tensor.h

// ---------------------------------------------------------------------
//
// Copyright (C) 1998 - 2018 by the deal.II authors
//
// This file is part of the deal.II library.
//
// The deal.II library is free software; you can use it, redistribute
// it, and/or modify it under the terms of the GNU Lesser General
// Public License as published by the Free Software Foundation; either
// version 2.1 of the License, or (at your option) any later version.
// The full text of the license can be found in the file LICENSE at
// the top level of the deal.II distribution.
//
// ---------------------------------------------------------------------

#ifndef dealii_tensor_h
#define dealii_tensor_h

#include <deal.II/base/config.h>
#include <deal.II/base/exceptions.h>
#include <deal.II/base/numbers.h>
#include <deal.II/base/table_indices.h>
#include <deal.II/base/tensor_accessors.h>
#include <deal.II/base/template_constraints.h>
#include <deal.II/base/utilities.h>

#ifdef DEAL_II_WITH_ADOLC
#include <adolc/adouble.h> // Taped double
#endif

#include <cmath>
#include <ostream>
#include <vector>


DEAL_II_NAMESPACE_OPEN

// Forward declarations:

template <int dim, typename Number> class Point;
template <int rank_, int dim, typename Number = double> class Tensor;
template <typename Number> class Vector;
template <typename Number> class VectorizedArray;

#ifndef DOXYGEN
// Overload invalid tensor types of negative rank that come up during
// overload resolution of operator* and related contraction variants.
template <int dim, typename Number>
class Tensor<-2, dim, Number>
{
};

template <int dim, typename Number>
class Tensor<-1, dim, Number>
{
};
#endif /* DOXYGEN */


template <int dim, typename Number>
class Tensor<0,dim,Number>
{
public:
  static const unsigned int dimension = dim;

  static const unsigned int rank = 0;

  static const unsigned int n_independent_components = 1;

  typedef typename numbers::NumberTraits<Number>::real_type real_type;

  typedef Number value_type;

  typedef Number array_type;

  DEAL_II_CUDA_HOST_DEV Tensor ();

  template <typename OtherNumber>
  Tensor (const Tensor<0,dim,OtherNumber> &initializer);

  template <typename OtherNumber>
  Tensor (const OtherNumber &initializer);

  Number *
  begin_raw();

  const Number *
  begin_raw() const;

  Number *
  end_raw();

  const Number *
  end_raw() const;

  DEAL_II_CUDA_HOST_DEV operator Number &();

  DEAL_II_CUDA_HOST_DEV operator const Number &() const;

  template <typename OtherNumber>
  Tensor &operator = (const Tensor<0,dim,OtherNumber> &rhs);

#ifdef __INTEL_COMPILER

  Tensor &operator = (const Tensor<0,dim,Number> &rhs);
#endif

  template <typename OtherNumber>
  Tensor &operator = (const OtherNumber &d);

  template <typename OtherNumber>
  bool operator == (const Tensor<0,dim,OtherNumber> &rhs) const;

  template <typename OtherNumber>
  bool operator != (const Tensor<0,dim,OtherNumber> &rhs) const;

  template <typename OtherNumber>
  Tensor &operator += (const Tensor<0,dim,OtherNumber> &rhs);

  template <typename OtherNumber>
  Tensor &operator -= (const Tensor<0,dim,OtherNumber> &rhs);

  template <typename OtherNumber>
  DEAL_II_CUDA_HOST_DEV Tensor &operator *= (const OtherNumber &factor);

  template <typename OtherNumber>
  Tensor &operator /= (const OtherNumber &factor);

  Tensor  operator - () const;

  void clear ();

  real_type norm () const;

  DEAL_II_CUDA_HOST_DEV real_type norm_square () const;

  template <class Archive>
  void serialize(Archive &ar, const unsigned int version);

  typedef Number tensor_type;

private:
  Number value;

  template <typename OtherNumber>
  void unroll_recursion(Vector<OtherNumber> &result,
                        unsigned int        &start_index) const;

  template <int, int, typename> friend class Tensor;
};



template <int rank_, int dim, typename Number>
class Tensor
{
public:
  static const unsigned int dimension = dim;

  static const unsigned int rank = rank_;

  static const unsigned int
  n_independent_components = Tensor<rank_-1,dim>::n_independent_components *dim;

  typedef typename Tensor<rank_-1,dim,Number>::tensor_type value_type;

  typedef typename Tensor<rank_-1,dim,Number>::array_type
  array_type[(dim != 0) ? dim : 1];
  // ... avoid a compiler warning in case of dim == 0 and ensure that the
  // array always has positive size.

  DEAL_II_CUDA_HOST_DEV Tensor ();

  explicit Tensor (const array_type &initializer);

  template <typename OtherNumber>
  Tensor (const Tensor<rank_,dim,OtherNumber> &initializer);

  template <typename OtherNumber>
  Tensor (const Tensor<1,dim,Tensor<rank_-1,dim,OtherNumber> > &initializer);

  template <typename OtherNumber>
  operator Tensor<1,dim,Tensor<rank_-1,dim,OtherNumber> > () const;

  DEAL_II_CUDA_HOST_DEV value_type &operator [] (const unsigned int i);

  DEAL_II_CUDA_HOST_DEV const value_type &operator[](const unsigned int i) const;

  const Number &operator [] (const TableIndices<rank_> &indices) const;

  Number &operator [] (const TableIndices<rank_> &indices);

  Number *
  begin_raw();

  const Number *
  begin_raw() const;

  Number *
  end_raw();

  const Number *
  end_raw() const;

  template <typename OtherNumber>
  Tensor &operator = (const Tensor<rank_,dim,OtherNumber> &rhs);

  Tensor &operator = (const Number &d);

  template <typename OtherNumber>
  bool operator == (const Tensor<rank_,dim,OtherNumber> &) const;

  template <typename OtherNumber>
  bool operator != (const Tensor<rank_,dim,OtherNumber> &) const;

  template <typename OtherNumber>
  Tensor &operator += (const Tensor<rank_,dim,OtherNumber> &);

  template <typename OtherNumber>
  Tensor &operator -= (const Tensor<rank_,dim,OtherNumber> &);

  template <typename OtherNumber>
  DEAL_II_CUDA_HOST_DEV Tensor &operator *= (const OtherNumber &factor);

  template <typename OtherNumber>
  Tensor &operator /= (const OtherNumber &factor);

  Tensor  operator - () const;

  void clear ();

  typename numbers::NumberTraits<Number>::real_type norm() const;

  DEAL_II_CUDA_HOST_DEV typename numbers::NumberTraits<Number>::real_type norm_square() const;

  template <typename OtherNumber>
  void unroll (Vector<OtherNumber> &result) const;

  static
  unsigned int
  component_to_unrolled_index(const TableIndices<rank_> &indices);

  static
  TableIndices<rank_> unrolled_to_component_indices(const unsigned int i);

  static std::size_t memory_consumption ();

  template <class Archive>
  void serialize(Archive &ar, const unsigned int version);

  typedef Tensor<rank_, dim, Number> tensor_type;

private:
  Tensor<rank_-1, dim, Number> values[(dim != 0) ? dim : 1];
  // ... avoid a compiler warning in case of dim == 0 and ensure that the
  // array always has positive size.

  template <typename OtherNumber>
  void unroll_recursion(Vector<OtherNumber> &result,
                        unsigned int        &start_index) const;

  template <int, int, typename> friend class Tensor;

  friend class Point<dim,Number>;
};


namespace internal
{
  template <int rank, int dim, typename T>
  struct NumberType<Tensor<rank,dim,T> >
  {
    static const Tensor<rank,dim,T> &value (const Tensor<rank,dim,T> &t)
    {
      return t;
    }

    static Tensor<rank,dim,T> value (const T &t)
    {
      Tensor<rank,dim,T> tmp;
      tmp=t;
      return tmp;
    }
  };

  template <int rank, int dim, typename T>
  struct NumberType<Tensor<rank,dim,VectorizedArray<T> > >
  {
    static const Tensor<rank,dim,VectorizedArray<T> > &value (const Tensor<rank,dim,VectorizedArray<T> > &t)
    {
      return t;
    }

    static Tensor<rank,dim,VectorizedArray<T> > value (const T &t)
    {
      Tensor<rank,dim,VectorizedArray<T> > tmp;
      tmp=internal::NumberType<VectorizedArray<T> >::value(t);
      return tmp;
    }

    static Tensor<rank,dim,VectorizedArray<T> > value (const VectorizedArray<T> &t)
    {
      Tensor<rank,dim,VectorizedArray<T> > tmp;
      tmp=t;
      return tmp;
    }
  };
}


/*---------------------- Inline functions: Tensor<0,dim> ---------------------*/


template <int dim,typename Number>
inline
DEAL_II_CUDA_HOST_DEV Tensor<0,dim,Number>::Tensor ()
// Some auto-differentiable numbers need explicit
// zero initialization.
  : value(internal::NumberType<Number>::value(0.0))
{
}



template <int dim, typename Number>
template <typename OtherNumber>
inline
Tensor<0,dim,Number>::Tensor (const OtherNumber &initializer)
{
  value = internal::NumberType<Number>::value(initializer);
}



template <int dim, typename Number>
template <typename OtherNumber>
inline
Tensor<0,dim,Number>::Tensor (const Tensor<0,dim,OtherNumber> &p)
{
  value = p.value;
}



template <int dim, typename Number>
inline
Number *
Tensor<0,dim,Number>::begin_raw()
{
  return std::addressof(value);
}



template <int dim, typename Number>
inline
const Number *
Tensor<0,dim,Number>::begin_raw() const
{
  return std::addressof(value);
}



template <int dim, typename Number>
inline
Number *
Tensor<0,dim,Number>::end_raw()
{
  return begin_raw()+n_independent_components;
}



template <int dim, typename Number>
inline
const Number *
Tensor<0,dim,Number>::end_raw() const
{
  return begin_raw()+n_independent_components;
}



template <int dim, typename Number>
inline
DEAL_II_CUDA_HOST_DEV Tensor<0,dim,Number>::operator Number &()
{
  // We cannot use Assert inside a CUDA kernel
#ifndef __CUDA_ARCH__
  Assert(dim != 0, ExcMessage("Cannot access an object of type Tensor<0,0,Number>"));
#endif
  return value;
}


template <int dim, typename Number>
inline
DEAL_II_CUDA_HOST_DEV Tensor<0,dim,Number>::operator const Number &() const
{
  // We cannot use Assert inside a CUDA kernel
#ifndef __CUDA_ARCH__
  Assert(dim != 0, ExcMessage("Cannot access an object of type Tensor<0,0,Number>"));
#endif
  return value;
}


template <int dim, typename Number>
template <typename OtherNumber>
inline
Tensor<0,dim,Number> &Tensor<0,dim,Number>::operator = (const Tensor<0,dim,OtherNumber> &p)
{
  value = internal::NumberType<Number>::value(p);
  return *this;
}


#ifdef __INTEL_COMPILER
template <int dim, typename Number>
inline
Tensor<0,dim,Number> &Tensor<0,dim,Number>::operator = (const Tensor<0,dim,Number> &p)
{
  value = p.value;
  return *this;
}
#endif


template <int dim, typename Number>
template <typename OtherNumber>
inline
Tensor<0,dim,Number> &Tensor<0,dim,Number>::operator = (const OtherNumber &d)
{
  value = internal::NumberType<Number>::value(d);
  return *this;
}


template <int dim, typename Number>
template <typename OtherNumber>
inline
bool Tensor<0,dim,Number>::operator == (const Tensor<0,dim,OtherNumber> &p) const
{
#ifdef DEAL_II_ADOLC_WITH_ADVANCED_BRANCHING
  Assert(!(std::is_same<Number,adouble>::value || std::is_same<OtherNumber,adouble>::value),
         ExcMessage("The Tensor equality operator for Adol-C taped numbers has not yet "
                    "been extended to support advanced branching."));
#endif

  return numbers::values_are_equal(value,p.value);
}


template <int dim, typename Number>
template <typename OtherNumber>
inline
bool Tensor<0,dim,Number>::operator != (const Tensor<0,dim,OtherNumber> &p) const
{
  return !((*this) == p);
}


template <int dim, typename Number>
template <typename OtherNumber>
inline
Tensor<0,dim,Number> &Tensor<0,dim,Number>::operator += (const Tensor<0,dim,OtherNumber> &p)
{
  value += p.value;
  return *this;
}


template <int dim, typename Number>
template <typename OtherNumber>
inline
Tensor<0,dim,Number> &Tensor<0,dim,Number>::operator -= (const Tensor<0,dim,OtherNumber> &p)
{
  value -= p.value;
  return *this;
}


template <int dim, typename Number>
template <typename OtherNumber>
inline
DEAL_II_CUDA_HOST_DEV Tensor<0,dim,Number> &Tensor<0,dim,Number>::operator *= (const OtherNumber &s)
{
  value *= s;
  return *this;
}


template <int dim, typename Number>
template <typename OtherNumber>
inline
Tensor<0,dim,Number> &Tensor<0,dim,Number>::operator /= (const OtherNumber &s)
{
  value /= s;
  return *this;
}


template <int dim, typename Number>
inline
Tensor<0,dim,Number> Tensor<0,dim,Number>::operator - () const
{
  return -value;
}


template <int dim, typename Number>
inline
typename Tensor<0,dim,Number>::real_type
Tensor<0,dim,Number>::norm () const
{
  Assert(dim != 0, ExcMessage("Cannot access an object of type Tensor<0,0,Number>"));
  return numbers::NumberTraits<Number>::abs (value);
}


template <int dim, typename Number>
inline
typename Tensor<0,dim,Number>::real_type
DEAL_II_CUDA_HOST_DEV Tensor<0,dim,Number>::norm_square () const
{
  // We cannot use Assert inside a CUDA kernel
#ifndef __CUDA_ARCH__
  Assert(dim != 0, ExcMessage("Cannot access an object of type Tensor<0,0,Number>"));
#endif
  return numbers::NumberTraits<Number>::abs_square (value);
}


template <int dim, typename Number>
template <typename OtherNumber>
inline
void
Tensor<0, dim, Number>::unroll_recursion (Vector<OtherNumber> &result,
                                          unsigned int        &index) const
{
  Assert(dim != 0, ExcMessage("Cannot unroll an object of type Tensor<0,0,Number>"));
  result[index] = value;
  ++index;
}


template <int dim, typename Number>
inline
void Tensor<0,dim,Number>::clear ()
{
  // Some auto-differentiable numbers need explicit
  // zero initialization.
  value = internal::NumberType<Number>::value(0.0);
}


template <int dim, typename Number>
template <class Archive>
inline
void Tensor<0,dim,Number>::serialize(Archive &ar, const unsigned int)
{
  ar &value;
}


/*-------------------- Inline functions: Tensor<rank,dim> --------------------*/


template <int rank_, int dim, typename Number>
inline DEAL_II_ALWAYS_INLINE
DEAL_II_CUDA_HOST_DEV Tensor<rank_,dim,Number>::Tensor ()
{
  // All members of the c-style array values are already default initialized
  // and thus all values are already set to zero recursively.
}


template <int rank_, int dim, typename Number>
inline DEAL_II_ALWAYS_INLINE
Tensor<rank_,dim,Number>::Tensor (const array_type &initializer)
{
  for (unsigned int i=0; i<dim; ++i)
    values[i] = Tensor<rank_-1, dim, Number>(initializer[i]);
}


template <int rank_, int dim, typename Number>
template <typename OtherNumber>
inline DEAL_II_ALWAYS_INLINE
Tensor<rank_,dim,Number>::Tensor (const Tensor<rank_,dim,OtherNumber> &initializer)
{
  for (unsigned int i=0; i!=dim; ++i)
    values[i] = Tensor<rank_-1,dim,Number>(initializer[i]);
}


template <int rank_, int dim, typename Number>
template <typename OtherNumber>
inline DEAL_II_ALWAYS_INLINE
Tensor<rank_,dim,Number>::Tensor
(const Tensor<1,dim,Tensor<rank_-1,dim,OtherNumber> > &initializer)
{
  for (unsigned int i=0; i<dim; ++i)
    values[i] = initializer[i];
}


template <int rank_, int dim, typename Number>
template <typename OtherNumber>
inline DEAL_II_ALWAYS_INLINE
Tensor<rank_,dim,Number>::
operator Tensor<1,dim,Tensor<rank_-1,dim,OtherNumber> > () const
{
  return Tensor<1,dim,Tensor<rank_-1,dim,Number> > (values);
}



namespace internal
{
  namespace TensorSubscriptor
  {
    template <typename ArrayElementType, int dim>
    inline DEAL_II_ALWAYS_INLINE
    DEAL_II_CUDA_HOST_DEV
    ArrayElementType &
    subscript (ArrayElementType *values,
               const unsigned int i,
               std::integral_constant<int, dim>)
    {
      // We cannot use Assert in a CUDA kernel
#ifndef __CUDA_ARCH__
      Assert (i<dim, ExcIndexRange(i, 0, dim));
#endif
      return values[i];
    }


    template <typename ArrayElementType>
    DEAL_II_CUDA_HOST_DEV
    ArrayElementType &
    subscript (ArrayElementType *,
               const unsigned int,
               std::integral_constant<int, 0>)
    {
      // We cannot use Assert in a CUDA kernel
#ifndef __CUDA_ARCH__
      Assert(false, ExcMessage("Cannot access elements of an object of type Tensor<rank,0,Number>."));
#endif
      static ArrayElementType t;
      return t;
    }
  }
}


template <int rank_, int dim, typename Number>
inline DEAL_II_ALWAYS_INLINE
DEAL_II_CUDA_HOST_DEV
typename Tensor<rank_,dim,Number>::value_type &
Tensor<rank_,dim,Number>::operator[] (const unsigned int i)
{
  return ::internal::TensorSubscriptor::subscript(values, i, std::integral_constant<int, dim>());
}


template <int rank_, int dim, typename Number>
inline DEAL_II_ALWAYS_INLINE
DEAL_II_CUDA_HOST_DEV
const typename Tensor<rank_,dim,Number>::value_type &
Tensor<rank_,dim,Number>::operator[] (const unsigned int i) const
{
  return ::internal::TensorSubscriptor::subscript(values, i, std::integral_constant<int, dim>());
}


template <int rank_, int dim, typename Number>
inline
const Number &
Tensor<rank_,dim,Number>::operator[] (const TableIndices<rank_> &indices) const
{
  Assert(dim != 0, ExcMessage("Cannot access an object of type Tensor<rank_,0,Number>"));

  return TensorAccessors::extract<rank_>(*this, indices);
}



template <int rank_, int dim, typename Number>
inline
Number &
Tensor<rank_,dim,Number>::operator[] (const TableIndices<rank_> &indices)
{
  Assert(dim != 0, ExcMessage("Cannot access an object of type Tensor<rank_,0,Number>"));

  return TensorAccessors::extract<rank_>(*this, indices);
}



template <int rank_, int dim, typename Number>
inline
Number *
Tensor<rank_,dim,Number>::begin_raw()
{
  return std::addressof(this->operator[](this->unrolled_to_component_indices(0)));
}



template <int rank_, int dim, typename Number>
inline
const Number *
Tensor<rank_,dim,Number>::begin_raw() const
{
  return std::addressof(this->operator[](this->unrolled_to_component_indices(0)));
}



template <int rank_, int dim, typename Number>
inline
Number *
Tensor<rank_,dim,Number>::end_raw()
{
  return begin_raw()+n_independent_components;
}



template <int rank_, int dim, typename Number>
inline
const Number *
Tensor<rank_,dim,Number>::end_raw() const
{
  return begin_raw()+n_independent_components;
}



template <int rank_, int dim, typename Number>
template <typename OtherNumber>
inline DEAL_II_ALWAYS_INLINE
Tensor<rank_,dim,Number> &
Tensor<rank_,dim,Number>::operator = (const Tensor<rank_,dim,OtherNumber> &t)
{
  if (dim > 0)
    std::copy (&t.values[0], &t.values[0]+dim, &values[0]);
  return *this;
}


template <int rank_, int dim, typename Number>
inline DEAL_II_ALWAYS_INLINE
Tensor<rank_,dim,Number> &
Tensor<rank_,dim,Number>::operator = (const Number &d)
{
  Assert (numbers::value_is_zero(d), ExcMessage ("Only assignment with zero is allowed"));
  (void) d;

  for (unsigned int i=0; i<dim; ++i)
    values[i] = internal::NumberType<Number>::value(0.0);
  return *this;
}


template <int rank_, int dim, typename Number>
template <typename OtherNumber>
inline
bool
Tensor<rank_,dim,Number>::operator == (const Tensor<rank_,dim,OtherNumber> &p) const
{
  for (unsigned int i=0; i<dim; ++i)
    if (values[i] != p.values[i])
      return false;
  return true;
}


// At some places in the library, we have Point<0> for formal reasons
// (e.g., we sometimes have Quadrature<dim-1> for faces, so we have
// Quadrature<0> for dim=1, and then we have Point<0>). To avoid warnings
// in the above function that the loop end check always fails, we
// implement this function here
template <>
template <>
inline
bool Tensor<1,0,double>::operator == (const Tensor<1,0,double> &) const
{
  return true;
}


template <int rank_, int dim, typename Number>
template <typename OtherNumber>
inline
bool
Tensor<rank_,dim,Number>::operator != (const Tensor<rank_,dim,OtherNumber> &p) const
{
  return !((*this) == p);
}


template <int rank_, int dim, typename Number>
template <typename OtherNumber>
inline
Tensor<rank_,dim,Number> &
Tensor<rank_,dim,Number>::operator += (const Tensor<rank_,dim,OtherNumber> &p)
{
  for (unsigned int i=0; i<dim; ++i)
    values[i] += p.values[i];
  return *this;
}


template <int rank_, int dim, typename Number>
template <typename OtherNumber>
inline
Tensor<rank_,dim,Number> &
Tensor<rank_,dim,Number>::operator -= (const Tensor<rank_,dim,OtherNumber> &p)
{
  for (unsigned int i=0; i<dim; ++i)
    values[i] -= p.values[i];
  return *this;
}


template <int rank_, int dim, typename Number>
template <typename OtherNumber>
inline
DEAL_II_CUDA_HOST_DEV
Tensor<rank_,dim,Number> &
Tensor<rank_,dim,Number>::operator *= (const OtherNumber &s)
{
  for (unsigned int i=0; i<dim; ++i)
    values[i] *= s;
  return *this;
}


template <int rank_, int dim, typename Number>
template <typename OtherNumber>
inline
Tensor<rank_,dim,Number> &
Tensor<rank_,dim,Number>::operator /= (const OtherNumber &s)
{
  for (unsigned int i=0; i<dim; ++i)
    values[i] /= s;
  return *this;
}


template <int rank_, int dim, typename Number>
inline
Tensor<rank_,dim,Number>
Tensor<rank_,dim,Number>::operator - () const
{
  Tensor<rank_,dim,Number> tmp;

  for (unsigned int i=0; i<dim; ++i)
    tmp.values[i] = -values[i];

  return tmp;
}


template <int rank_, int dim, typename Number>
inline
typename numbers::NumberTraits<Number>::real_type
Tensor<rank_,dim,Number>::norm () const
{
  return std::sqrt (norm_square());
}


template <int rank_, int dim, typename Number>
inline
DEAL_II_CUDA_HOST_DEV
typename numbers::NumberTraits<Number>::real_type
Tensor<rank_,dim,Number>::norm_square () const
{
  typename numbers::NumberTraits<Number>::real_type s
    = internal::NumberType<typename numbers::NumberTraits<Number>::real_type>::value(0.0);
  for (unsigned int i=0; i<dim; ++i)
    s += values[i].norm_square();

  return s;
}


template <int rank_, int dim, typename Number>
template <typename OtherNumber>
inline
void
Tensor<rank_, dim, Number>::unroll (Vector<OtherNumber> &result) const
{
  AssertDimension (result.size(),(Utilities::fixed_power<rank_, unsigned int>(dim)));

  unsigned int index = 0;
  unroll_recursion (result, index);
}


template <int rank_, int dim, typename Number>
template <typename OtherNumber>
inline
void
Tensor<rank_, dim, Number>::unroll_recursion (Vector<OtherNumber> &result,
                                              unsigned int        &index) const
{
  for (unsigned int i=0; i<dim; ++i)
    values[i].unroll_recursion(result, index);
}


template <int rank_, int dim, typename Number>
inline
unsigned int
Tensor<rank_, dim, Number>::component_to_unrolled_index(const TableIndices<rank_> &indices)
{
  unsigned int index = 0;
  for (int r = 0; r < rank_; ++r)
    index = index * dim + indices[r];

  return index;
}


template <int rank_, int dim, typename Number>
inline
TableIndices<rank_>
Tensor<rank_, dim, Number>::unrolled_to_component_indices(const unsigned int i)
{
  Assert (i < n_independent_components,
          ExcIndexRange (i, 0, n_independent_components));

  TableIndices<rank_>   indices;

  unsigned int remainder = i;
  for (int r=rank_-1; r>=0; --r)
    {
      indices[r] = (remainder % dim);
      remainder /= dim;
    }
  Assert (remainder == 0, ExcInternalError());

  return indices;
}


template <int rank_, int dim, typename Number>
inline
void Tensor<rank_,dim,Number>::clear ()
{
  for (unsigned int i=0; i<dim; ++i)
    values[i] = internal::NumberType<Number>::value(0.0);
}


template <int rank_, int dim, typename Number>
inline
std::size_t
Tensor<rank_,dim,Number>::memory_consumption ()
{
  return sizeof(Tensor<rank_,dim,Number>);
}


template <int rank_, int dim, typename Number>
template <class Archive>
inline
void
Tensor<rank_,dim,Number>::serialize(Archive &ar, const unsigned int)
{
  ar &values;
}


/* ----------------- Non-member functions operating on tensors. ------------ */


template <int rank_, int dim, typename Number>
inline
std::ostream &operator << (std::ostream &out, const Tensor<rank_,dim,Number> &p)
{
  for (unsigned int i = 0; i < dim; ++i)
    {
      out << p[i];
      if (i != dim - 1)
        out << ' ';
    }

  return out;
}


template <int dim, typename Number>
inline
std::ostream &operator << (std::ostream &out, const Tensor<0,dim,Number> &p)
{
  out << static_cast<const Number &>(p);
  return out;
}





template <int dim, typename Number, typename Other>
inline DEAL_II_ALWAYS_INLINE
typename ProductType<Other, Number>::type
operator * (const Other                &object,
            const Tensor<0,dim,Number> &t)
{
  return object * static_cast<const Number &>(t);
}



template <int dim, typename Number, typename Other>
inline DEAL_II_ALWAYS_INLINE
typename ProductType<Number, Other>::type
operator * (const Tensor<0,dim,Number> &t,
            const Other                &object)
{
  return static_cast<const Number &>(t) * object;
}


template <int dim, typename Number, typename OtherNumber>
inline DEAL_II_ALWAYS_INLINE
typename ProductType<Number, OtherNumber>::type
operator * (const Tensor<0, dim, Number>      &src1,
            const Tensor<0, dim, OtherNumber> &src2)
{
  return static_cast<const Number &>(src1) *
         static_cast<const OtherNumber &>(src2);
}


template <int dim, typename Number, typename OtherNumber>
inline DEAL_II_ALWAYS_INLINE
Tensor<0,dim,typename ProductType<Number, typename EnableIfScalar<OtherNumber>::type>::type>
operator / (const Tensor<0,dim,Number> &t,
            const OtherNumber          &factor)
{
  return static_cast<const Number &>(t) / factor;
}


template <int dim, typename Number, typename OtherNumber>
inline DEAL_II_ALWAYS_INLINE
Tensor<0, dim, typename ProductType<Number, OtherNumber>::type>
operator+ (const Tensor<0,dim,Number>      &p,
           const Tensor<0,dim,OtherNumber> &q)
{
  return static_cast<const Number &>(p) + static_cast<const OtherNumber &>(q);
}


template <int dim, typename Number, typename OtherNumber>
inline DEAL_II_ALWAYS_INLINE
Tensor<0, dim, typename ProductType<Number, OtherNumber>::type>
operator- (const Tensor<0,dim,Number>      &p,
           const Tensor<0,dim,OtherNumber> &q)
{
  return static_cast<const Number &>(p) - static_cast<const OtherNumber &>(q);
}


template <int rank, int dim,
          typename Number,
          typename OtherNumber>
inline DEAL_II_ALWAYS_INLINE
Tensor<rank,dim,typename ProductType<Number, typename EnableIfScalar<OtherNumber>::type>::type>
operator * (const Tensor<rank,dim,Number> &t,
            const OtherNumber             &factor)
{
  // recurse over the base objects
  Tensor<rank,dim,typename ProductType<Number,OtherNumber>::type> tt;
  for (unsigned int d=0; d<dim; ++d)
    tt[d] = t[d] * factor;
  return tt;
}


template <int rank, int dim,
          typename Number,
          typename OtherNumber>
inline DEAL_II_ALWAYS_INLINE
Tensor<rank,dim,typename ProductType<typename EnableIfScalar<Number>::type, OtherNumber>::type>
operator * (const Number                       &factor,
            const Tensor<rank,dim,OtherNumber> &t)
{
  // simply forward to the operator above
  return t * factor;
}


template <int rank, int dim,
          typename Number,
          typename OtherNumber>
inline
Tensor<rank,dim,typename ProductType<Number, typename EnableIfScalar<OtherNumber>::type>::type>
operator / (const Tensor<rank,dim,Number> &t,
            const OtherNumber             &factor)
{
  // recurse over the base objects
  Tensor<rank,dim,typename ProductType<Number,OtherNumber>::type> tt;
  for (unsigned int d=0; d<dim; ++d)
    tt[d] = t[d] / factor;
  return tt;
}


template <int rank, int dim, typename Number, typename OtherNumber>
inline DEAL_II_ALWAYS_INLINE
Tensor<rank, dim, typename ProductType<Number, OtherNumber>::type>
operator+ (const Tensor<rank,dim,Number> &p,
           const Tensor<rank,dim,OtherNumber> &q)
{
  Tensor<rank, dim, typename ProductType<Number, OtherNumber>::type> tmp (p);

  for (unsigned int i=0; i<dim; ++i)
    tmp[i] += q[i];

  return tmp;
}


template <int rank, int dim, typename Number, typename OtherNumber>
inline DEAL_II_ALWAYS_INLINE
Tensor<rank, dim, typename ProductType<Number, OtherNumber>::type>
operator- (const Tensor<rank,dim,Number> &p,
           const Tensor<rank,dim,OtherNumber> &q)
{
  Tensor<rank, dim, typename ProductType<Number, OtherNumber>::type> tmp (p);

  for (unsigned int i=0; i<dim; ++i)
    tmp[i] -= q[i];

  return tmp;
}





template <int rank_1, int rank_2, int dim,
          typename Number, typename OtherNumber>
inline DEAL_II_ALWAYS_INLINE
typename Tensor<rank_1 + rank_2 - 2, dim, typename ProductType<Number, OtherNumber>::type>::tensor_type
operator * (const Tensor<rank_1, dim, Number> &src1,
            const Tensor<rank_2, dim, OtherNumber> &src2)
{
  typename Tensor<rank_1 + rank_2 - 2, dim, typename ProductType<Number, OtherNumber>::type>::tensor_type result;

  TensorAccessors::internal::ReorderedIndexView<0, rank_2, const Tensor<rank_2, dim, OtherNumber> >
  reordered = TensorAccessors::reordered_index_view<0, rank_2>(src2);
  TensorAccessors::contract<1, rank_1, rank_2, dim>(result, src1, reordered);

  return result;
}


template <int index_1, int index_2,
          int rank_1, int rank_2, int dim,
          typename Number, typename OtherNumber>
inline DEAL_II_ALWAYS_INLINE
typename Tensor<rank_1 + rank_2 - 2, dim, typename ProductType<Number, OtherNumber>::type>::tensor_type
contract (const Tensor<rank_1, dim, Number> &src1,
          const Tensor<rank_2, dim, OtherNumber> &src2)
{
  Assert(0 <= index_1 && index_1 < rank_1,
         ExcMessage("The specified index_1 must lie within the range [0,rank_1)"));
  Assert(0 <= index_2 && index_2 < rank_2,
         ExcMessage("The specified index_2 must lie within the range [0,rank_2)"));

  using namespace TensorAccessors;
  using namespace TensorAccessors::internal;

  // Reorder index_1 to the end of src1:
  ReorderedIndexView<index_1, rank_1, const Tensor<rank_1, dim, Number> >
  reord_01 = reordered_index_view<index_1, rank_1>(src1);

  // Reorder index_2 to the end of src2:
  ReorderedIndexView<index_2, rank_2, const Tensor<rank_2, dim, OtherNumber> >
  reord_02 = reordered_index_view<index_2, rank_2>(src2);

  typename Tensor<rank_1 + rank_2 - 2, dim, typename ProductType<Number, OtherNumber>::type>::tensor_type
  result;
  TensorAccessors::contract<1, rank_1, rank_2, dim>(result, reord_01, reord_02);
  return result;
}


template <int index_1, int index_2, int index_3, int index_4,
          int rank_1, int rank_2, int dim,
          typename Number, typename OtherNumber>
inline
typename Tensor<rank_1 + rank_2 - 4, dim, typename ProductType<Number, OtherNumber>::type>::tensor_type
double_contract (const Tensor<rank_1, dim, Number> &src1,
                 const Tensor<rank_2, dim, OtherNumber> &src2)
{
  Assert(0 <= index_1 && index_1 < rank_1,
         ExcMessage("The specified index_1 must lie within the range [0,rank_1)"));
  Assert(0 <= index_3 && index_3 < rank_1,
         ExcMessage("The specified index_3 must lie within the range [0,rank_1)"));
  Assert(index_1 != index_3,
         ExcMessage("index_1 and index_3 must not be the same"));
  Assert(0 <= index_2 && index_2 < rank_2,
         ExcMessage("The specified index_2 must lie within the range [0,rank_2)"));
  Assert(0 <= index_4 && index_4 < rank_2,
         ExcMessage("The specified index_4 must lie within the range [0,rank_2)"));
  Assert(index_2 != index_4,
         ExcMessage("index_2 and index_4 must not be the same"));

  using namespace TensorAccessors;
  using namespace TensorAccessors::internal;

  // Reorder index_1 to the end of src1:
  ReorderedIndexView<index_1, rank_1, const Tensor<rank_1, dim, Number> >
  reord_1 = TensorAccessors::reordered_index_view<index_1, rank_1>(src1);

  // Reorder index_2 to the end of src2:
  ReorderedIndexView<index_2, rank_2, const Tensor<rank_2, dim, OtherNumber> >
  reord_2 = TensorAccessors::reordered_index_view<index_2, rank_2>(src2);

  // Now, reorder index_3 to the end of src1. We have to make sure to
  // preserve the orginial ordering: index_1 has been removed. If
  // index_3 > index_1, we have to use (index_3 - 1) instead:
  ReorderedIndexView<(index_3 < index_1 ? index_3 : index_3 - 1), rank_1, ReorderedIndexView<index_1, rank_1, const Tensor<rank_1, dim, Number> > >
  reord_3 = TensorAccessors::reordered_index_view<index_3 < index_1 ? index_3 : index_3 - 1, rank_1>(reord_1);

  // Now, reorder index_4 to the end of src2. We have to make sure to
  // preserve the orginial ordering: index_2 has been removed. If
  // index_4 > index_2, we have to use (index_4 - 1) instead:
  ReorderedIndexView<(index_4 < index_2 ? index_4 : index_4 - 1), rank_2, ReorderedIndexView<index_2, rank_2, const Tensor<rank_2, dim, OtherNumber> > >
  reord_4 = TensorAccessors::reordered_index_view<index_4 < index_2 ? index_4 : index_4 - 1, rank_2>(reord_2);

  typename Tensor<rank_1 + rank_2 - 4, dim, typename ProductType<Number, OtherNumber>::type>::tensor_type
  result;
  TensorAccessors::contract<2, rank_1, rank_2, dim>(result, reord_3, reord_4);
  return result;
}


template <int rank, int dim, typename Number, typename OtherNumber>
inline DEAL_II_ALWAYS_INLINE
typename ProductType<Number, OtherNumber>::type
scalar_product (const Tensor<rank, dim, Number> &left,
                const Tensor<rank, dim, OtherNumber> &right)
{
  typename ProductType<Number, OtherNumber>::type result;
  TensorAccessors::contract<rank, rank, rank, dim>(result, left, right);
  return result;
}


template <template <int, int, typename> class TensorT1,
          template <int, int, typename> class TensorT2,
          template <int, int, typename> class TensorT3,
          int rank_1, int rank_2, int dim,
          typename T1, typename T2, typename T3>
typename ProductType<T1, typename ProductType<T2, T3>::type>::type
contract3 (const TensorT1<rank_1, dim, T1>          &left,
           const TensorT2<rank_1 + rank_2, dim, T2> &middle,
           const TensorT3<rank_2, dim, T3>          &right)
{
  typedef typename ProductType<T1, typename ProductType<T2, T3>::type>::type
  return_type;
  return TensorAccessors::contract3<rank_1, rank_2, dim, return_type>(
           left, middle, right);
}


template <int rank_1, int rank_2, int dim,
          typename Number, typename OtherNumber>
inline DEAL_II_ALWAYS_INLINE
Tensor<rank_1 + rank_2, dim, typename ProductType<Number, OtherNumber>::type>
outer_product(const Tensor<rank_1, dim, Number> &src1,
              const Tensor<rank_2, dim, OtherNumber> &src2)
{
  typename Tensor<rank_1 + rank_2, dim, typename ProductType<Number, OtherNumber>::type>::tensor_type result;
  TensorAccessors::contract<0, rank_1, rank_2, dim>(result, src1, src2);
  return result;
}





template <int dim, typename Number>
inline DEAL_II_ALWAYS_INLINE
Tensor<1,dim,Number>
cross_product_2d (const Tensor<1,dim,Number> &src)
{
  Assert (dim==2, ExcInternalError());

  Tensor<1, dim, Number> result;

  result[0] = src[1];
  result[1] = -src[0];

  return result;
}


template <int dim, typename Number>
inline DEAL_II_ALWAYS_INLINE
Tensor<1,dim,Number>
cross_product_3d (const Tensor<1,dim,Number> &src1,
                  const Tensor<1,dim,Number> &src2)
{
  Assert (dim==3, ExcInternalError());

  Tensor<1, dim, Number> result;

  result[0] = src1[1]*src2[2] - src1[2]*src2[1];
  result[1] = src1[2]*src2[0] - src1[0]*src2[2];
  result[2] = src1[0]*src2[1] - src1[1]*src2[0];

  return result;
}





template <int dim, typename Number>
inline
Number determinant (const Tensor<2,dim,Number> &t)
{
  // Compute the determinant using the Laplace expansion of the
  // determinant. We expand along the last row.
  Number det = internal::NumberType<Number>::value(0.0);

  for (unsigned int k=0; k<dim; ++k)
    {
      Tensor<2,dim-1,Number> minor;
      for (unsigned int i=0; i<dim-1; ++i)
        for (unsigned int j=0; j<dim-1; ++j)
          minor[i][j] = t[i][j<k ? j : j+1];

      const Number cofactor = ((k % 2 == 0) ? -1. : 1.) * determinant(minor);

      det += t[dim-1][k] * cofactor;
    }

  return ((dim % 2 == 0) ? 1. : -1.) * det;
}

template <typename Number>
inline
Number determinant (const Tensor<2,1,Number> &t)
{
  return t[0][0];
}


template <int dim, typename Number>
inline DEAL_II_ALWAYS_INLINE
Number trace (const Tensor<2,dim,Number> &d)
{
  Number t=d[0][0];
  for (unsigned int i=1; i<dim; ++i)
    t += d[i][i];
  return t;
}


template <int dim, typename Number>
inline
Tensor<2,dim,Number>
invert (const Tensor<2,dim,Number> &)
{
  Number return_tensor [dim][dim];

  // if desired, take over the
  // inversion of a 4x4 tensor
  // from the FullMatrix
  AssertThrow (false, ExcNotImplemented());

  return Tensor<2,dim,Number>(return_tensor);
}


#ifndef DOXYGEN

template <typename Number>
inline
Tensor<2,1,Number>
invert (const Tensor<2,1,Number> &t)
{
  Number return_tensor [1][1];

  return_tensor[0][0] = internal::NumberType<Number>::value(1.0/t[0][0]);

  return Tensor<2,1,Number>(return_tensor);
}


template <typename Number>
inline
Tensor<2,2,Number>
invert (const Tensor<2,2,Number> &t)
{
  Tensor<2,2,Number> return_tensor;

  // this is Maple output,
  // thus a bit unstructured
  const Number inv_det_t = internal::NumberType<Number>::value(1.0/(t[0][0]*t[1][1]-t[1][0]*t[0][1]));
  return_tensor[0][0] = t[1][1];
  return_tensor[0][1] = -t[0][1];
  return_tensor[1][0] = -t[1][0];
  return_tensor[1][1] = t[0][0];
  return_tensor *= inv_det_t;

  return return_tensor;
}


template <typename Number>
inline
Tensor<2,3,Number>
invert (const Tensor<2,3,Number> &t)
{
  Tensor<2,3,Number> return_tensor;

  const Number t4 = internal::NumberType<Number>::value(t[0][0]*t[1][1]),
               t6 = internal::NumberType<Number>::value(t[0][0]*t[1][2]),
               t8 = internal::NumberType<Number>::value(t[0][1]*t[1][0]),
               t00 = internal::NumberType<Number>::value(t[0][2]*t[1][0]),
               t01 = internal::NumberType<Number>::value(t[0][1]*t[2][0]),
               t04 = internal::NumberType<Number>::value(t[0][2]*t[2][0]),
               inv_det_t = internal::NumberType<Number>::value(
                             1.0/(t4*t[2][2]-t6*t[2][1]-t8*t[2][2]+
                                  t00*t[2][1]+t01*t[1][2]-t04*t[1][1]));
  return_tensor[0][0] = internal::NumberType<Number>::value(t[1][1]*t[2][2])-internal::NumberType<Number>::value(t[1][2]*t[2][1]);
  return_tensor[0][1] = internal::NumberType<Number>::value(t[0][2]*t[2][1])-internal::NumberType<Number>::value(t[0][1]*t[2][2]);
  return_tensor[0][2] = internal::NumberType<Number>::value(t[0][1]*t[1][2])-internal::NumberType<Number>::value(t[0][2]*t[1][1]);
  return_tensor[1][0] = internal::NumberType<Number>::value(t[1][2]*t[2][0])-internal::NumberType<Number>::value(t[1][0]*t[2][2]);
  return_tensor[1][1] = internal::NumberType<Number>::value(t[0][0]*t[2][2])-t04;
  return_tensor[1][2] = t00-t6;
  return_tensor[2][0] = internal::NumberType<Number>::value(t[1][0]*t[2][1])-internal::NumberType<Number>::value(t[1][1]*t[2][0]);
  return_tensor[2][1] = t01-internal::NumberType<Number>::value(t[0][0]*t[2][1]);
  return_tensor[2][2] = internal::NumberType<Number>::value(t4-t8);
  return_tensor *= inv_det_t;

  return return_tensor;
}

#endif /* DOXYGEN */


template <int dim, typename Number>
inline DEAL_II_ALWAYS_INLINE
Tensor<2,dim,Number>
transpose (const Tensor<2,dim,Number> &t)
{
  Tensor<2, dim, Number> tt;
  for (unsigned int i=0; i<dim; ++i)
    {
      tt[i][i] = t[i][i];
      for (unsigned int j=i+1; j<dim; ++j)
        {
          tt[i][j] = t[j][i];
          tt[j][i] = t[i][j];
        };
    }
  return tt;
}


template <int dim, typename Number>
inline
Tensor<2,dim,Number>
adjugate (const Tensor<2,dim,Number> &t)
{
  return determinant(t)*invert(t);
}


template <int dim, typename Number>
inline
Tensor<2,dim,Number>
cofactor (const Tensor<2,dim,Number> &t)
{
  return transpose(adjugate(t));
}


template <int dim, typename Number>
inline
Number
l1_norm (const Tensor<2,dim,Number> &t)
{
  Number max = internal::NumberType<Number>::value(0.0);
  for (unsigned int j=0; j<dim; ++j)
    {
      Number sum = internal::NumberType<Number>::value(0.0);
      for (unsigned int i=0; i<dim; ++i)
        sum += std::fabs(t[i][j]);

      if (sum > max)
        max = sum;
    }

  return max;
}


template <int dim, typename Number>
inline
Number
linfty_norm (const Tensor<2,dim,Number> &t)
{
  Number max = internal::NumberType<Number>::value(0.0);
  for (unsigned int i=0; i<dim; ++i)
    {
      Number sum = internal::NumberType<Number>::value(0.0);
      for (unsigned int j=0; j<dim; ++j)
        sum += std::fabs(t[i][j]);

      if (sum > max)
        max = sum;
    }

  return max;
}



#ifndef DOXYGEN


#ifdef DEAL_II_ADOLC_WITH_ADVANCED_BRANCHING

// Specialization of functions for Adol-C number types when
// the advanced branching feature is used
template <int dim>
inline
adouble
l1_norm (const Tensor<2,dim,adouble> &t)
{
  adouble max = internal::NumberType<adouble>::value(0.0);
  for (unsigned int j=0; j<dim; ++j)
    {
      adouble sum = internal::NumberType<adouble>::value(0.0);
      for (unsigned int i=0; i<dim; ++i)
        sum += std::fabs(t[i][j]);

      condassign(max, (sum > max), sum, max);
    }

  return max;
}


template <int dim>
inline
adouble
linfty_norm (const Tensor<2,dim,adouble> &t)
{
  adouble max = internal::NumberType<adouble>::value(0.0);
  for (unsigned int i=0; i<dim; ++i)
    {
      adouble sum = internal::NumberType<adouble>::value(0.0);
      for (unsigned int j=0; j<dim; ++j)
        sum += std::fabs(t[i][j]);

      condassign(max, (sum > max), sum, max);
    }

  return max;
}

#endif // DEAL_II_ADOLC_WITH_ADVANCED_BRANCHING


#endif // DOXYGEN

DEAL_II_NAMESPACE_CLOSE

// include deprecated non-member functions operating on Tensor
#include <deal.II/base/tensor_deprecated.h>

#endif