Differentiation

Differentiation is performed by descendants of o2scl::deriv_base. These allow one to calculate either first, second, and third derivatives. A GSL-based routine is used in o2scl::deriv_gsl, and the CERNLIB routine is used in o2scl::deriv_cern. For functions which are tabulated over equally-spaced abscissas, the class o2scl::deriv_eqi is provided which applies the formulas from Abramowitz and Stegun at a specified order. The class o2scl::deriv_cern is slower and sometimes more accurate, but also fails more often than o2scl::deriv_gsl, which never calls the error handler.

Warning: For o2scl::deriv_gsl and o2scl::deriv_cern, the second and third derivatives are calculated by naive repeated application of the code for the first derivative and can be particularly troublesome if the function is not sufficiently smooth. Error estimation is not provided for second and third derivatives.

# Differentiation example

This example computes first and second derivatives of

with both o2scl::deriv_gsl and o2scl::deriv_cern .

/* Example: ex_deriv.cpp
-------------------------------------------------------------------
An example to demonstrate numerical differentiation
*/
#include <cmath>
#include <boost/numeric/ublas/vector.hpp>
#include <o2scl/test_mgr.h>
#include <o2scl/funct.h>
#include <o2scl/deriv_gsl.h>
#include <o2scl/deriv_cern.h>
using namespace std;
using namespace o2scl;
class cl {
public:
// This is the function we'll take the derivative of
double function(double x) {
return sin(2.0*x)+0.5;
}
};
int main(void) {
// The class and associated function
cl acl;
funct11 f1=std::bind(std::mem_fn<double(double)>(&cl::function),
&acl,std::placeholders::_1);
// Note that the GSL derivative routine requires an initial stepsize
gd.h=1.0e-3;
// Compute the first derivative using the deriv_gsl class and
// verify that the answer is correct
double d1=gd.deriv(1.0,f1);
t.test_rel(d1,2.0*cos(2.0),1.0e-10,"deriv_gsl");
// Compute the first derivative using the deriv_cern class and
// verify that the answer is correct
double d2=cd.deriv(1.0,f1);
t.test_rel(d2,2.0*cos(2.0),1.0e-10,"deriv_cern");
// Compute the second derivative also
double d3=gd.deriv2(1.0,f1);
t.test_rel(d3,-4.0*sin(2.0),5.0e-7,"deriv_gsl");
double d4=cd.deriv2(1.0,f1);
t.test_rel(d4,-4.0*sin(2.0),1.0e-8,"deriv_cern");
t.report();
return 0;
}
// End of example

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