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// $Id: auto_derivative_function.h 14881 2007-08-02 15:11:31Z kanschat $
// Version: $Name$
//
// Copyright (C) 2001, 2002, 2003, 2004, 2005, 2006, 2007 by the deal.II authors
//
// This file is subject to QPL and may not be distributed
// without copyright and license information. Please refer
// to the file deal.II/doc/license.html for the text and
// further information on this license.
//
//---------------------------------------------------------------------------
#ifndef __deal2__auto_derivative_function_h
#define __deal2__auto_derivative_function_h
#include <base/exceptions.h>
#include <base/function.h>
DEAL_II_NAMESPACE_OPEN
/**
* This class automatically computes the gradient of a function by
* employing numerical difference quotients. This only, if the user
* function does not provide the gradient function himself.
*
* The following example of an user defined function overloads and
* implements only the value() function but not the gradient()
* function. If the gradient() function is invoked then the gradient
* function implemented by the AutoDerivativeFunction is called,
* where the latter function imployes numerical difference quotients.
*
* @code
* class UserFunction: public AutoDerivativeFunction
* { // access to one component at one point
* double value (const Point<dim> &p, const
* unsigned int component = 0) const
* { // Implementation .... };
* } user_function;
*
* // gradient by employing difference quotients.
* Tensor<1,dim> grad=user_function.gradient(some_point);
* @endcode
*
* If the user overloads and implements also the gradient function,
* then, of course, the users gradient function is called.
*
* Note, that the usage of the value() and gradient() functions
* explained above, also applies to the value_list() and
* gradient_list() functions as well as to the vector valued
* versions of these functions, see e.g. vector_value(),
* vector_gradient(), vector_value_list() and
* vector_gradient_list().
*
* The gradient() and gradient_list() functions make use of the
* Function::value() function. The vector_gradient() and
* vector_gradient_list() make use of the Function::vector_value()
* function. Make sure that the user defined function implements the
* value() function and the vector_value() function, respectively.
*
* Furthermore note, that an object of this class does <b>not</b> represent
* the derivative of a function, like FunctionDerivative, that
* gives a directional derivate by calling the value() function. In
* fact, this class (the AutoDerivativeFunction class) can
* substitute the Function class as base class for user defined
* classes. This class implements the gradient() functions for
* automatic computation of numerical difference quotients and serves
* as intermediate class between the base Function class and the
* user defined function class.
*
* @ingroup functions
* @author Ralf Hartmann, 2001
*/
template <int dim>
class AutoDerivativeFunction : public Function<dim>
{
public:
/**
* Names of difference formulas.
*/
enum DifferenceFormula
{
/**
* The symmetric Euler
* formula of second order:
* @f[
* u'(t) \approx
* \frac{u(t+h) -
* u(t-h)}{2h}.
* @f]
*/
Euler,
/**
* The upwind Euler
* formula of first order:
* @f[
* u'(t) \approx
* \frac{u(t) -
* u(t-h)}{h}.
* @f]
*/
UpwindEuler,
/**
* The fourth order scheme
* @f[
* u'(t) \approx
* \frac{u(t-2h) - 8u(t-h)
* + 8u(t+h) - u(t+2h)}{12h}.
* @f]
*/
FourthOrder
};
/**
* Constructor. Takes the
* difference step size
* <tt>h</tt>. It's within the user's
* responsibility to choose an
* appropriate value here. <tt>h</tt>
* should be chosen taking into
* account the absolute value as
* well as the amount of local
* variation of the function.
* Setting <tt>h=1e-6</tt> might be a
* good choice for functions with
* an absolute value of about 1,
* that furthermore does not vary
* to much.
*
* <tt>h</tt> can be changed later
* using the set_h() function.
*
* Sets DifferenceFormula
* <tt>formula</tt> to the default
* <tt>Euler</tt> formula of the
* set_formula()
* function. Change this preset
* formula by calling the
* set_formula() function.
*/
AutoDerivativeFunction (const double h,
const unsigned int n_components = 1,
const double initial_time = 0.0);
/**
* Virtual destructor; absolutely
* necessary in this case.
*/
virtual ~AutoDerivativeFunction ();
/**
* Choose the difference formula.
* See the enum #DifferenceFormula
* for available choices.
*/
void set_formula (const DifferenceFormula formula = Euler);
/**
* Takes the difference step size
* <tt>h</tt>. It's within the user's
* responsibility to choose an
* appropriate value here. <tt>h</tt>
* should be chosen taking into
* account the absolute value of
* as well as the amount of local
* variation of the function.
* Setting <tt>h=1e-6</tt> might be a
* good choice for functions with
* an absolute value of about 1,
* that furthermore does not vary
* to much.
*/
void set_h (const double h);
/**
* Return the gradient of the
* specified component of the
* function at the given point.
*
* Computes numerical difference
* quotients using the preset
* #DifferenceFormula.
*/
virtual Tensor<1,dim> gradient (const Point<dim> &p,
const unsigned int component = 0) const;
/**
* Return the gradient of all
* components of the
* function at the given point.
*
* Computes numerical difference
* quotients using the preset
* #DifferenceFormula.
*/
virtual void vector_gradient (const Point<dim> &p,
std::vector<Tensor<1,dim> > &gradients) const;
/**
* Set <tt>gradients</tt> to the
* gradients of the specified
* component of the function at
* the <tt>points</tt>. It is assumed
* that <tt>gradients</tt> already has the
* right size, i.e. the same
* size as the <tt>points</tt> array.
*
* Computes numerical difference
* quotients using the preset
* #DifferenceFormula.
*/
virtual void gradient_list (const std::vector<Point<dim> > &points,
std::vector<Tensor<1,dim> > &gradients,
const unsigned int component = 0) const;
/**
* Set <tt>gradients</tt> to the gradients of
* the function at the <tt>points</tt>,
* for all components.
* It is assumed that <tt>gradients</tt>
* already has the right size, i.e.
* the same size as the <tt>points</tt> array.
*
* The outer loop over
* <tt>gradients</tt> is over the points
* in the list, the inner loop
* over the different components
* of the function.
*
* Computes numerical difference
* quotients using the preset
* #DifferenceFormula.
*/
virtual void vector_gradient_list (const std::vector<Point<dim> > &points,
std::vector<std::vector<Tensor<1,dim> > > &gradients) const;
/**
* Returns a
* #DifferenceFormula of the
* order <tt>ord</tt> at minimum.
*/
static
DifferenceFormula
get_formula_of_order (const unsigned int ord);
/**
* Exception.
*/
DeclException0(ExcInvalidFormula);
private:
/**
* Step size of the difference
* formula. Set by the set_h()
* function.
*/
double h;
/**
* Includes the unit vectors
* scaled by <tt>h</tt>.
*/
std::vector<Tensor<1,dim> > ht;
/**
* Difference formula. Set by the
* set_formula() function.
*/
DifferenceFormula formula;
};
DEAL_II_NAMESPACE_CLOSE
#endif
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