Class deriv_cern (o2scl)

O2scl : Class List

template<class func_t = funct>
class deriv_cern : public o2scl::deriv_base<funct>

Numerical differentiation routine (CERNLIB)

This uses Romberg extrapolation to compute the derivative with the finite-differencing formula

\[ f^{\prime}(x) = [f(x+h)-f(x-h)]/(2h) \]

If deriv_base::verbose is greater than zero, then each iteration prints out the extrapolation table, and if deriv_base::verbose is greater than 1, then a keypress is required at the end of each iteration.

For sufficiently difficult functions, the derivative computation can fail, and will call the error handler and return zero with zero error.

Based on the CERNLIB routine DERIV, which was based on [Rutishauser63] and is documented at http://wwwasdoc.web.cern.ch/wwwasdoc/shortwrupsdir/d401/top.html

An example demonstrating the usage of this class is given in the Differentiation example.

If deriv_base::verbose is greater than zero, at each iteration this class prints something similar to

deriv_cern, iteration: 1
(hh, a[k], derivative) list: 
-4.193459e-05 4.387643e-14 8.775286e-01
-2.995402e-05 4.387792e-14 8.775585e-01
-1.048405e-05 4.387845e-14 8.775690e-01
-7.488654e-06 4.387882e-14 8.775765e-01
-2.621038e-06 4.387895e-14 8.775791e-01
-1.872173e-06 4.387905e-14 8.775810e-01
-6.552611e-07 4.387908e-14 8.775817e-01
-4.680438e-07 4.387910e-14 8.775821e-01
-1.638153e-07 4.387911e-14 8.775823e-01

If deriv_base::verbose is greater than 1, a keypress is required after each iteration.

Idea for Future:

All of the coefficients appear to be fractions which could be replaced with exact representation?

Record the number of function calls?

Note

Second and third derivatives are computed by naive nested applications of the formula for the first derivative. No uncertainty for these derivatives is provided.

Storage for the fixed coefficients

double dx[10]

double w[10][4]

Public Functions

inline deriv_cern()

inline virtual int deriv_err(double x, func_t &func, double &dfdx, double &err)

Calculate the first derivative of func w.r.t. x and the uncertainty.

inline virtual const char *type()

Return string denoting type (“deriv_cern”)

Public Members

double delta

A scaling factor (default 1.0)

double eps

Extrapolation tolerance (default is \( 5 \times 10^{14} \))

Protected Functions

template<class func2_t>
inline int deriv_tlate(double x, func2_t &func, double &dfdx, double &err)

Internal template version of the derivative function.

inline virtual int deriv_err_int(double x, funct &func, double &dfdx, double &err)

Calculate the first derivative of func w.r.t. x.

This is an internal version of deriv() which is used in computing second and third derivatives