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
| const ROOT::Math::IGenFunction* | fFunction | pointer to function |
| bool | fFunctionCopied | flag to control if function is copied in the class |
| double | fLastError | error estimate of last derivative calculation |
| double | fStepSize | step size used for derivative calculation |
| Inheritance Chart: | |||||
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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
Construct from function and step size
Returns the estimate of the absolute Error of the last derivative calculation.
{ 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.
{ return Derivative1(*fFunction,x,fStepSize); }
First Derivative calculation passing function object and step-size
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.
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.
Set function for derivative calculation (copy the function if option has been enabled in the constructor)
\@param f Function to be differentiated
Set step size for derivative calculation
\@param h step size for calculation
{ fStepSize = h; }