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RichardsonDerivator.h
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1// @(#)root/mathcore:$Id$
2// Authors: David Gonzalez Maline 01/2008
3
4/**********************************************************************
5 * *
6 * Copyright (c) 2006 , LCG ROOT MathLib Team *
7 * *
8 * *
9 **********************************************************************/
10
11// Header file for RichardsonDerivator
12//
13// Created by: David Gonzalez Maline : Mon Feb 4 2008
14//
15
16#ifndef ROOT_Math_RichardsonDerivator
17#define ROOT_Math_RichardsonDerivator
18
19#include <Math/IFunction.h>
20
21/**
22 @defgroup Deriv Numerical Differentiation
23 Classes for numerical differentiation
24 @ingroup NumAlgo
25*/
26
27
28namespace ROOT {
29namespace Math {
30
31//___________________________________________________________________________________________
32/**
33 User class for calculating the derivatives of a function. It can calculate first (method Derivative1),
34 second (method Derivative2) and third (method Derivative3) of a function.
35
36 It uses the Richardson extrapolation method for function derivation in a given interval.
37 The method use 2 derivative estimates (one computed with step h and one computed with step h/2)
38 to compute a third, more accurate estimation. It is equivalent to the
39 <a href = http://en.wikipedia.org/wiki/Five-point_stencil>5-point method</a>,
40 which can be obtained with a Taylor expansion.
41 A step size should be given, depending on x and f(x).
42 An optimal step size value minimizes the truncation error of the expansion and the rounding
43 error in evaluating x+h and f(x+h). A too small h will yield a too large rounding error while a too large
44 h will give a large truncation error in the derivative approximation.
45 A good discussion can be found in discussed in
46 <a href=http://www.nrbook.com/a/bookcpdf/c5-7.pdf>Chapter 5.7</a> of Numerical Recipes in C.
47 By default a value of 0.001 is uses, acceptable in many cases.
48
49 This class is implemented using code previously in TF1::Derivate{,2,3}(). Now TF1 uses this class.
50
51 @ingroup Deriv
52
53 */
54
56public:
57
58 /** Destructor: Removes function if needed. */
60
61 /** Default Constructor.
62 Give optionally the step size for derivation. By default is 0.001, which is fine for x ~ 1
63 Increase if x is in average larger or decrease if x is smaller
64 */
65 RichardsonDerivator(double h = 0.001);
66
67 /** Construct from function and step size
68 */
69 RichardsonDerivator(const ROOT::Math::IGenFunction & f, double h = 0.001, bool copyFunc = false);
70
71 /**
72 Copy constructor
73 */
75
76 /**
77 Assignment operator
78 */
80
81
82 /** Returns the estimate of the absolute Error of the last derivative calculation. */
83 double Error () const { return fLastError; }
84
85
86 /**
87 Returns the first derivative of the function at point x,
88 computed by Richardson's extrapolation method (use 2 derivative estimates
89 to compute a third, more accurate estimation)
90 first, derivatives with steps h and h/2 are computed by central difference formulas
91 \f[
92 D(h) = \frac{f(x+h) - f(x-h)}{2h}
93 \f]
94 the final estimate
95 \f[
96 D = \frac{4D(h/2) - D(h)}{3}
97 \f]
98 "Numerical Methods for Scientists and Engineers", H.M.Antia, 2nd edition"
99
100 the argument eps may be specified to control the step size (precision).
101 the step size is taken as eps*(xmax-xmin).
102 the default value (0.001) should be good enough for the vast majority
103 of functions. Give a smaller value if your function has many changes
104 of the second derivative in the function range.
105
106 Getting the error via TF1::DerivativeError:
107 (total error = roundoff error + interpolation error)
108 the estimate of the roundoff error is taken as follows:
109 \f[
110 err = k\sqrt{f(x)^{2} + x^{2}deriv^{2}}\sqrt{\sum ai^{2}},
111 \f]
112 where k is the double precision, ai are coefficients used in
113 central difference formulas
114 interpolation error is decreased by making the step size h smaller.
115 */
116 double Derivative1 (double x) { return Derivative1(*fFunction,x,fStepSize); }
117 double operator() (double x) { return Derivative1(*fFunction,x,fStepSize); }
118
119 /**
120 First Derivative calculation passing function object and step-size
121 */
122 double Derivative1(const IGenFunction & f, double x, double h);
123
124 /// Computation of the first derivative using a forward formula
125 double DerivativeForward(double x) {
127 }
128
129 /// Computation of the first derivative using a forward formula
130 double DerivativeForward(const IGenFunction &f, double x, double h);
131
132 /// Computation of the first derivative using a backward formula
133 double DerivativeBackward(double x) {
135 }
136
137 /// Computation of the first derivative using a forward formula
138 double DerivativeBackward(const IGenFunction &f, double x, double h) {
139 return DerivativeForward(f, x, -h);
140 }
141
142 /**
143 Returns the second derivative of the function at point x,
144 computed by Richardson's extrapolation method (use 2 derivative estimates
145 to compute a third, more accurate estimation)
146 first, derivatives with steps h and h/2 are computed by central difference formulas
147 \f[
148 D(h) = \frac{f(x+h) - 2f(x) + f(x-h)}{h^{2}}
149 \f]
150 the final estimate
151 \f[
152 D = \frac{4D(h/2) - D(h)}{3}
153 \f]
154 "Numerical Methods for Scientists and Engineers", H.M.Antia, 2nd edition"
155
156 the argument eps may be specified to control the step size (precision).
157 the step size is taken as eps*(xmax-xmin).
158 the default value (0.001) should be good enough for the vast majority
159 of functions. Give a smaller value if your function has many changes
160 of the second derivative in the function range.
161
162 Getting the error via TF1::DerivativeError:
163 (total error = roundoff error + interpolation error)
164 the estimate of the roundoff error is taken as follows:
165 \f[
166 err = k\sqrt{f(x)^{2} + x^{2}deriv^{2}}\sqrt{\sum ai^{2}},
167 \f]
168 where k is the double precision, ai are coefficients used in
169 central difference formulas
170 interpolation error is decreased by making the step size h smaller.
171 */
172 double Derivative2 (double x) {
173 return Derivative2( *fFunction, x, fStepSize);
174 }
175
176 /**
177 Second Derivative calculation passing function and step-size
178 */
179 double Derivative2(const IGenFunction & f, double x, double h);
180
181 /**
182 Returns the third derivative of the function at point x,
183 computed by Richardson's extrapolation method (use 2 derivative estimates
184 to compute a third, more accurate estimation)
185 first, derivatives with steps h and h/2 are computed by central difference formulas
186 \f[
187 D(h) = \frac{f(x+2h) - 2f(x+h) + 2f(x-h) - f(x-2h)}{2h^{3}}
188 \f]
189 the final estimate
190 \f[
191 D = \frac{4D(h/2) - D(h)}{3}
192 \f]
193 "Numerical Methods for Scientists and Engineers", H.M.Antia, 2nd edition"
194
195 the argument eps may be specified to control the step size (precision).
196 the step size is taken as eps*(xmax-xmin).
197 the default value (0.001) should be good enough for the vast majority
198 of functions. Give a smaller value if your function has many changes
199 of the second derivative in the function range.
200
201 Getting the error via TF1::DerivativeError:
202 (total error = roundoff error + interpolation error)
203 the estimate of the roundoff error is taken as follows:
204 \f[
205 err = k\sqrt{f(x)^{2} + x^{2}deriv^{2}}\sqrt{\sum ai^{2}},
206 \f]
207 where k is the double precision, ai are coefficients used in
208 central difference formulas
209 interpolation error is decreased by making the step size h smaller.
210 */
211 double Derivative3 (double x) {
212 return Derivative3(*fFunction, x, fStepSize);
213 }
214
215 /**
216 Third Derivative calculation passing function and step-size
217 */
218 double Derivative3(const IGenFunction & f, double x, double h);
219
220 /** Set function for derivative calculation (copy the function if option has been enabled in the constructor)
221
222 \@param f Function to be differentiated
223 */
224 void SetFunction (const IGenFunction & f);
225
226 /** Set step size for derivative calculation
227
228 \@param h step size for calculation
229 */
230 void SetStepSize (double h) { fStepSize = h; }
231
232protected:
233
234 bool fFunctionCopied; // flag to control if function is copied in the class
235 double fStepSize; // step size used for derivative calculation
236 double fLastError; // error estimate of last derivative calculation
237 const IGenFunction* fFunction; // pointer to function
238
239};
240
241} // end namespace Math
242
243} // end namespace ROOT
244
245#endif /* ROOT_Math_RichardsonDerivator */
246
#define f(i)
Definition: RSha256.hxx:104
#define h(i)
Definition: RSha256.hxx:106
Interface (abstract class) for generic functions objects of one-dimension Provides a method to evalua...
Definition: IFunction.h:135
User class for calculating the derivatives of a function.
double DerivativeBackward(const IGenFunction &f, double x, double h)
Computation of the first derivative using a forward formula.
double Derivative2(double x)
Returns the second derivative of the function at point x, computed by Richardson's extrapolation meth...
double Error() const
Returns the estimate of the absolute Error of the last derivative calculation.
void SetStepSize(double h)
Set step size for derivative calculation.
double DerivativeBackward(double x)
Computation of the first derivative using a backward formula.
~RichardsonDerivator()
Destructor: Removes function if needed.
RichardsonDerivator(double h=0.001)
Default Constructor.
double DerivativeForward(double x)
Computation of the first derivative using a forward formula.
RichardsonDerivator & operator=(const RichardsonDerivator &rhs)
Assignment operator.
double Derivative3(double x)
Returns the third derivative of the function at point x, computed by Richardson's extrapolation metho...
double Derivative1(double x)
Returns the first derivative of the function at point x, computed by Richardson's extrapolation metho...
void SetFunction(const IGenFunction &f)
Set function for derivative calculation (copy the function if option has been enabled in the construc...
Double_t x[n]
Definition: legend1.C:17
Namespace for new Math classes and functions.
tbb::task_arena is an alias of tbb::interface7::task_arena, which doesn't allow to forward declare tb...