# Differentiation¶

O2scl

## Differentiation introduction¶

A GSL-based routine for derivatives is provided in deriv_gsl, and a CERNLIB-based routine is given in deriv_cern. These classes allow one to calculate either first, second, and third derivatives. For functions which are tabulated over equally-spaced abscissas, the class deriv_eqi is provided which applies the formulas from Abramowitz and Stegun at a specified order. The class deriv_cern is slower and sometimes more accurate, but also fails more often than deriv_gsl, which never calls the error handler. The GSL derivative class deriv_gsl supports numerical derivatives of functions which operate on multiprecision numbers (see also Multiprecision Support). The class deriv_multip_gsl uses multiprecision to automatically compute a derivative to within a requested precision.

The estimatation of second and third derivatives by deriv_gsl and deriv_cern can be particularly troublesome if the function is not sufficiently smooth. Error estimation is not provided for second and third derivatives. If one has more information about the function, then second and third derivatives are often better computed by fitting to a model and then taking the second or third derivative of the model instead.

## Differentiation example¶

This example computes first and second derivatives of

$y(x) = \sin (2 x) + \frac{1}{2}$

with both deriv_gsl and deriv_cern.

/*
-------------------------------------------------------------------

Copyright (C) 2006-2022, Andrew W. Steiner

This file is part of O2scl.

O2scl is free software; you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation; either version 3 of the License, or
(at your option) any later version.

O2scl is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
GNU General Public License for more details.

You should have received a copy of the GNU General Public License
along with O2scl. If not, see <http://www.gnu.org/licenses/>.

-------------------------------------------------------------------
*/

/* 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) {

test_mgr t;
t.set_output_level(2);

// The class and associated function
cl acl;
funct f1=std::bind(std::mem_fn<double(double)>(&cl::function),
&acl,std::placeholders::_1);

deriv_gsl<> gd;
// Note that the GSL derivative routine requires an initial stepsize
gd.h=1.0e-3;
deriv_cern<> cd;

// 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