type_tensor.h

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// The libMesh Finite Element Library.
// Copyright (C) 2002-2026 Benjamin S. Kirk, John W. Peterson, Roy H. Stogner

// This library is free software; you can 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.

// This library 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
// Lesser General Public License for more details.

// You should have received a copy of the GNU Lesser General Public
// License along with this library; if not, write to the Free Software
// Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA



#ifndef LIBMESH_TYPE_TENSOR_H
#define LIBMESH_TYPE_TENSOR_H

// Local includes
#include "libmesh/libmesh_common.h"
#include "libmesh/type_vector.h"

// C++ includes
#include <cstdlib> // *must* precede <cmath> for proper std:abs() on PGI, Sun Studio CC
#include <cmath>
#include <tuple>

#ifdef LIBMESH_HAVE_METAPHYSICL
#include "metaphysicl/dualnumber_decl.h"
#endif

namespace libMesh
{

// Forward declarations
template <typename T> class TypeTensorColumn;
template <typename T> class ConstTypeTensorColumn;
template <unsigned int N, typename T> class TypeNTensor;

template <typename T>
class TypeTensor
{
  template <typename T2>
  friend class TypeTensor;

public:
  TypeTensor  ();

protected:
  explicit TypeTensor  (const T & xx,
                        const T & xy=0,
                        const T & xz=0,
                        const T & yx=0,
                        const T & yy=0,
                        const T & yz=0,
                        const T & zx=0,
                        const T & zy=0,
                        const T & zz=0);


  template <typename Scalar>
  explicit TypeTensor  (const Scalar & xx,
                        const Scalar & xy=0,
                        const Scalar & xz=0,
                        const Scalar & yx=0,
                        const Scalar & yy=0,
                        const Scalar & yz=0,
                        const Scalar & zx=0,
                        const Scalar & zy=0,
                        typename
                        std::enable_if<ScalarTraits<Scalar>::value,
                        const Scalar>::type & zz=0);

  template<typename T2>
  TypeTensor(const TypeVector<T2> & vx);

  template<typename T2>
  TypeTensor(const TypeVector<T2> & vx,
             const TypeVector<T2> & vy);

  template<typename T2>
  TypeTensor(const TypeVector<T2> & vx,
             const TypeVector<T2> & vy,
             const TypeVector<T2> & vz);

public:

  typedef T value_type;

  typedef std::tuple<unsigned int, unsigned int> index_type;

  template<typename T2>
  TypeTensor(const TypeTensor<T2> & p);

  ~TypeTensor();

  template<typename T2>
  void assign (const TypeTensor<T2> &);

  template <typename Scalar>
  typename std::enable_if<
    ScalarTraits<Scalar>::value,
    TypeTensor &>::type
  operator = (const Scalar & libmesh_dbg_var(p))
  { libmesh_assert_equal_to (p, Scalar(0)); this->zero(); return *this; }

  const T & operator () (const unsigned int i, const unsigned int j) const;

  T & operator () (const unsigned int i, const unsigned int j);

  ConstTypeTensorColumn<T> slice (const unsigned int i) const;

  TypeTensorColumn<T> slice (const unsigned int i);

  TypeVector<T> row(const unsigned int r) const;

  TypeVector<T> column(const unsigned int r) const;

  template<typename T2>
  TypeTensor<typename CompareTypes<T, T2>::supertype>
  operator + (const TypeTensor<T2> &) const;

  template<typename T2>
  const TypeTensor<T> & operator += (const TypeTensor<T2> &);

  template<typename T2>
  void add (const TypeTensor<T2> &);

  template <typename T2>
  void add_scaled (const TypeTensor<T2> &, const T &);

  template<typename T2>
  TypeTensor<typename CompareTypes<T, T2>::supertype>
  operator - (const TypeTensor<T2> &) const;

  template<typename T2>
  const TypeTensor<T> & operator -= (const TypeTensor<T2> &);

  template<typename T2>
  void subtract (const TypeTensor<T2> &);

  template <typename T2>
  void subtract_scaled (const TypeTensor<T2> &, const T &);

  TypeTensor<T> operator - () const;

  template <typename Scalar>
  auto
  operator * (const Scalar & scalar) const -> typename std::enable_if<
    ScalarTraits<Scalar>::value,
    TypeTensor<decltype(T() * scalar)>>::type;

  template <typename Scalar, typename std::enable_if<
                               ScalarTraits<Scalar>::value, int>::type = 0>
  const TypeTensor<T> & operator *= (const Scalar & factor)
    {
      for (unsigned int i=0; i<LIBMESH_DIM*LIBMESH_DIM; i++)
        _coords[i] *= factor;

      return *this;
    }

  template <typename Scalar>
  typename std::enable_if<
    ScalarTraits<Scalar>::value,
    TypeTensor<typename CompareTypes<T, Scalar>::supertype>>::type
  operator / (const Scalar &) const;

  const TypeTensor<T> & operator /= (const T &);

  template <typename T2>
  TypeTensor<typename CompareTypes<T, T2>::supertype>
  operator * (const TypeTensor<T2> &) const;

  template <typename T2>
  const TypeTensor<T> & operator *= (const TypeTensor<T2> &);

  template <typename T2>
  typename CompareTypes<T,T2>::supertype
  contract (const TypeTensor<T2> &) const;

  template <typename T2>
  TypeVector<typename CompareTypes<T,T2>::supertype>
  operator * (const TypeVector<T2> &) const;

  template <typename T2>
  TypeVector<typename CompareTypes<T,T2>::supertype>
  left_multiply (const TypeVector<T2> & p) const;

  TypeTensor<T> transpose() const;

  TypeTensor<T> inverse() const;

  void solve(const TypeVector<T> & b, TypeVector<T> & x) const;

  auto norm() const;

  auto norm_sq() const;

  bool is_zero() const;

  T det() const;

  T tr() const;

  void zero();

  bool operator == (const TypeTensor<T> & rhs) const;

  bool operator < (const TypeTensor<T> & rhs) const;

  bool operator > (const TypeTensor<T> & rhs) const;

  void print(std::ostream & os = libMesh::out) const;

  friend std::ostream & operator << (std::ostream & os, const TypeTensor<T> & t)
  {
    t.print(os);
    return os;
  }

  void write_unformatted (std::ostream & out_stream,
                          const bool newline = true) const;

  bool is_hpd(Real rel_tol = TOLERANCE*TOLERANCE) const;

protected:

  T _coords[LIBMESH_DIM*LIBMESH_DIM];
};


template <typename T>
class TypeTensorColumn
{
public:

  TypeTensorColumn(TypeTensor<T> & tensor,
                   unsigned int j) :
    _tensor(&tensor), _j(j) {}

  T & operator () (const unsigned int i)
  { return (*_tensor)(i,_j); }

  T & slice (const unsigned int i)
  { return (*_tensor)(i,_j); }

  TypeTensorColumn<T> & operator = (const TypeVector<T> & rhs)
  {
    for (unsigned int i=0; i != LIBMESH_DIM; ++i)
      (*this)(i) = rhs(i);
    return *this;
  }

private:
  TypeTensor<T> * _tensor;
  const unsigned int _j;
};


template <typename T>
class ConstTypeTensorColumn
{
public:

  ConstTypeTensorColumn(const TypeTensor<T> & tensor,
                        unsigned int j) :
    _tensor(&tensor), _j(j) {}

  const T & operator () (const unsigned int i) const
  { return (*_tensor)(i,_j); }

  const T & slice (const unsigned int i) const
  { return (*_tensor)(i,_j); }

private:
  const TypeTensor<T> * _tensor;
  const unsigned int _j;
};

//------------------------------------------------------
// Inline functions
template <typename T>
inline
TypeTensor<T>::TypeTensor ()
{
  _coords[0] = {};

#if LIBMESH_DIM > 1
  _coords[1] = {};
  _coords[2] = {};
  _coords[3] = {};
#endif

#if LIBMESH_DIM > 2
  _coords[4] = {};
  _coords[5] = {};
  _coords[6] = {};
  _coords[7] = {};
  _coords[8] = {};
#endif
}



template <typename T>
inline
TypeTensor<T>::TypeTensor (const T & xx,
                           const T & xy,
                           const T & xz,
                           const T & yx,
                           const T & yy,
                           const T & yz,
                           const T & zx,
                           const T & zy,
                           const T & zz)
{
  _coords[0] = xx;

#if LIBMESH_DIM == 2
  _coords[1] = xy;
  _coords[2] = yx;
  _coords[3] = yy;
#elif LIBMESH_DIM == 1
  libmesh_assert_equal_to (xy, 0);
  libmesh_assert_equal_to (yx, 0);
  libmesh_assert_equal_to (yy, 0);
  libmesh_ignore(xy, yx, yy);
#endif

#if LIBMESH_DIM == 3
  _coords[1] = xy;
  _coords[2] = xz;
  _coords[3] = yx;
  _coords[4] = yy;
  _coords[5] = yz;
  _coords[6] = zx;
  _coords[7] = zy;
  _coords[8] = zz;
#else
  libmesh_assert_equal_to (xz, 0);
  libmesh_assert_equal_to (yz, 0);
  libmesh_assert_equal_to (zx, 0);
  libmesh_assert_equal_to (zy, 0);
  libmesh_assert_equal_to (zz, 0);
  libmesh_ignore(xz, yz, zx, zy, zz);
#endif
}


template <typename T>
template <typename Scalar>
inline
TypeTensor<T>::TypeTensor (const Scalar & xx,
                           const Scalar & xy,
                           const Scalar & xz,
                           const Scalar & yx,
                           const Scalar & yy,
                           const Scalar & yz,
                           const Scalar & zx,
                           const Scalar & zy,
                           typename
                           std::enable_if<ScalarTraits<Scalar>::value,
                           const Scalar>::type & zz)
{
  _coords[0] = xx;

#if LIBMESH_DIM == 2
  _coords[1] = xy;
  _coords[2] = yx;
  _coords[3] = yy;
#elif LIBMESH_DIM == 1
  libmesh_assert_equal_to (xy, 0);
  libmesh_assert_equal_to (yx, 0);
  libmesh_assert_equal_to (yy, 0);
  libmesh_ignore(xy, yx, yy);
#endif

#if LIBMESH_DIM == 3
  _coords[1] = xy;
  _coords[2] = xz;
  _coords[3] = yx;
  _coords[4] = yy;
  _coords[5] = yz;
  _coords[6] = zx;
  _coords[7] = zy;
  _coords[8] = zz;
#else
  libmesh_assert_equal_to (xz, 0);
  libmesh_assert_equal_to (yz, 0);
  libmesh_assert_equal_to (zx, 0);
  libmesh_assert_equal_to (zy, 0);
  libmesh_assert_equal_to (zz, 0);
  libmesh_ignore(xz, yz, zx, zy, zz);
#endif
}



template <typename T>
template<typename T2>
inline
TypeTensor<T>::TypeTensor (const TypeTensor <T2> & p)
{
  // copy the nodes from vector p to me
  for (unsigned int i=0; i<LIBMESH_DIM*LIBMESH_DIM; i++)
    _coords[i] = p._coords[i];
}


template <typename T>
template <typename T2>
TypeTensor<T>::TypeTensor(const TypeVector<T2> & vx)
{
  libmesh_assert_equal_to (LIBMESH_DIM, 1);
  _coords[0] = vx(0);
}

template <typename T>
template <typename T2>
TypeTensor<T>::TypeTensor(const TypeVector<T2> & vx,
                          const TypeVector<T2> & vy)
{
  libmesh_assert_equal_to (LIBMESH_DIM, 2);
#if LIBMESH_DIM > 1
  _coords[0] = vx(0);
  _coords[1] = vx(1);
  _coords[2] = vy(0);
  _coords[3] = vy(1);
#else
  libmesh_ignore(vx, vy);
#endif
}

template <typename T>
template <typename T2>
TypeTensor<T>::TypeTensor(const TypeVector<T2> & vx,
                          const TypeVector<T2> & vy,
                          const TypeVector<T2> & vz)
{
  libmesh_assert_equal_to (LIBMESH_DIM, 3);
#if LIBMESH_DIM > 2
  _coords[0] = vx(0);
  _coords[1] = vx(1);
  _coords[2] = vx(2);
  _coords[3] = vy(0);
  _coords[4] = vy(1);
  _coords[5] = vy(2);
  _coords[6] = vz(0);
  _coords[7] = vz(1);
  _coords[8] = vz(2);
#else
  libmesh_ignore(vx, vy, vz);
#endif
}




template <typename T>
inline
TypeTensor<T>::~TypeTensor ()
{
}



template <typename T>
template<typename T2>
inline
void TypeTensor<T>::assign (const TypeTensor<T2> & p)
{
  for (unsigned int i=0; i<LIBMESH_DIM*LIBMESH_DIM; i++)
    _coords[i] = p._coords[i];
}



template <typename T>
inline
const T & TypeTensor<T>::operator () (const unsigned int i,
                                      const unsigned int j) const
{
  libmesh_assert_less (i, 3);
  libmesh_assert_less (j, 3);

#if LIBMESH_DIM < 3
  const static T my_zero = 0;
  if (i >= LIBMESH_DIM || j >= LIBMESH_DIM)
    return my_zero;
#endif

  return _coords[i*LIBMESH_DIM+j];
}



template <typename T>
inline
T & TypeTensor<T>::operator () (const unsigned int i,
                                const unsigned int j)
{
#if LIBMESH_DIM < 3

  libmesh_error_msg_if(i >= LIBMESH_DIM || j >= LIBMESH_DIM,
                       "ERROR:  You are assigning to a tensor component that is out of range for the compiled LIBMESH_DIM!");

#endif

  libmesh_assert_less (i, LIBMESH_DIM);
  libmesh_assert_less (j, LIBMESH_DIM);

  return _coords[i*LIBMESH_DIM+j];
}


template <typename T>
inline
ConstTypeTensorColumn<T>
TypeTensor<T>::slice (const unsigned int i) const
{
  libmesh_assert_less (i, LIBMESH_DIM);
  return ConstTypeTensorColumn<T>(*this, i);
}


template <typename T>
inline
TypeTensorColumn<T>
TypeTensor<T>::slice (const unsigned int i)
{
  libmesh_assert_less (i, LIBMESH_DIM);
  return TypeTensorColumn<T>(*this, i);
}


template <typename T>
inline
TypeVector<T>
TypeTensor<T>::row(const unsigned int r) const
{
  TypeVector<T> return_vector;

  for (unsigned int j=0; j<LIBMESH_DIM; j++)
    return_vector._coords[j] = _coords[r*LIBMESH_DIM + j];

  return return_vector;
}


template <typename T>
inline
TypeVector<T>
TypeTensor<T>::column(const unsigned int r) const
{
  TypeVector<T> return_vector;

  for (unsigned int i=0; i<LIBMESH_DIM; i++)
    return_vector._coords[i] = _coords[i*LIBMESH_DIM + r];

  return return_vector;
}


template <typename T>
template<typename T2>
inline
TypeTensor<typename CompareTypes<T, T2>::supertype>
TypeTensor<T>::operator + (const TypeTensor<T2> & p) const
{

#if LIBMESH_DIM == 1
  return TypeTensor<typename CompareTypes<T, T2>::supertype>(_coords[0] + p._coords[0]);
#endif

#if LIBMESH_DIM == 2
  return TypeTensor<typename CompareTypes<T, T2>::supertype>(_coords[0] + p._coords[0],
                                                             _coords[1] + p._coords[1],
                                                             0.,
                                                             _coords[2] + p._coords[2],
                                                             _coords[3] + p._coords[3]);
#endif

#if LIBMESH_DIM == 3
  return TypeTensor<typename CompareTypes<T, T2>::supertype>(_coords[0] + p._coords[0],
                                                             _coords[1] + p._coords[1],
                                                             _coords[2] + p._coords[2],
                                                             _coords[3] + p._coords[3],
                                                             _coords[4] + p._coords[4],
                                                             _coords[5] + p._coords[5],
                                                             _coords[6] + p._coords[6],
                                                             _coords[7] + p._coords[7],
                                                             _coords[8] + p._coords[8]);
#endif

}



template <typename T>
template<typename T2>
inline
const TypeTensor<T> & TypeTensor<T>::operator += (const TypeTensor<T2> & p)
{
  this->add (p);

  return *this;
}



template <typename T>
template<typename T2>
inline
void TypeTensor<T>::add (const TypeTensor<T2> & p)
{
  for (unsigned int i=0; i<LIBMESH_DIM*LIBMESH_DIM; i++)
    _coords[i] += p._coords[i];
}



template <typename T>
template <typename T2>
inline
void TypeTensor<T>::add_scaled (const TypeTensor<T2> & p, const T & factor)
{
  for (unsigned int i=0; i<LIBMESH_DIM*LIBMESH_DIM; i++)
    _coords[i] += factor*p._coords[i];

}



template <typename T>
template<typename T2>
inline
TypeTensor<typename CompareTypes<T, T2>::supertype>
TypeTensor<T>::operator - (const TypeTensor<T2> & p) const
{

#if LIBMESH_DIM == 1
  return TypeTensor<typename CompareTypes<T, T2>::supertype>(_coords[0] - p._coords[0]);
#endif

#if LIBMESH_DIM == 2
  return TypeTensor<typename CompareTypes<T, T2>::supertype>(_coords[0] - p._coords[0],
                                                             _coords[1] - p._coords[1],
                                                             0.,
                                                             _coords[2] - p._coords[2],
                                                             _coords[3] - p._coords[3]);
#endif

#if LIBMESH_DIM == 3
  return TypeTensor<typename CompareTypes<T, T2>::supertype>(_coords[0] - p._coords[0],
                                                             _coords[1] - p._coords[1],
                                                             _coords[2] - p._coords[2],
                                                             _coords[3] - p._coords[3],
                                                             _coords[4] - p._coords[4],
                                                             _coords[5] - p._coords[5],
                                                             _coords[6] - p._coords[6],
                                                             _coords[7] - p._coords[7],
                                                             _coords[8] - p._coords[8]);
#endif

}



template <typename T>
template <typename T2>
inline
const TypeTensor<T> & TypeTensor<T>::operator -= (const TypeTensor<T2> & p)
{
  this->subtract (p);

  return *this;
}



template <typename T>
template <typename T2>
inline
void TypeTensor<T>::subtract (const TypeTensor<T2> & p)
{
  for (unsigned int i=0; i<LIBMESH_DIM*LIBMESH_DIM; i++)
    _coords[i] -= p._coords[i];
}



template <typename T>
template <typename T2>
inline
void TypeTensor<T>::subtract_scaled (const TypeTensor<T2> & p, const T & factor)
{
  for (unsigned int i=0; i<LIBMESH_DIM*LIBMESH_DIM; i++)
    _coords[i] -= factor*p._coords[i];
}



template <typename T>
inline
TypeTensor<T> TypeTensor<T>::operator - () const
{

#if LIBMESH_DIM == 1
  return TypeTensor(-_coords[0]);
#endif

#if LIBMESH_DIM == 2
  return TypeTensor(-_coords[0],
                    -_coords[1],
                    -_coords[2],
                    -_coords[3]);
#endif

#if LIBMESH_DIM == 3
  return TypeTensor(-_coords[0],
                    -_coords[1],
                    -_coords[2],
                    -_coords[3],
                    -_coords[4],
                    -_coords[5],
                    -_coords[6],
                    -_coords[7],
                    -_coords[8]);
#endif

}



template <typename T>
template <typename Scalar>
inline
auto
TypeTensor<T>::operator * (const Scalar & factor) const -> typename std::enable_if<
  ScalarTraits<Scalar>::value,
  TypeTensor<decltype(T() * factor)>>::type
{
  typedef decltype((*this)(0, 0) * factor) TS;


#if LIBMESH_DIM == 1
  return TypeTensor<TS>(_coords[0]*factor);
#endif

#if LIBMESH_DIM == 2
  return TypeTensor<TS>(_coords[0]*factor,
                        _coords[1]*factor,
                        _coords[2]*factor,
                        _coords[3]*factor);
#endif

#if LIBMESH_DIM == 3
  return TypeTensor<TS>(_coords[0]*factor,
                        _coords[1]*factor,
                        _coords[2]*factor,
                        _coords[3]*factor,
                        _coords[4]*factor,
                        _coords[5]*factor,
                        _coords[6]*factor,
                        _coords[7]*factor,
                        _coords[8]*factor);
#endif
}



template <typename T, typename Scalar>
inline
typename std::enable_if<
  ScalarTraits<Scalar>::value,
  TypeTensor<typename CompareTypes<T, Scalar>::supertype>>::type
operator * (const Scalar & factor,
            const TypeTensor<T> & t)
{
  return t * factor;
}

template <typename T>
template <typename Scalar>
inline
typename std::enable_if<
  ScalarTraits<Scalar>::value,
  TypeTensor<typename CompareTypes<T, Scalar>::supertype>>::type
TypeTensor<T>::operator / (const Scalar & factor) const
{
  libmesh_assert_not_equal_to (factor, static_cast<T>(0.));

  typedef typename CompareTypes<T, Scalar>::supertype TS;

#if LIBMESH_DIM == 1
  return TypeTensor<TS>(_coords[0]/factor);
#endif

#if LIBMESH_DIM == 2
  return TypeTensor<TS>(_coords[0]/factor,
                        _coords[1]/factor,
                        _coords[2]/factor,
                        _coords[3]/factor);
#endif

#if LIBMESH_DIM == 3
  return TypeTensor<TS>(_coords[0]/factor,
                        _coords[1]/factor,
                        _coords[2]/factor,
                        _coords[3]/factor,
                        _coords[4]/factor,
                        _coords[5]/factor,
                        _coords[6]/factor,
                        _coords[7]/factor,
                        _coords[8]/factor);
#endif

}



template <typename T>
inline
TypeTensor<T> TypeTensor<T>::transpose() const
{
#if LIBMESH_DIM == 1
  return TypeTensor(_coords[0]);
#endif

#if LIBMESH_DIM == 2
  return TypeTensor(_coords[0],
                    _coords[2],
                    _coords[1],
                    _coords[3]);
#endif

#if LIBMESH_DIM == 3
  return TypeTensor(_coords[0],
                    _coords[3],
                    _coords[6],
                    _coords[1],
                    _coords[4],
                    _coords[7],
                    _coords[2],
                    _coords[5],
                    _coords[8]);
#endif
}



template <typename T>
inline
TypeTensor<T> TypeTensor<T>::inverse() const
{
#if LIBMESH_DIM == 1
  if (_coords[0] == static_cast<T>(0.))
    libmesh_convergence_failure();
  return TypeTensor(1. / _coords[0]);
#endif

#if LIBMESH_DIM == 2
  // Get convenient reference to this.
  const TypeTensor<T> & A = *this;

  // Use temporary variables, avoid multiple accesses
  T a = A(0,0), b = A(0,1),
    c = A(1,0), d = A(1,1);

  // Make sure det = ad - bc is not zero
  T my_det = a*d - b*c;

  if (my_det == static_cast<T>(0.))
    libmesh_convergence_failure();

  return TypeTensor(d/my_det, -b/my_det, -c/my_det, a/my_det);
#endif

#if LIBMESH_DIM == 3
  // Get convenient reference to this.
  const TypeTensor<T> & A = *this;

  T a11 = A(0,0), a12 = A(0,1), a13 = A(0,2),
    a21 = A(1,0), a22 = A(1,1), a23 = A(1,2),
    a31 = A(2,0), a32 = A(2,1), a33 = A(2,2);

  T my_det = a11*(a33*a22-a32*a23) - a21*(a33*a12-a32*a13) + a31*(a23*a12-a22*a13);

  if (my_det == static_cast<T>(0.))
    libmesh_convergence_failure();

  // Inline comment characters are for lining up columns.
  return TypeTensor(  (a33*a22-a32*a23)/my_det, -(a33*a12-a32*a13)/my_det,  (a23*a12-a22*a13)/my_det,
                     -(a33*a21-a31*a23)/my_det,  (a33*a11-a31*a13)/my_det, -(a23*a11-a21*a13)/my_det,
                      (a32*a21-a31*a22)/my_det, -(a32*a11-a31*a12)/my_det,  (a22*a11-a21*a12)/my_det);
#endif
}



template <typename T>
inline
void TypeTensor<T>::solve(const TypeVector<T> & b, TypeVector<T> & x) const
{
#if LIBMESH_DIM == 1
  if (_coords[0] == static_cast<T>(0.))
    libmesh_convergence_failure();
  x(0) = b(0) / _coords[0];
#endif

#if LIBMESH_DIM == 2
  T my_det = _coords[0]*_coords[3] - _coords[1]*_coords[2];

  if (my_det == static_cast<T>(0.))
    libmesh_convergence_failure();

  T my_det_inv = 1./my_det;

  x(0) = my_det_inv*( _coords[3]*b(0) - _coords[1]*b(1));
  x(1) = my_det_inv*(-_coords[2]*b(0) + _coords[0]*b(1));
#endif

#if LIBMESH_DIM == 3
  T my_det =
    //           a11*(a33       *a22        - a32       *a23)
      _coords[0]*(_coords[8]*_coords[4] - _coords[7]*_coords[5])
    //          -a21*(a33       *a12        - a32       *a13)
     -_coords[3]*(_coords[8]*_coords[1] - _coords[7]*_coords[2]) +
    //          +a31*(a23       *a12        - a22       *a13)
     +_coords[6]*(_coords[5]*_coords[1] - _coords[4]*_coords[2]);

  if (my_det == static_cast<T>(0.))
    libmesh_convergence_failure();

  T my_det_inv = 1./my_det;
  x(0) = my_det_inv*((_coords[8]*_coords[4] - _coords[7]*_coords[5])*b(0) -
                     (_coords[8]*_coords[1] - _coords[7]*_coords[2])*b(1) +
                     (_coords[5]*_coords[1] - _coords[4]*_coords[2])*b(2));

  x(1) = my_det_inv*((_coords[6]*_coords[5] - _coords[8]*_coords[3])*b(0) +
                     (_coords[8]*_coords[0] - _coords[6]*_coords[2])*b(1) -
                     (_coords[5]*_coords[0] - _coords[3]*_coords[2])*b(2));

  x(2) = my_det_inv*((_coords[7]*_coords[3] - _coords[6]*_coords[4])*b(0) -
                     (_coords[7]*_coords[0] - _coords[6]*_coords[1])*b(1) +
                     (_coords[4]*_coords[0] - _coords[3]*_coords[1])*b(2));
#endif
}



template <typename T>
inline
const TypeTensor<T> & TypeTensor<T>::operator /= (const T & factor)
{
  libmesh_assert_not_equal_to (factor, static_cast<T>(0.));

  for (unsigned int i=0; i<LIBMESH_DIM*LIBMESH_DIM; i++)
    _coords[i] /= factor;

  return *this;
}




template <typename T>
template <typename T2>
inline
TypeVector<typename CompareTypes<T,T2>::supertype>
TypeTensor<T>::operator * (const TypeVector<T2> & p) const
{
  TypeVector<typename CompareTypes<T,T2>::supertype> returnval;
  for (unsigned int i=0; i<LIBMESH_DIM; i++)
    for (unsigned int j=0; j<LIBMESH_DIM; j++)
      returnval(i) += (*this)(i,j)*p(j);

  return returnval;
}

template <typename T>
template <typename T2>
inline
TypeVector<typename CompareTypes<T,T2>::supertype>
TypeTensor<T>::left_multiply (const TypeVector<T2> & p) const
{
  TypeVector<typename CompareTypes<T,T2>::supertype> returnval;
  for (unsigned int i=0; i<LIBMESH_DIM; i++)
    for (unsigned int j=0; j<LIBMESH_DIM; j++)
      returnval(i) += p(j)*(*this)(j,i);

  return returnval;
}

template <typename T, typename T2>
inline
TypeVector<typename CompareTypes<T,T2>::supertype>
operator * (const TypeVector<T> & a, const TypeTensor<T2> & b)
{
  return b.left_multiply(a);
}

template <typename T>
template <typename T2>
inline
TypeTensor<typename CompareTypes<T, T2>::supertype>
TypeTensor<T>::operator * (const TypeTensor<T2> & p) const
{
  TypeTensor<typename CompareTypes<T, T2>::supertype> returnval;
  for (unsigned int i=0; i<LIBMESH_DIM; i++)
    for (unsigned int j=0; j<LIBMESH_DIM; j++)
      for (unsigned int k=0; k<LIBMESH_DIM; k++)
        returnval(i,j) += (*this)(i,k)*p(k,j);

  return returnval;
}

template <typename T>
template <typename T2>
inline
const TypeTensor<T> & TypeTensor<T>::operator *= (const TypeTensor<T2> & p)
{
  TypeTensor<T> temp;
  for (unsigned int i=0; i<LIBMESH_DIM; i++)
    for (unsigned int j=0; j<LIBMESH_DIM; j++)
      for (unsigned int k=0; k<LIBMESH_DIM; k++)
        temp(i,j) += (*this)(i,k)*p(k,j);

  this->assign(temp);
  return *this;
}


template <typename T>
template <typename T2>
inline
typename CompareTypes<T,T2>::supertype
TypeTensor<T>::contract (const TypeTensor<T2> & t) const
{
  typename CompareTypes<T,T2>::supertype sum = 0.;
  for (unsigned int i=0; i<LIBMESH_DIM*LIBMESH_DIM; i++)
    sum += _coords[i]*t._coords[i];
  return sum;
}



template <typename T>
inline
auto TypeTensor<T>::norm() const
{
  using std::sqrt;
  return sqrt(this->norm_sq());
}


template <typename T>
inline
bool TypeTensor<T>::is_zero() const
{
  for (const auto & val : _coords)
    if (val != T(0))
      return false;
  return true;
}

template <typename T>
inline
T TypeTensor<T>::det() const
{
#if LIBMESH_DIM == 1
  return _coords[0];
#endif

#if LIBMESH_DIM == 2
  return (_coords[0] * _coords[3]
          - _coords[1] * _coords[2]);
#endif

#if LIBMESH_DIM == 3
  return (_coords[0] * _coords[4] * _coords[8]
          + _coords[1] * _coords[5] * _coords[6]
          + _coords[2] * _coords[3] * _coords[7]
          - _coords[0] * _coords[5] * _coords[7]
          - _coords[1] * _coords[3] * _coords[8]
          - _coords[2] * _coords[4] * _coords[6]);
#endif
}

template <typename T>
inline
T TypeTensor<T>::tr() const
{
#if LIBMESH_DIM == 1
  return _coords[0];
#endif

#if LIBMESH_DIM == 2
  return _coords[0] + _coords[3];
#endif

#if LIBMESH_DIM == 3
  return _coords[0] + _coords[4] + _coords[8];
#endif
}

template <typename T>
inline
void TypeTensor<T>::zero()
{
  for (unsigned int i=0; i<LIBMESH_DIM*LIBMESH_DIM; i++)
    _coords[i] = 0.;
}



template <typename T>
inline
auto TypeTensor<T>::norm_sq () const
{
  Real sum = 0.;
  for (unsigned int i=0; i<LIBMESH_DIM*LIBMESH_DIM; i++)
    sum += TensorTools::norm_sq(_coords[i]);
  return sum;
}



template <typename T>
inline
bool TypeTensor<T>::operator == (const TypeTensor<T> & rhs) const
{
#if LIBMESH_DIM == 1
  return (std::abs(_coords[0] - rhs._coords[0])
          < TOLERANCE);
#endif

#if LIBMESH_DIM == 2
  return ((std::abs(_coords[0] - rhs._coords[0]) +
           std::abs(_coords[1] - rhs._coords[1]) +
           std::abs(_coords[2] - rhs._coords[2]) +
           std::abs(_coords[3] - rhs._coords[3]))
          < 4.*TOLERANCE);
#endif

#if LIBMESH_DIM == 3
  return ((std::abs(_coords[0] - rhs._coords[0]) +
           std::abs(_coords[1] - rhs._coords[1]) +
           std::abs(_coords[2] - rhs._coords[2]) +
           std::abs(_coords[3] - rhs._coords[3]) +
           std::abs(_coords[4] - rhs._coords[4]) +
           std::abs(_coords[5] - rhs._coords[5]) +
           std::abs(_coords[6] - rhs._coords[6]) +
           std::abs(_coords[7] - rhs._coords[7]) +
           std::abs(_coords[8] - rhs._coords[8]))
          < 9.*TOLERANCE);
#endif

}

template <typename T>
void TypeTensor<T>::print(std::ostream & os) const
{
#if LIBMESH_DIM == 1

  os << "x=" << (*this)(0,0) << std::endl;

#endif
#if LIBMESH_DIM == 2

  os << "(xx,xy)=("
     << std::setw(8) << (*this)(0,0) << ", "
     << std::setw(8) << (*this)(0,1) << ")"
     << std::endl;
  os << "(yx,yy)=("
     << std::setw(8) << (*this)(1,0) << ", "
     << std::setw(8) << (*this)(1,1) << ")"
     << std::endl;

#endif
#if LIBMESH_DIM == 3

  os <<  "(xx,xy,xz)=("
     << std::setw(8) << (*this)(0,0) << ", "
     << std::setw(8) << (*this)(0,1) << ", "
     << std::setw(8) << (*this)(0,2) << ")"
     << std::endl;
  os <<  "(yx,yy,yz)=("
     << std::setw(8) << (*this)(1,0) << ", "
     << std::setw(8) << (*this)(1,1) << ", "
     << std::setw(8) << (*this)(1,2) << ")"
     << std::endl;
  os <<  "(zx,zy,zz)=("
     << std::setw(8) << (*this)(2,0) << ", "
     << std::setw(8) << (*this)(2,1) << ", "
     << std::setw(8) << (*this)(2,2) << ")"
     << std::endl;
#endif
}

template <typename T, typename T2>
inline
TypeTensor<typename CompareTypes<T, T2>::supertype>
outer_product(const TypeVector<T> & a, const TypeVector<T2> & b)
{
  TypeTensor<typename CompareTypes<T, T2>::supertype> ret;
  for (unsigned int i=0; i<LIBMESH_DIM; i++)
    for (unsigned int j=0; j<LIBMESH_DIM; j++)
      ret(i,j) = a(i) * libmesh_conj(b(j));

  return ret;
}


} // namespace libMesh

#ifdef LIBMESH_HAVE_METAPHYSICL
namespace MetaPhysicL
{
template <typename T>
struct RawType<libMesh::TypeTensor<T>>
{
  typedef libMesh::TypeTensor<typename RawType<T>::value_type> value_type;

  static value_type value (const libMesh::TypeTensor<T> & in)
  {
    value_type ret;
    for (unsigned int i = 0; i < LIBMESH_DIM; ++i)
      for (unsigned int j = 0; j < LIBMESH_DIM; ++j)
        ret(i,j) = raw_value(in(i,j));

    return ret;
  }
};

template <typename T, typename U>
struct ReplaceAlgebraicType<libMesh::TypeTensor<T>, U>
{
  typedef U type;
};
}
#endif

#endif // LIBMESH_TYPE_TENSOR_H