RichardsonDerivator.h

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// @(#)root/mathcore:$Id$
// Authors: David Gonzalez Maline    01/2008
 
/**********************************************************************
 *                                                                    *
 * Copyright (c) 2006 , LCG ROOT MathLib Team                         *
 *                                                                    *
 *                                                                    *
 **********************************************************************/
 
// Header file for RichardsonDerivator
//
// Created by: David Gonzalez Maline  : Mon Feb 4 2008
//
 
#ifndef ROOT_Math_RichardsonDerivator
#define ROOT_Math_RichardsonDerivator
 
#include <Math/IFunction.h>
 
/**
    @defgroup Deriv Numerical Differentiation
    Classes for numerical differentiation
    @ingroup NumAlgo
*/
 
 
namespace ROOT {
namespace Math {
 
//___________________________________________________________________________________________
/**
   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
   <a href = http://en.wikipedia.org/wiki/Five-point_stencil>5-point method</a>,
   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
   <a href=http://www.nrbook.com/a/bookcpdf/c5-7.pdf>Chapter 5.7</a>  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.
 
   @ingroup Deriv
 
 */
 
class RichardsonDerivator {
public:
 
   /** Destructor: Removes function if needed. */
   ~RichardsonDerivator();
 
   /** 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
    */
   RichardsonDerivator(double h = 0.001);
 
   /** Construct from function and step size
    */
   RichardsonDerivator(const ROOT::Math::IGenFunction & f, double h = 0.001, bool copyFunc = false);
 
   /**
      Copy constructor
    */
   RichardsonDerivator(const RichardsonDerivator & rhs);
 
   /**
      Assignment operator
    */
   RichardsonDerivator & operator= ( const RichardsonDerivator & rhs);
 
 
   /** Returns the estimate of the absolute Error of the last derivative calculation. */
   double Error () const {  return fLastError; }
 
 
   /**
      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
     \f[
      D(h) = \frac{f(x+h) - f(x-h)}{2h}
     \f]
      the final estimate
     \f[
      D = \frac{4D(h/2) - D(h)}{3}
     \f]
       "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:
     \f[
         err = k\sqrt{f(x)^{2} + x^{2}deriv^{2}}\sqrt{\sum ai^{2}},
     \f]
      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.
   */
   double Derivative1 (double x) { return Derivative1(*fFunction,x,fStepSize); }
   double operator() (double x) { return Derivative1(*fFunction,x,fStepSize); }
 
   /**
      First Derivative calculation passing function object and step-size
    */
   double Derivative1(const IGenFunction & f, double x, double h);
 
   /// Computation of the first derivative using a forward formula
   double DerivativeForward(double x) {
      return DerivativeForward(*fFunction, x, fStepSize);
   }
 
   /// Computation of the first derivative using a forward formula
   double DerivativeForward(const IGenFunction &f, double x, double h);
 
      /// Computation of the first derivative using a backward formula
   double DerivativeBackward(double x) {
      return DerivativeBackward(*fFunction, x, fStepSize);
   }
 
   /// Computation of the first derivative using a forward formula
   double DerivativeBackward(const IGenFunction &f, double x, double h) {
      return DerivativeForward(f, x, -h);
   }
 
   /**
      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
     \f[
         D(h) = \frac{f(x+h) - 2f(x) + f(x-h)}{h^{2}}
     \f]
      the final estimate
     \f[
         D = \frac{4D(h/2) - D(h)}{3}
     \f]
       "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:
     \f[
         err = k\sqrt{f(x)^{2} + x^{2}deriv^{2}}\sqrt{\sum ai^{2}},
     \f]
      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.
   */
   double Derivative2 (double x) {
      return Derivative2( *fFunction, x, fStepSize);
   }
 
   /**
      Second Derivative calculation passing function and step-size
    */
   double Derivative2(const IGenFunction & f, double x, double h);
 
   /**
      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
     \f[
         D(h) = \frac{f(x+2h) - 2f(x+h) + 2f(x-h) - f(x-2h)}{2h^{3}}
     \f]
      the final estimate
     \f[
         D = \frac{4D(h/2) - D(h)}{3}
     \f]
       "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:
     \f[
         err = k\sqrt{f(x)^{2} + x^{2}deriv^{2}}\sqrt{\sum ai^{2}},
     \f]
      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.
   */
   double Derivative3 (double x) {
      return Derivative3(*fFunction, x, fStepSize);
   }
 
   /**
      Third Derivative calculation passing function and step-size
    */
   double Derivative3(const IGenFunction & f, double x, double h);
 
   /** Set function for derivative calculation (copy the function if option has been enabled in the constructor)
 
       \@param f Function to be differentiated
   */
   void SetFunction (const IGenFunction & f);
 
   /** Set step size for derivative calculation
 
       \@param h step size for calculation
   */
   void SetStepSize (double h) { fStepSize = h; }
 
protected:
 
   bool fFunctionCopied;     ///< flag to control if function is copied in the class
   double fStepSize;         ///< step size used for derivative calculation
   double fLastError;        ///<  error estimate of last derivative calculation
   const IGenFunction* fFunction;  ///< pointer to function
 
};
 
} // end namespace Math
 
} // end namespace ROOT
 
#endif /* ROOT_Math_RichardsonDerivator */