1 | // This file is part of Eigen, a lightweight C++ template library
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2 | // for linear algebra.
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3 | //
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4 | // Copyright (C) 2011 Gael Guennebaud <gael.guennebaud@inria.fr>
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5 | // Copyright (C) 2012 Désiré Nuentsa-Wakam <desire.nuentsa_wakam@inria.fr>
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6 | //
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7 | // This Source Code Form is subject to the terms of the Mozilla
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8 | // Public License v. 2.0. If a copy of the MPL was not distributed
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9 | // with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
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10 |
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11 | #ifndef EIGEN_BICGSTAB_H
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12 | #define EIGEN_BICGSTAB_H
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13 |
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14 | namespace Eigen {
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15 |
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16 | namespace internal {
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17 |
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18 | /** \internal Low-level bi conjugate gradient stabilized algorithm
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19 | * \param mat The matrix A
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20 | * \param rhs The right hand side vector b
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21 | * \param x On input and initial solution, on output the computed solution.
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22 | * \param precond A preconditioner being able to efficiently solve for an
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23 | * approximation of Ax=b (regardless of b)
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24 | * \param iters On input the max number of iteration, on output the number of performed iterations.
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25 | * \param tol_error On input the tolerance error, on output an estimation of the relative error.
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26 | * \return false in the case of numerical issue, for example a break down of BiCGSTAB.
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27 | */
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28 | template<typename MatrixType, typename Rhs, typename Dest, typename Preconditioner>
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29 | bool bicgstab(const MatrixType& mat, const Rhs& rhs, Dest& x,
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30 | const Preconditioner& precond, int& iters,
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31 | typename Dest::RealScalar& tol_error)
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32 | {
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33 | using std::sqrt;
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34 | using std::abs;
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35 | typedef typename Dest::RealScalar RealScalar;
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36 | typedef typename Dest::Scalar Scalar;
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37 | typedef Matrix<Scalar,Dynamic,1> VectorType;
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38 | RealScalar tol = tol_error;
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39 | int maxIters = iters;
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40 |
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41 | int n = mat.cols();
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42 | VectorType r = rhs - mat * x;
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43 | VectorType r0 = r;
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44 |
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45 | RealScalar r0_sqnorm = r0.squaredNorm();
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46 | RealScalar rhs_sqnorm = rhs.squaredNorm();
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47 | if(rhs_sqnorm == 0)
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48 | {
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49 | x.setZero();
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50 | return true;
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51 | }
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52 | Scalar rho = 1;
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53 | Scalar alpha = 1;
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54 | Scalar w = 1;
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55 |
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56 | VectorType v = VectorType::Zero(n), p = VectorType::Zero(n);
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57 | VectorType y(n), z(n);
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58 | VectorType kt(n), ks(n);
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59 |
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60 | VectorType s(n), t(n);
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61 |
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62 | RealScalar tol2 = tol*tol;
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63 | RealScalar eps2 = NumTraits<Scalar>::epsilon()*NumTraits<Scalar>::epsilon();
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64 | int i = 0;
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65 | int restarts = 0;
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66 |
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67 | while ( r.squaredNorm()/rhs_sqnorm > tol2 && i<maxIters )
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68 | {
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69 | Scalar rho_old = rho;
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70 |
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71 | rho = r0.dot(r);
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72 | if (abs(rho) < eps2*r0_sqnorm)
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73 | {
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74 | // The new residual vector became too orthogonal to the arbitrarily choosen direction r0
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75 | // Let's restart with a new r0:
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76 | r0 = r;
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77 | rho = r0_sqnorm = r.squaredNorm();
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78 | if(restarts++ == 0)
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79 | i = 0;
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80 | }
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81 | Scalar beta = (rho/rho_old) * (alpha / w);
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82 | p = r + beta * (p - w * v);
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83 |
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84 | y = precond.solve(p);
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85 |
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86 | v.noalias() = mat * y;
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87 |
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88 | alpha = rho / r0.dot(v);
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89 | s = r - alpha * v;
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90 |
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91 | z = precond.solve(s);
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92 | t.noalias() = mat * z;
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93 |
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94 | RealScalar tmp = t.squaredNorm();
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95 | if(tmp>RealScalar(0))
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96 | w = t.dot(s) / tmp;
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97 | else
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98 | w = Scalar(0);
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99 | x += alpha * y + w * z;
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100 | r = s - w * t;
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101 | ++i;
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102 | }
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103 | tol_error = sqrt(r.squaredNorm()/rhs_sqnorm);
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104 | iters = i;
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105 | return true;
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106 | }
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107 |
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108 | }
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109 |
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110 | template< typename _MatrixType,
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111 | typename _Preconditioner = DiagonalPreconditioner<typename _MatrixType::Scalar> >
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112 | class BiCGSTAB;
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113 |
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114 | namespace internal {
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115 |
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116 | template< typename _MatrixType, typename _Preconditioner>
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117 | struct traits<BiCGSTAB<_MatrixType,_Preconditioner> >
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118 | {
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119 | typedef _MatrixType MatrixType;
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120 | typedef _Preconditioner Preconditioner;
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121 | };
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122 |
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123 | }
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124 |
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125 | /** \ingroup IterativeLinearSolvers_Module
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126 | * \brief A bi conjugate gradient stabilized solver for sparse square problems
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127 | *
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128 | * This class allows to solve for A.x = b sparse linear problems using a bi conjugate gradient
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129 | * stabilized algorithm. The vectors x and b can be either dense or sparse.
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130 | *
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131 | * \tparam _MatrixType the type of the sparse matrix A, can be a dense or a sparse matrix.
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132 | * \tparam _Preconditioner the type of the preconditioner. Default is DiagonalPreconditioner
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133 | *
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134 | * The maximal number of iterations and tolerance value can be controlled via the setMaxIterations()
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135 | * and setTolerance() methods. The defaults are the size of the problem for the maximal number of iterations
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136 | * and NumTraits<Scalar>::epsilon() for the tolerance.
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137 | *
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138 | * This class can be used as the direct solver classes. Here is a typical usage example:
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139 | * \code
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140 | * int n = 10000;
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141 | * VectorXd x(n), b(n);
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142 | * SparseMatrix<double> A(n,n);
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143 | * // fill A and b
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144 | * BiCGSTAB<SparseMatrix<double> > solver;
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145 | * solver.compute(A);
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146 | * x = solver.solve(b);
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147 | * std::cout << "#iterations: " << solver.iterations() << std::endl;
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148 | * std::cout << "estimated error: " << solver.error() << std::endl;
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149 | * // update b, and solve again
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150 | * x = solver.solve(b);
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151 | * \endcode
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152 | *
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153 | * By default the iterations start with x=0 as an initial guess of the solution.
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154 | * One can control the start using the solveWithGuess() method.
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155 | *
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156 | * \sa class SimplicialCholesky, DiagonalPreconditioner, IdentityPreconditioner
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157 | */
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158 | template< typename _MatrixType, typename _Preconditioner>
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159 | class BiCGSTAB : public IterativeSolverBase<BiCGSTAB<_MatrixType,_Preconditioner> >
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160 | {
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161 | typedef IterativeSolverBase<BiCGSTAB> Base;
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162 | using Base::mp_matrix;
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163 | using Base::m_error;
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164 | using Base::m_iterations;
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165 | using Base::m_info;
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166 | using Base::m_isInitialized;
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167 | public:
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168 | typedef _MatrixType MatrixType;
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169 | typedef typename MatrixType::Scalar Scalar;
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170 | typedef typename MatrixType::Index Index;
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171 | typedef typename MatrixType::RealScalar RealScalar;
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172 | typedef _Preconditioner Preconditioner;
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173 |
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174 | public:
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175 |
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176 | /** Default constructor. */
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177 | BiCGSTAB() : Base() {}
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178 |
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179 | /** Initialize the solver with matrix \a A for further \c Ax=b solving.
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180 | *
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181 | * This constructor is a shortcut for the default constructor followed
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182 | * by a call to compute().
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183 | *
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184 | * \warning this class stores a reference to the matrix A as well as some
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185 | * precomputed values that depend on it. Therefore, if \a A is changed
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186 | * this class becomes invalid. Call compute() to update it with the new
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187 | * matrix A, or modify a copy of A.
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188 | */
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189 | template<typename MatrixDerived>
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190 | explicit BiCGSTAB(const EigenBase<MatrixDerived>& A) : Base(A.derived()) {}
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191 |
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192 | ~BiCGSTAB() {}
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193 |
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194 | /** \returns the solution x of \f$ A x = b \f$ using the current decomposition of A
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195 | * \a x0 as an initial solution.
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196 | *
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197 | * \sa compute()
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198 | */
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199 | template<typename Rhs,typename Guess>
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200 | inline const internal::solve_retval_with_guess<BiCGSTAB, Rhs, Guess>
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201 | solveWithGuess(const MatrixBase<Rhs>& b, const Guess& x0) const
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202 | {
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203 | eigen_assert(m_isInitialized && "BiCGSTAB is not initialized.");
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204 | eigen_assert(Base::rows()==b.rows()
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205 | && "BiCGSTAB::solve(): invalid number of rows of the right hand side matrix b");
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206 | return internal::solve_retval_with_guess
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207 | <BiCGSTAB, Rhs, Guess>(*this, b.derived(), x0);
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208 | }
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209 |
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210 | /** \internal */
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211 | template<typename Rhs,typename Dest>
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212 | void _solveWithGuess(const Rhs& b, Dest& x) const
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213 | {
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214 | bool failed = false;
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215 | for(int j=0; j<b.cols(); ++j)
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216 | {
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217 | m_iterations = Base::maxIterations();
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218 | m_error = Base::m_tolerance;
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219 |
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220 | typename Dest::ColXpr xj(x,j);
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221 | if(!internal::bicgstab(*mp_matrix, b.col(j), xj, Base::m_preconditioner, m_iterations, m_error))
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222 | failed = true;
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223 | }
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224 | m_info = failed ? NumericalIssue
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225 | : m_error <= Base::m_tolerance ? Success
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226 | : NoConvergence;
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227 | m_isInitialized = true;
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228 | }
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229 |
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230 | /** \internal */
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231 | template<typename Rhs,typename Dest>
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232 | void _solve(const Rhs& b, Dest& x) const
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233 | {
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234 | // x.setZero();
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235 | x = b;
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236 | _solveWithGuess(b,x);
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237 | }
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238 |
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239 | protected:
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240 |
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241 | };
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242 |
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243 |
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244 | namespace internal {
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245 |
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246 | template<typename _MatrixType, typename _Preconditioner, typename Rhs>
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247 | struct solve_retval<BiCGSTAB<_MatrixType, _Preconditioner>, Rhs>
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248 | : solve_retval_base<BiCGSTAB<_MatrixType, _Preconditioner>, Rhs>
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249 | {
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250 | typedef BiCGSTAB<_MatrixType, _Preconditioner> Dec;
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251 | EIGEN_MAKE_SOLVE_HELPERS(Dec,Rhs)
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252 |
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253 | template<typename Dest> void evalTo(Dest& dst) const
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254 | {
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255 | dec()._solve(rhs(),dst);
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256 | }
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257 | };
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258 |
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259 | } // end namespace internal
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260 |
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261 | } // end namespace Eigen
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262 |
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263 | #endif // EIGEN_BICGSTAB_H
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