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NekLinAlgAlgorithms.hpp
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2 //
3 // File: NekLinAlgAlgorithms.hpp
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10 // Department of Aeronautics, Imperial College London (UK), and Scientific
11 // Computing and Imaging Institute, University of Utah (USA).
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31 //
32 // Description: Linear Algebra Algorithms
33 //
34 ///////////////////////////////////////////////////////////////////////////////
35 
36 #ifndef NEKTAR_LIB_UTILITIES_LINEAR_ALGEBRA_NEK_LIN_ALG_ALGORITHMS_HPP
37 #define NEKTAR_LIB_UTILITIES_LINEAR_ALGEBRA_NEK_LIN_ALG_ALGORITHMS_HPP
38 
41 #include <vector>
42 
43 namespace Nektar
44 {
45  /// \brief Calculates the orthogonal, normalized vectors from
46  /// linearly independent input.
47  /// \return A list of ortogonalized vectors corresponding to the
48  /// input vectors. If the size of the output doesn't
49  /// match the size of the input then an error occurred.
50  /// This algorithm is taken from "Parallel Scientific Computing in
51  /// C++ and MPI", Karniadakis and Kirby, page 55.
52  template<typename DataType>
53  std::vector<NekVector<DataType> >
55  {
56  typedef NekVector<DataType> VectorType;
58 
59  //typename dim = x[0].GetDimension();
60  unsigned int dim = x[0].GetDimension();
61  std::vector<VectorType> q(x.size(), VectorType());
62 
63  // This matrix holds the r_ij values. Using the matrix object
64  // is a convenience since it provides a 2D access to a table
65  // of values.
66  MatrixType r(dim, dim);
67  r(0,0) = x[0].L2Norm();
68 
69  if( r(0,0) == DataType(0) )
70  {
71  return q;
72  }
73 
74  q[0] = x[0]/r(0,0);
75 
76  for(unsigned int j = 1; j < x.size(); ++j)
77  {
78  for(unsigned int i = 0; i <= j-1; ++i)
79  {
80  r(i,j) = q[i].Dot(x[j]);
81  }
82 
83  VectorType y = x[j];
84  for(unsigned int i = 0; i <= j-1; ++i)
85  {
86  y = y - r(i,j)*q[i];
87  }
88 
89  r(j,j) = y.L2Norm();
90  if( r(j,j) == DataType(0) )
91  {
92  return q;
93  }
94 
95  q[j] = y/r(j,j);
96  }
97 
98  return q;
99  }
100 }
101 
102 #endif //NEKTAR_LIB_UTILITIES_LINEAR_ALGEBRA_NEK_LIN_ALG_ALGORITHMS_HPP
103 
std::vector< NekVector< DataType > > GramSchmidtOrthogonalization(const std::vector< NekVector< DataType > > &x)
Calculates the orthogonal, normalized vectors from linearly independent input.