text: /* -*- C++ -*- ------------------------------------------------------------
   
  Copyright (c) 2007 Jesse Anders and Demian Nave http://cmldev.net/
  
  The Configurable Math Library (CML) is distributed under the terms of the
  Boost Software License, v1.0 (see cml/LICENSE for details).
  
   *-----------------------------------------------------------------------*/
  /** @file
   *  @brief Compute the inverse of a matrix by LU factorization.
   */
  
  #ifndef matrix_inverse_h
  #define matrix_inverse_h
  
  #include <vector>
  #include <cml/matrix/lu.h>
  
  namespace cml {
  namespace detail {
  
  /* Need to use a functional, since template functions cannot be
   * specialized. _tag is used to specialize based upon dimension:
   */
  template<typename MatT, int _tag> struct inverse_f;
  
  /* @todo: Reciprocal optimization for division by determinant.
   */
  
  /* 2x2 inverse.  Despite being marked for fixed_size matrices, this can
   * be used for dynamic-sized ones also:
   */
  template<typename MatT>
  struct inverse_f<MatT,2>
  {
      typename MatT::temporary_type operator()(const MatT& M) const
      {
          typedef typename MatT::temporary_type temporary_type;
          typedef typename temporary_type::value_type value_type;
  
          /* Matrix containing the inverse: */
          temporary_type Z;
          cml::et::detail::Resize(Z,2,2);
  
          /* Compute determinant and inverse: */
          value_type D = value_type(1) / (M(0,0)*M(1,1) - M(0,1)*M(1,0));
          Z(0,0) =   M(1,1)*D; Z(0,1) = - M(0,1)*D;
          Z(1,0) = - M(1,0)*D; Z(1,1) =   M(0,0)*D;
  
          return Z;
      }
  };
  
  /* 3x3 inverse.  Despite being marked for fixed_size matrices, this can
   * be used for dynamic-sized ones also:
   */
  template<typename MatT>
  struct inverse_f<MatT,3>
  {
      /*     [00 01 02]
       * M = [10 11 12]
       *     [20 21 22]
       */
      typename MatT::temporary_type operator()(const MatT& M) const
      {
          /* Shorthand. */
          typedef typename MatT::value_type value_type;
  
          /* Compute cofactors for each entry: */
          value_type m_00 = M(1,1)*M(2,2) - M(1,2)*M(2,1);
          value_type m_01 = M(1,2)*M(2,0) - M(1,0)*M(2,2);
          value_type m_02 = M(1,0)*M(2,1) - M(1,1)*M(2,0);
  
          value_type m_10 = M(0,2)*M(2,1) - M(0,1)*M(2,2);
          value_type m_11 = M(0,0)*M(2,2) - M(0,2)*M(2,0);
          value_type m_12 = M(0,1)*M(2,0) - M(0,0)*M(2,1);
  
          value_type m_20 = M(0,1)*M(1,2) - M(0,2)*M(1,1);
          value_type m_21 = M(0,2)*M(1,0) - M(0,0)*M(1,2);
          value_type m_22 = M(0,0)*M(1,1) - M(0,1)*M(1,0);
  
          /* Compute determinant from the minors: */
          value_type D =
              value_type(1) / (M(0,0)*m_00 + M(0,1)*m_01 + M(0,2)*m_02);
  
          /* Matrix containing the inverse: */
          typename MatT::temporary_type Z;
          cml::et::detail::Resize(Z,3,3);
  
          /* Assign the inverse as (1/D) * (cofactor matrix)^T: */
          Z(0,0) = m_00*D;  Z(0,1) = m_10*D;  Z(0,2) = m_20*D;
          Z(1,0) = m_01*D;  Z(1,1) = m_11*D;  Z(1,2) = m_21*D;
          Z(2,0) = m_02*D;  Z(2,1) = m_12*D;  Z(2,2) = m_22*D;
  
          return Z;
      }
  };
  
  /* 4x4 inverse.  Despite being marked for fixed_size matrices, this can
   * be used for dynamic-sized ones also:
   */
  template<typename MatT>
  struct inverse_f<MatT,4>
  {
      /*     [00 01 02 03]
       * M = [10 11 12 13]
       *     [20 21 22 23]
       *     [30 31 32 33]
       *
       *       |11 12 13|         |10 12 13|
       * C00 = |21 22 23|   C01 = |20 22 23|
       *       |31 32 33|         |30 32 33|
       *
       *       |10 11 13|         |10 11 12|
       * C02 = |20 21 23|   C03 = |20 21 22|
       *       |30 31 33|         |30 31 32|
       */
      typename MatT::temporary_type operator()(const MatT& M) const
      {
          /* Shorthand. */
          typedef typename MatT::value_type value_type;
  
          /* Common cofactors, rows 0,1: */
          value_type m_22_33_23_32 = M(2,2)*M(3,3) - M(2,3)*M(3,2);
          value_type m_23_30_20_33 = M(2,3)*M(3,0) - M(2,0)*M(3,3);
          value_type m_20_31_21_30 = M(2,0)*M(3,1) - M(2,1)*M(3,0);
          value_type m_21_32_22_31 = M(2,1)*M(3,2) - M(2,2)*M(3,1);
          value_type m_23_31_21_33 = M(2,3)*M(3,1) - M(2,1)*M(3,3);
          value_type m_20_32_22_30 = M(2,0)*M(3,2) - M(2,2)*M(3,0);
  
          /* Compute minors: */
          value_type d00
              = M(1,1)*m_22_33_23_32+M(1,2)*m_23_31_21_33+M(1,3)*m_21_32_22_31;
  
          value_type d01
              = M(1,0)*m_22_33_23_32+M(1,2)*m_23_30_20_33+M(1,3)*m_20_32_22_30;
  
          value_type d02
              = M(1,0)*-m_23_31_21_33+M(1,1)*m_23_30_20_33+M(1,3)*m_20_31_21_30;
  
          value_type d03
              = M(1,0)*m_21_32_22_31+M(1,1)*-m_20_32_22_30+M(1,2)*m_20_31_21_30;
  
          /* Compute minors: */
          value_type d10
              = M(0,1)*m_22_33_23_32+M(0,2)*m_23_31_21_33+M(0,3)*m_21_32_22_31;
  
          value_type d11
              = M(0,0)*m_22_33_23_32+M(0,2)*m_23_30_20_33+M(0,3)*m_20_32_22_30;
  
          value_type d12
              = M(0,0)*-m_23_31_21_33+M(0,1)*m_23_30_20_33+M(0,3)*m_20_31_21_30;
  
          value_type d13
              = M(0,0)*m_21_32_22_31+M(0,1)*-m_20_32_22_30+M(0,2)*m_20_31_21_30;
  
          /* Common cofactors, rows 2,3: */
          value_type m_02_13_03_12 = M(0,2)*M(1,3) - M(0,3)*M(1,2);
          value_type m_03_10_00_13 = M(0,3)*M(1,0) - M(0,0)*M(1,3);
          value_type m_00_11_01_10 = M(0,0)*M(1,1) - M(0,1)*M(1,0);
          value_type m_01_12_02_11 = M(0,1)*M(1,2) - M(0,2)*M(1,1);
          value_type m_03_11_01_13 = M(0,3)*M(1,1) - M(0,1)*M(1,3);
          value_type m_00_12_02_10 = M(0,0)*M(1,2) - M(0,2)*M(1,0);
  
          /* Compute minors (uses row 3 as the multipliers instead of row 0,
           * which uses the same signs as row 0):
           */
          value_type d20
              = M(3,1)*m_02_13_03_12+M(3,2)*m_03_11_01_13+M(3,3)*m_01_12_02_11;
  
          value_type d21
              = M(3,0)*m_02_13_03_12+M(3,2)*m_03_10_00_13+M(3,3)*m_00_12_02_10;
  
          value_type d22
              = M(3,0)*-m_03_11_01_13+M(3,1)*m_03_10_00_13+M(3,3)*m_00_11_01_10;
  
          value_type d23
              = M(3,0)*m_01_12_02_11+M(3,1)*-m_00_12_02_10+M(3,2)*m_00_11_01_10;
  
          /* Compute minors: */
          value_type d30
              = M(2,1)*m_02_13_03_12+M(2,2)*m_03_11_01_13+M(2,3)*m_01_12_02_11;
  
          value_type d31
              = M(2,0)*m_02_13_03_12+M(2,2)*m_03_10_00_13+M(2,3)*m_00_12_02_10;
  
          value_type d32
              = M(2,0)*-m_03_11_01_13+M(2,1)*m_03_10_00_13+M(2,3)*m_00_11_01_10;
  
          value_type d33
              = M(2,0)*m_01_12_02_11+M(2,1)*-m_00_12_02_10+M(2,2)*m_00_11_01_10;
  
          /* Finally, compute determinant from the minors, and assign the
           * inverse as (1/D) * (cofactor matrix)^T:
           */
          typename MatT::temporary_type Z;
          cml::et::detail::Resize(Z,4,4);
  
          value_type D = value_type(1) /
              (M(0,0)*d00 - M(0,1)*d01 + M(0,2)*d02 - M(0,3)*d03);
          Z(0,0) = +d00*D; Z(0,1) = -d10*D; Z(0,2) = +d20*D; Z(0,3) = -d30*D;
          Z(1,0) = -d01*D; Z(1,1) = +d11*D; Z(1,2) = -d21*D; Z(1,3) = +d31*D;
          Z(2,0) = +d02*D; Z(2,1) = -d12*D; Z(2,2) = +d22*D; Z(2,3) = -d32*D;
          Z(3,0) = -d03*D; Z(3,1) = +d13*D; Z(3,2) = -d23*D; Z(3,3) = +d33*D;
  
          return Z;
      }
  };
  
  /* If more extensive general linear algebra functionality is offered in
   * future versions it may be useful to make the elementary row and column
   * operations separate functions. For now they're simply performed in place,
   * but the commented-out lines of code show where the calls to these functions
   * should go if and when they become available.
   */
   
  /* @todo: In-place version, and address memory allocation for pivot vector.
   */
  
  /* General NxN inverse by Gauss-Jordan elimination with full pivoting:  */
  template<typename MatT, int _tag>
  struct inverse_f
  {
      typename MatT::temporary_type operator()(const MatT& M) const
      {
          /* Shorthand. */
          typedef typename MatT::value_type value_type;
          
          /* Size of matrix */
          size_t N = M.rows();
  
          /* Matrix containing the inverse: */
          typename MatT::temporary_type Z;
          cml::et::detail::Resize(Z,N,N);
          Z = M;
  
          /* For tracking pivots */
          std::vector<size_t> row_index(N);
          std::vector<size_t> col_index(N);
          std::vector<size_t> pivoted(N,0);
  
          /* For each column */
          for (size_t i = 0; i < N; ++i) {
          
              /* Find the pivot */
              size_t row = 0, col = 0;
              value_type max = value_type(0);
              for (size_t j = 0; j < N; ++j) {
                  if (!pivoted[j]) {
                      for (size_t k = 0; k < N; ++k) {
                          if (!pivoted[k]) {
                              value_type mag = std::fabs(Z(j,k));
                              if (mag > max) {
                                  max = mag;
                                  row = j;
                                  col = k;
                              }
                          }
                      }
                  }
              }
  
              /* TODO: Check max against epsilon here to catch singularity */
  
              row_index[i] = row;
              col_index[i] = col;
  
              /* Swap rows if necessary */
              if (row != col) {
                  /*Z.row_op_swap(row,col);*/
                  for (size_t j = 0; j < Z.cols(); ++j) {
                      std::swap(Z(row,j),Z(col,j));
                  }
              }
              
              /* Process pivot row */
              pivoted[col] = true;
              value_type pivot = Z(col,col);
              Z(col,col) = value_type(1);
              /*Z.row_op_mult(col,value_type(1)/pivot);*/
              value_type k = value_type(1)/pivot;
              for (size_t j = 0; j < Z.cols(); ++j) {
                  Z(col,j) *= k;
              }
  
              /* Process other rows */
              for (size_t j = 0; j < N; ++j) {
                  if (j != col) {
                      value_type mult = -Z(j,col);
                      Z(j,col) = value_type(0);
                      /*Z.row_op_add_mult(col,j,mult);*/
                      for (size_t k = 0; k < Z.cols(); ++k) {
                          Z(j,k) += mult * Z(col,k);
                      }
                  }
              }
          }
  
          /* Swap columns if necessary */
          for (int i = N-1; i >= 0; --i) {
              if (row_index[i] != col_index[i]) {
                  /*Z.col_op_swap(row_index[i],col_index[i]);*/
                  for (size_t j = 0; j < Z.rows(); ++j) {
                      std::swap(Z(j,row_index[i]),Z(j,col_index[i]));
                  }
              }
          }
  
          /* Return result */
          return Z;
      }
  };
  
  /* Inversion by LU factorization is turned off for now due to lack of
   * pivoting in the implementation, but we may switch back to it at some future
   * time.
   */
   
  #if 0
  
  /* General NxN inverse by LU factorization:  */
  template<typename MatT, int _tag>
  struct inverse_f
  {
      typename MatT::temporary_type operator()(const MatT& M) const
      {
          /* Shorthand. */
          typedef typename MatT::value_type value_type;
  
          /* Compute LU factorization: */
          size_t N = M.rows();
          typename MatT::temporary_type LU;
          cml::et::detail::Resize(LU,N,N);
          LU = lu(M);
  
          /* Matrix containing the inverse: */
          typename MatT::temporary_type Z;
          cml::et::detail::Resize(Z,N,N);
  
          typename MatT::col_vector_type v, x;
          cml::et::detail::Resize(v,N);
          cml::et::detail::Resize(x,N);
          for(size_t i = 0; i < N; ++i)
              v[i] = value_type(0);
          /* XXX Need a fill() function here. */
  
          /* Use lu_solve to solve M*x = v for x, where v = [0 ... 1 ... 0]^T: */
          for(size_t i = 0; i < N; ++i) {
              v[i] = 1.;
              x = lu_solve(LU,v);
  
              /* x is column i of the inverse of LU: */
              for(size_t k = 0; k < N; ++ k) {
                  Z(k,i) = x[k];
              }
              v[i] = 0.;
          }
  
          return Z;
      }
  
  };
  
  #endif
  
  /* Note: force_NxN is for checking general NxN inversion against the special-
   * case 2x2, 3x3 and 4x4 code. I'm leaving it in for now since we may need to
   * test the NxN code further if the implementation changes. At some future
   * time when the implementation is stable, everything related to force_NxN can
   * be taken out.
   */
  
  /* Note: Commenting the force_NxN stuff out, but leaving the code here in
   * case we need to do more testing in the future.
   */
  
  /* Generator for the inverse functional for fixed-size matrices: */
  template<typename MatT> typename MatT::temporary_type
  inverse(const MatT& M, fixed_size_tag/*, bool force_NxN*/)
  {
      /* Require a square matrix: */
      cml::et::CheckedSquare(M, fixed_size_tag());
      
      /*
      if (force_NxN) {
          return inverse_f<MatT,0>()(M);
      } else {
      */
      return inverse_f<MatT,MatT::array_rows>()(M);
      /*
      }
      */
  }
  
  /* Generator for the inverse functional for dynamic-size matrices: */
  template<typename MatT> typename MatT::temporary_type
  inverse(const MatT& M, dynamic_size_tag/*, bool force_NxN*/)
  {
      /* Require a square matrix: */
      cml::et::CheckedSquare(M, dynamic_size_tag());
      
      /*
      if (force_NxN) { 
          return inverse_f<MatT,0>()(M);
      } else {
      */
      /* Dispatch based upon the matrix dimension: */
      switch(M.rows()) {
          case 2:  return inverse_f<MatT,2>()(M);     //   2x2
          case 3:  return inverse_f<MatT,3>()(M);     //   3x3
          case 4:  return inverse_f<MatT,4>()(M);     //   4x4
          default: return inverse_f<MatT,0>()(M);     // > 4x4 (or 1x1)
      }
      /*
      }
      */
  }
  
  } // namespace detail
  
  /** Inverse of a matrix. */
  template<typename E, class AT, typename BO, typename L> inline
  typename matrix<E,AT,BO,L>::temporary_type
  inverse(const matrix<E,AT,BO,L>& M/*, bool force_NxN = false*/)
  {
      typedef typename matrix<E,AT,BO,L>::size_tag size_tag;
      return detail::inverse(M,size_tag()/*,force_NxN*/);
  }
  
  /** Inverse of a matrix expression. */
  template<typename XprT> inline
  typename et::MatrixXpr<XprT>::temporary_type
  inverse(const et::MatrixXpr<XprT>& e/*, bool force_NxN = false*/)
  {
      typedef typename et::MatrixXpr<XprT>::size_tag size_tag;
      return detail::inverse(e,size_tag()/*,force_NxN*/);
  }
  
  } // namespace cml
  
  #endif
  
  // -------------------------------------------------------------------------
  // vim:ft=cpp
  