eigen/Eigen/src/Core/Dot.h

// This file is part of Eigen, a lightweight C++ template library
// for linear algebra. Eigen itself is part of the KDE project.
//
// Copyright (C) 2006-2008 Benoit Jacob <jacob@math.jussieu.fr>
//
// Eigen 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 2 or (at your option) any later version.
//
// Eigen 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 Eigen; if not, write to the Free Software Foundation, Inc., 51
// Franklin St, Fifth Floor, Boston, MA 02110-1301 USA.
//
// As a special exception, if other files instantiate templates or use macros
// or functions from this file, or you compile this file and link it
// with other works to produce a work based on this file, this file does not
// by itself cause the resulting work to be covered by the GNU General Public
// License. This exception does not invalidate any other reasons why a work
// based on this file might be covered by the GNU General Public License.

#ifndef EIGEN_DOT_H
#define EIGEN_DOT_H

template<int Index, int Size, typename Derived1, typename Derived2>
struct DotUnroller
{
  static void run(const Derived1 &v1, const Derived2& v2, typename Derived1::Scalar &dot)
  {
    DotUnroller<Index-1, Size, Derived1, Derived2>::run(v1, v2, dot);
    dot += v1.coeff(Index) * ei_conj(v2.coeff(Index));
  }
};

template<int Size, typename Derived1, typename Derived2>
struct DotUnroller<0, Size, Derived1, Derived2>
{
  static void run(const Derived1 &v1, const Derived2& v2, typename Derived1::Scalar &dot)
  {
    dot = v1.coeff(0) * ei_conj(v2.coeff(0));
  }
};

template<int Index, typename Derived1, typename Derived2>
struct DotUnroller<Index, Dynamic, Derived1, Derived2>
{
  static void run(const Derived1&, const Derived2&, typename Derived1::Scalar&) {}
};

// prevent buggy user code from causing an infinite recursion
template<int Index, typename Derived1, typename Derived2>
struct DotUnroller<Index, 0, Derived1, Derived2>
{
  static void run(const Derived1&, const Derived2&, typename Derived1::Scalar&) {}
};

/** \returns the dot product of *this with other.
  *
  * \only_for_vectors
  * 
  * \note If the scalar type is complex numbers, then this function returns the hermitian
  * (sesquilinear) dot product, linear in the first variable and anti-linear in the
  * second variable.
  *
  * \sa norm2(), norm()
  */
template<typename Scalar, typename Derived>
template<typename OtherDerived>
Scalar MatrixBase<Scalar, Derived>::dot(const OtherDerived& other) const
{
  assert(Traits::IsVectorAtCompileTime
      && OtherDerived::Traits::IsVectorAtCompileTime
      && size() == other.size());
  Scalar res;
  if(EIGEN_UNROLLED_LOOPS
  && Traits::SizeAtCompileTime != Dynamic
  && Traits::SizeAtCompileTime <= 16)
    DotUnroller<Traits::SizeAtCompileTime-1, Traits::SizeAtCompileTime,
                Derived, OtherDerived>
      ::run(*static_cast<const Derived*>(this), other, res);
  else
  {
    res = (*this).coeff(0) * ei_conj(other.coeff(0));
    for(int i = 1; i < size(); i++)
      res += (*this).coeff(i)* ei_conj(other.coeff(i));
  }
  return res;
}

/** \returns the squared norm of *this, i.e. the dot product of *this with itself.
  *
  * \only_for_vectors
  *
  * \sa dot(), norm()
  */
template<typename Scalar, typename Derived>
typename NumTraits<Scalar>::Real MatrixBase<Scalar, Derived>::norm2() const
{
  return ei_real(dot(*this));
}

/** \returns the norm of *this, i.e. the square root of the dot product of *this with itself.
  *
  * \only_for_vectors
  *
  * \sa dot(), norm2()
  */
template<typename Scalar, typename Derived>
typename NumTraits<Scalar>::Real MatrixBase<Scalar, Derived>::norm() const
{
  return ei_sqrt(norm2());
}

/** \returns an expression of the quotient of *this by its own norm.
  *
  * \only_for_vectors
  *
  * \sa norm()
  */
template<typename Scalar, typename Derived>
const ScalarMultiple<typename NumTraits<Scalar>::Real, Derived>
MatrixBase<Scalar, Derived>::normalized() const
{
  return (*this) / norm();
}

/** \returns true if *this is approximately orthogonal to \a other,
  *          within the precision given by \a prec.
  *
  * Example: \include MatrixBase_isOrtho_vector.cpp
  * Output: \verbinclude MatrixBase_isOrtho_vector.out
  */
template<typename Scalar, typename Derived>
template<typename OtherDerived>
bool MatrixBase<Scalar, Derived>::isOrtho
(const OtherDerived& other,
 typename NumTraits<Scalar>::Real prec) const
{
  return ei_abs2(dot(other)) <= prec * prec * norm2() * other.norm2();
}

/** \returns true if *this is approximately an unitary matrix,
  *          within the precision given by \a prec. In the case where the \a Scalar
  *          type is real numbers, a unitary matrix is an orthogonal matrix, whence the name.
  *
  * \note This can be used to check whether a family of vectors forms an orthonormal basis.
  *       Indeed, \c m.isOrtho() returns true if and only if the columns of m form an
  *       orthonormal basis.
  *
  * Example: \include MatrixBase_isOrtho_matrix.cpp
  * Output: \verbinclude MatrixBase_isOrtho_matrix.out
  */
template<typename Scalar, typename Derived>
bool MatrixBase<Scalar, Derived>::isOrtho
(typename NumTraits<Scalar>::Real prec) const
{
  for(int i = 0; i < cols(); i++)
  {
    if(!ei_isApprox(col(i).norm2(), static_cast<Scalar>(1), prec))
      return false;
    for(int j = 0; j < i; j++)
      if(!ei_isMuchSmallerThan(col(i).dot(col(j)), static_cast<Scalar>(1), prec))
        return false;
  }
  return true;
}
#endif // EIGEN_DOT_H