalar, 2 >::y_ ( int i ) const

inlineprotected

Returns the y coordinate of the i-th sampling point.

Member Data Documentation

◆ m_

template<class Scalar >

Vector Dumux::Spline< Scalar, 2 >::m_

protected

◆ xPos_

template<class Scalar >

Vector Dumux::Spline< Scalar, 2 >::xPos_

protected

◆ yPos_

template<class Scalar >

Vector Dumux::Spline< Scalar, 2 >::yPos_

protected

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

spline.hh

Description

Public Member Functions

Protected Member Functions

Protected Attributes

Constructor & Destructor Documentation

◆ Spline() [1/4]

◆ Spline() [2/4]

◆ Spline() [3/4]

◆ Spline() [4/4]

Member Function Documentation

◆ a_()

◆ applies()

◆ assignFromArrayList_()

◆ assignFromTupleList_()

◆ assignSamplingPoints_()

◆ assignXY_()

◆ b_()

◆ c_()

◆ d_()

◆ eval()

◆ eval_()

◆ evalDerivative()

◆ evalDerivative_()

◆ h_()

◆ intersect()

◆ intersectInterval()

◆ intersectSegment_()

◆ makeFullSystem_()

◆ makeNaturalSystem_()

◆ makePeriodicSystem_()

◆ moment_()

◆ monotonic() [1/2]

◆ monotonic() [2/2]

◆ monotonic_()

◆ numSamples()

◆ numSamples_()

◆ printCSV()

◆ segmentIdx_()

◆ set()

◆ setArrayOfPoints()

◆ setContainerOfPoints()

◆ setContainerOfTuples()

◆ setXYArrays()

◆ setXYContainers()

◆ x_()

◆ xMax()

◆ xMin()

◆ y_()

Member Data Documentation

◆ m_

◆ xPos_

◆ yPos_