User class for calculating the derivatives of a function.

It can calculate first (method Derivative1), second (method Derivative2) and third (method Derivative3) of a function.

It uses the Richardson extrapolation method for function derivation in a given interval. The method use 2 derivative estimates (one computed with step h and one computed with step h/2) to compute a third, more accurate estimation. It is equivalent to the 5-point method, which can be obtained with a Taylor expansion. A step size should be given, depending on x and f(x). An optimal step size value minimizes the truncation error of the expansion and the rounding error in evaluating x+h and f(x+h). A too small h will yield a too large rounding error while a too large h will give a large truncation error in the derivative approximation. A good discussion can be found in discussed in Chapter 5.7 of Numerical Recipes in C. By default a value of 0.001 is uses, acceptable in many cases.

This class is implemented using code previously in TF1::Derivate{,2,3}(). Now TF1 uses this class.

Definition at line 55 of file RichardsonDerivator.h.

Public Member Functions
	RichardsonDerivator (double h=0.001)
	Default Constructor. More...

	RichardsonDerivator (const ROOT::Math::IGenFunction &f, double h=0.001, bool copyFunc=false)
	Construct from function and step size. More...

	RichardsonDerivator (const RichardsonDerivator &rhs)
	Copy constructor. More...

	~RichardsonDerivator ()
	Destructor: Removes function if needed. More...

double	Derivative1 (double x)
	Returns the first derivative of the function at point x, computed by Richardson's extrapolation method (use 2 derivative estimates to compute a third, more accurate estimation) first, derivatives with steps h and h/2 are computed by central difference formulas \[ D(h) = \frac{f(x+h) - f(x-h)}{2h} \] the final estimate \[ D = \frac{4D(h/2) - D(h)}{3} \] "Numerical Methods for Scientists and Engineers", H.M.Antia, 2nd edition". More...

double	Derivative1 (const IGenFunction &f, double x, double h)
	First Derivative calculation passing function object and step-size. More...

double	Derivative2 (double x)
	Returns the second derivative of the function at point x, computed by Richardson's extrapolation method (use 2 derivative estimates to compute a third, more accurate estimation) first, derivatives with steps h and h/2 are computed by central difference formulas \[ D(h) = \frac{f(x+h) - 2f(x) + f(x-h)}{h^{2}} \] the final estimate \[ D = \frac{4D(h/2) - D(h)}{3} \] "Numerical Methods for Scientists and Engineers", H.M.Antia, 2nd edition". More...

double	Derivative2 (const IGenFunction &f, double x, double h)
	Second Derivative calculation passing function and step-size. More...

double	Derivative3 (double x)
	Returns the third derivative of the function at point x, computed by Richardson's extrapolation method (use 2 derivative estimates to compute a third, more accurate estimation) first, derivatives with steps h and h/2 are computed by central difference formulas \[ D(h) = \frac{f(x+2h) - 2f(x+h) + 2f(x-h) - f(x-2h)}{2h^{3}} \] the final estimate \[ D = \frac{4D(h/2) - D(h)}{3} \] "Numerical Methods for Scientists and Engineers", H.M.Antia, 2nd edition". More...

double	Derivative3 (const IGenFunction &f, double x, double h)
	Third Derivative calculation passing function and step-size. More...

double	DerivativeBackward (double x)
	Computation of the first derivative using a backward formula. More...

double	DerivativeBackward (const IGenFunction &f, double x, double h)
	Computation of the first derivative using a forward formula. More...

double	DerivativeForward (double x)
	Computation of the first derivative using a forward formula. More...

double	DerivativeForward (const IGenFunction &f, double x, double h)
	Computation of the first derivative using a forward formula. More...

double	Error () const
	Returns the estimate of the absolute Error of the last derivative calculation. More...

double	operator() (double x)

RichardsonDerivator &	operator= (const RichardsonDerivator &rhs)
	Assignment operator. More...

void	SetFunction (const IGenFunction &f)
	Set function for derivative calculation (copy the function if option has been enabled in the constructor) More...

void	SetStepSize (double h)
	Set step size for derivative calculation. More...

Protected Attributes
const IGenFunction *	fFunction

bool	fFunctionCopied

double	fLastError

double	fStepSize

#include <Math/RichardsonDerivator.h>

Constructor & Destructor Documentation

◆ ~RichardsonDerivator()

ROOT::Math::RichardsonDerivator::~RichardsonDerivator ( )

Destructor: Removes function if needed.

Definition at line 42 of file RichardsonDerivator.cxx.

◆ RichardsonDerivator() [1/3]

ROOT::Math::RichardsonDerivator::RichardsonDerivator ( double h = 0.001 )

Default Constructor.

Give optionally the step size for derivation. By default is 0.001, which is fine for x ~ 1 Increase if x is in average larger or decrease if x is smaller

Definition at line 22 of file RichardsonDerivator.cxx.

◆ RichardsonDerivator() [2/3]

ROOT::Math::RichardsonDerivator::RichardsonDerivator	(	const ROOT::Math::IGenFunction &	f,
		double	h = `0.001`,
		bool	copyFunc = `false`
	)

Construct from function and step size.

Definition at line 30 of file RichardsonDerivator.cxx.

◆ RichardsonDerivator() [3/3]

ROOT::Math::RichardsonDerivator::RichardsonDerivator ( const RichardsonDerivator & rhs )

Copy constructor.

Definition at line 49 of file RichardsonDerivator.cxx.

Member Function Documentation

◆ Derivative1() [1/2]

double ROOT::Math::RichardsonDerivator::Derivative1 ( double x )

inline

Returns the first derivative of the function at point x, computed by Richardson's extrapolation method (use 2 derivative estimates to compute a third, more accurate estimation) first, derivatives with steps h and h/2 are computed by central difference formulas

\[ D(h) = \frac{f(x+h) - f(x-h)}{2h} \]

the final estimate

\[ D = \frac{4D(h/2) - D(h)}{3} \]

"Numerical Methods for Scientists and Engineers", H.M.Antia, 2nd edition".

the argument eps may be specified to control the step size (precision). the step size is taken as eps*(xmax-xmin). the default value (0.001) should be good enough for the vast majority of functions. Give a smaller value if your function has many changes of the second derivative in the function range.

Getting the error via TF1::DerivativeError: (total error = roundoff error + interpolation error) the estimate of the roundoff error is taken as follows:

\[ err = k\sqrt{f(x)^{2} + x^{2}deriv^{2}}\sqrt{\sum ai^{2}}, \]

where k is the double precision, ai are coefficients used in central difference formulas interpolation error is decreased by making the step size h smaller.

Definition at line 116 of file RichardsonDerivator.h.

◆ Derivative1() [2/2]

double ROOT::Math::RichardsonDerivator::Derivative1	(	const IGenFunction &	f,
		double	x,
		double	h
	)

First Derivative calculation passing function object and step-size.

Definition at line 80 of file RichardsonDerivator.cxx.

◆ Derivative2() [1/2]

double ROOT::Math::RichardsonDerivator::Derivative2 ( double x )

inline

Returns the second derivative of the function at point x, computed by Richardson's extrapolation method (use 2 derivative estimates to compute a third, more accurate estimation) first, derivatives with steps h and h/2 are computed by central difference formulas

\[ D(h) = \frac{f(x+h) - 2f(x) + f(x-h)}{h^{2}} \]

the final estimate

\[ D = \frac{4D(h/2) - D(h)}{3} \]

"Numerical Methods for Scientists and Engineers", H.M.Antia, 2nd edition".

the argument eps may be specified to control the step size (precision). the step size is taken as eps*(xmax-xmin). the default value (0.001) should be good enough for the vast majority of functions. Give a smaller value if your function has many changes of the second derivative in the function range.

Getting the error via TF1::DerivativeError: (total error = roundoff error + interpolation error) the estimate of the roundoff error is taken as follows:

\[ err = k\sqrt{f(x)^{2} + x^{2}deriv^{2}}\sqrt{\sum ai^{2}}, \]

where k is the double precision, ai are coefficients used in central difference formulas interpolation error is decreased by making the step size h smaller.

Definition at line 172 of file RichardsonDerivator.h.

◆ Derivative2() [2/2]

double ROOT::Math::RichardsonDerivator::Derivative2	(	const IGenFunction &	f,
		double	x,
		double	h
	)

Second Derivative calculation passing function and step-size.

Definition at line 157 of file RichardsonDerivator.cxx.

◆ Derivative3() [1/2]

double ROOT::Math::RichardsonDerivator::Derivative3 ( double x )

inline

Returns the third derivative of the function at point x, computed by Richardson's extrapolation method (use 2 derivative estimates to compute a third, more accurate estimation) first, derivatives with steps h and h/2 are computed by central difference formulas

\[ D(h) = \frac{f(x+2h) - 2f(x+h) + 2f(x-h) - f(x-2h)}{2h^{3}} \]

the final estimate

\[ D = \frac{4D(h/2) - D(h)}{3} \]

"Numerical Methods for Scientists and Engineers", H.M.Antia, 2nd edition".

the argument eps may be specified to control the step size (precision). the step size is taken as eps*(xmax-xmin). the default value (0.001) should be good enough for the vast majority of functions. Give a smaller value if your function has many changes of the second derivative in the function range.

Getting the error via TF1::DerivativeError: (total error = roundoff error + interpolation error) the estimate of the roundoff error is taken as follows:

\[ err = k\sqrt{f(x)^{2} + x^{2}deriv^{2}}\sqrt{\sum ai^{2}}, \]

where k is the double precision, ai are coefficients used in central difference formulas interpolation error is decreased by making the step size h smaller.

Definition at line 211 of file RichardsonDerivator.h.

◆ Derivative3() [2/2]

double ROOT::Math::RichardsonDerivator::Derivative3	(	const IGenFunction &	f,
		double	x,
		double	h
	)

Third Derivative calculation passing function and step-size.

Definition at line 178 of file RichardsonDerivator.cxx.

◆ DerivativeBackward() [1/2]

double ROOT::Math::RichardsonDerivator::DerivativeBackward ( double x )

inline

Computation of the first derivative using a backward formula.

Definition at line 133 of file RichardsonDerivator.h.

◆ DerivativeBackward() [2/2]

double ROOT::Math::RichardsonDerivator::DerivativeBackward	(	const IGenFunction &	f,
		double	x,
		double	h
	)

inline

Computation of the first derivative using a forward formula.

Definition at line 138 of file RichardsonDerivator.h.

◆ DerivativeForward() [1/2]

double ROOT::Math::RichardsonDerivator::DerivativeForward ( double x )

inline

Computation of the first derivative using a forward formula.

Definition at line 125 of file RichardsonDerivator.h.

◆ DerivativeForward() [2/2]

double ROOT::Math::RichardsonDerivator::DerivativeForward	(	const IGenFunction &	f,
		double	x,
		double	h
	)

Computation of the first derivative using a forward formula.

Definition at line 116 of file RichardsonDerivator.cxx.

◆ Error()

double ROOT::Math::RichardsonDerivator::Error ( ) const

inline

Returns the estimate of the absolute Error of the last derivative calculation.

Definition at line 83 of file RichardsonDerivator.h.

◆ operator()()

double ROOT::Math::RichardsonDerivator::operator() ( double x )

inline

Definition at line 117 of file RichardsonDerivator.h.

◆ operator=()

RichardsonDerivator & ROOT::Math::RichardsonDerivator::operator= ( const RichardsonDerivator & rhs )

Assignment operator.

Definition at line 59 of file RichardsonDerivator.cxx.

◆ SetFunction()

void ROOT::Math::RichardsonDerivator::SetFunction ( const IGenFunction & f )

Set function for derivative calculation (copy the function if option has been enabled in the constructor)

@param f Function to be differentiated

Definition at line 70 of file RichardsonDerivator.cxx.

◆ SetStepSize()

void ROOT::Math::RichardsonDerivator::SetStepSize ( double h )

inline

Set step size for derivative calculation.

@param h step size for calculation

Definition at line 230 of file RichardsonDerivator.h.

Member Data Documentation

◆ fFunction

const IGenFunction* ROOT::Math::RichardsonDerivator::fFunction

protected

Definition at line 237 of file RichardsonDerivator.h.

◆ fFunctionCopied

bool ROOT::Math::RichardsonDerivator::fFunctionCopied

protected

Definition at line 234 of file RichardsonDerivator.h.

◆ fLastError

double ROOT::Math::RichardsonDerivator::fLastError

protected

Definition at line 236 of file RichardsonDerivator.h.

◆ fStepSize

double ROOT::Math::RichardsonDerivator::fStepSize

protected

Definition at line 235 of file RichardsonDerivator.h.

Libraries for ROOT::Math::RichardsonDerivator:

[legend]

Public Member Functions

Protected Attributes

Constructor & Destructor Documentation

◆ ~RichardsonDerivator()

◆ RichardsonDerivator() [1/3]

◆ RichardsonDerivator() [2/3]

◆ RichardsonDerivator() [3/3]

Member Function Documentation

◆ Derivative1() [1/2]

◆ Derivative1() [2/2]

◆ Derivative2() [1/2]

◆ Derivative2() [2/2]

◆ Derivative3() [1/2]

◆ Derivative3() [2/2]

◆ DerivativeBackward() [1/2]

◆ DerivativeBackward() [2/2]

◆ DerivativeForward() [1/2]

◆ DerivativeForward() [2/2]

◆ Error()

◆ operator()()

◆ operator=()

◆ SetFunction()

◆ SetStepSize()

Member Data Documentation

◆ fFunction

◆ fFunctionCopied

◆ fLastError

◆ fStepSize