cart-elc

ConjugateGradient.h (8887B)

---

 // This file is part of Eigen, a lightweight C++ template library
 // for linear algebra.
 //
 // Copyright (C) 2011-2014 Gael Guennebaud <gael.guennebaud@inria.fr>
 //
 // This Source Code Form is subject to the terms of the Mozilla
 // Public License v. 2.0. If a copy of the MPL was not distributed
 // with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
 
 #ifndef EIGEN_CONJUGATE_GRADIENT_H
 #define EIGEN_CONJUGATE_GRADIENT_H
 
 namespace Eigen { 
 
 namespace internal {
 
 /** \internal Low-level conjugate gradient algorithm
   * \param mat The matrix A
   * \param rhs The right hand side vector b
   * \param x On input and initial solution, on output the computed solution.
   * \param precond A preconditioner being able to efficiently solve for an
   *                approximation of Ax=b (regardless of b)
   * \param iters On input the max number of iteration, on output the number of performed iterations.
   * \param tol_error On input the tolerance error, on output an estimation of the relative error.
   */
 template<typename MatrixType, typename Rhs, typename Dest, typename Preconditioner>
 EIGEN_DONT_INLINE
 void conjugate_gradient(const MatrixType& mat, const Rhs& rhs, Dest& x,
                         const Preconditioner& precond, Index& iters,
                         typename Dest::RealScalar& tol_error)
 {
   using std::sqrt;
   using std::abs;
   typedef typename Dest::RealScalar RealScalar;
   typedef typename Dest::Scalar Scalar;
   typedef Matrix<Scalar,Dynamic,1> VectorType;
   
   RealScalar tol = tol_error;
   Index maxIters = iters;
   
   Index n = mat.cols();
 
   VectorType residual = rhs - mat * x; //initial residual
 
   RealScalar rhsNorm2 = rhs.squaredNorm();
   if(rhsNorm2 == 0) 
   {
     x.setZero();
     iters = 0;
     tol_error = 0;
     return;
   }
   const RealScalar considerAsZero = (std::numeric_limits<RealScalar>::min)();
   RealScalar threshold = numext::maxi(RealScalar(tol*tol*rhsNorm2),considerAsZero);
   RealScalar residualNorm2 = residual.squaredNorm();
   if (residualNorm2 < threshold)
   {
     iters = 0;
     tol_error = sqrt(residualNorm2 / rhsNorm2);
     return;
   }
 
   VectorType p(n);
   p = precond.solve(residual);      // initial search direction
 
   VectorType z(n), tmp(n);
   RealScalar absNew = numext::real(residual.dot(p));  // the square of the absolute value of r scaled by invM
   Index i = 0;
   while(i < maxIters)
   {
     tmp.noalias() = mat * p;                    // the bottleneck of the algorithm
 
     Scalar alpha = absNew / p.dot(tmp);         // the amount we travel on dir
     x += alpha * p;                             // update solution
     residual -= alpha * tmp;                    // update residual
     
     residualNorm2 = residual.squaredNorm();
     if(residualNorm2 < threshold)
       break;
     
     z = precond.solve(residual);                // approximately solve for "A z = residual"
 
     RealScalar absOld = absNew;
     absNew = numext::real(residual.dot(z));     // update the absolute value of r
     RealScalar beta = absNew / absOld;          // calculate the Gram-Schmidt value used to create the new search direction
     p = z + beta * p;                           // update search direction
     i++;
   }
   tol_error = sqrt(residualNorm2 / rhsNorm2);
   iters = i;
 }
 
 }
 
 template< typename _MatrixType, int _UpLo=Lower,
           typename _Preconditioner = DiagonalPreconditioner<typename _MatrixType::Scalar> >
 class ConjugateGradient;
 
 namespace internal {
 
 template< typename _MatrixType, int _UpLo, typename _Preconditioner>
 struct traits<ConjugateGradient<_MatrixType,_UpLo,_Preconditioner> >
 {
   typedef _MatrixType MatrixType;
   typedef _Preconditioner Preconditioner;
 };
 
 }
 
 /** \ingroup IterativeLinearSolvers_Module
   * \brief A conjugate gradient solver for sparse (or dense) self-adjoint problems
   *
   * This class allows to solve for A.x = b linear problems using an iterative conjugate gradient algorithm.
   * The matrix A must be selfadjoint. The matrix A and the vectors x and b can be either dense or sparse.
   *
   * \tparam _MatrixType the type of the matrix A, can be a dense or a sparse matrix.
   * \tparam _UpLo the triangular part that will be used for the computations. It can be Lower,
   *               \c Upper, or \c Lower|Upper in which the full matrix entries will be considered.
   *               Default is \c Lower, best performance is \c Lower|Upper.
   * \tparam _Preconditioner the type of the preconditioner. Default is DiagonalPreconditioner
   *
   * \implsparsesolverconcept
   *
   * The maximal number of iterations and tolerance value can be controlled via the setMaxIterations()
   * and setTolerance() methods. The defaults are the size of the problem for the maximal number of iterations
   * and NumTraits<Scalar>::epsilon() for the tolerance.
   * 
   * The tolerance corresponds to the relative residual error: |Ax-b|/|b|
   * 
   * \b Performance: Even though the default value of \c _UpLo is \c Lower, significantly higher performance is
   * achieved when using a complete matrix and \b Lower|Upper as the \a _UpLo template parameter. Moreover, in this
   * case multi-threading can be exploited if the user code is compiled with OpenMP enabled.
   * See \ref TopicMultiThreading for details.
   * 
   * This class can be used as the direct solver classes. Here is a typical usage example:
     \code
     int n = 10000;
     VectorXd x(n), b(n);
     SparseMatrix<double> A(n,n);
     // fill A and b
     ConjugateGradient<SparseMatrix<double>, Lower|Upper> cg;
     cg.compute(A);
     x = cg.solve(b);
     std::cout << "#iterations:     " << cg.iterations() << std::endl;
     std::cout << "estimated error: " << cg.error()      << std::endl;
     // update b, and solve again
     x = cg.solve(b);
     \endcode
   * 
   * By default the iterations start with x=0 as an initial guess of the solution.
   * One can control the start using the solveWithGuess() method.
   * 
   * ConjugateGradient can also be used in a matrix-free context, see the following \link MatrixfreeSolverExample example \endlink.
   *
   * \sa class LeastSquaresConjugateGradient, class SimplicialCholesky, DiagonalPreconditioner, IdentityPreconditioner
   */
 template< typename _MatrixType, int _UpLo, typename _Preconditioner>
 class ConjugateGradient : public IterativeSolverBase<ConjugateGradient<_MatrixType,_UpLo,_Preconditioner> >
 {
   typedef IterativeSolverBase<ConjugateGradient> Base;
   using Base::matrix;
   using Base::m_error;
   using Base::m_iterations;
   using Base::m_info;
   using Base::m_isInitialized;
 public:
   typedef _MatrixType MatrixType;
   typedef typename MatrixType::Scalar Scalar;
   typedef typename MatrixType::RealScalar RealScalar;
   typedef _Preconditioner Preconditioner;
 
   enum {
     UpLo = _UpLo
   };
 
 public:
 
   /** Default constructor. */
   ConjugateGradient() : Base() {}
 
   /** Initialize the solver with matrix \a A for further \c Ax=b solving.
     * 
     * This constructor is a shortcut for the default constructor followed
     * by a call to compute().
     * 
     * \warning this class stores a reference to the matrix A as well as some
     * precomputed values that depend on it. Therefore, if \a A is changed
     * this class becomes invalid. Call compute() to update it with the new
     * matrix A, or modify a copy of A.
     */
   template<typename MatrixDerived>
   explicit ConjugateGradient(const EigenBase<MatrixDerived>& A) : Base(A.derived()) {}
 
   ~ConjugateGradient() {}
 
   /** \internal */
   template<typename Rhs,typename Dest>
   void _solve_vector_with_guess_impl(const Rhs& b, Dest& x) const
   {
     typedef typename Base::MatrixWrapper MatrixWrapper;
     typedef typename Base::ActualMatrixType ActualMatrixType;
     enum {
       TransposeInput  =   (!MatrixWrapper::MatrixFree)
                       &&  (UpLo==(Lower|Upper))
                       &&  (!MatrixType::IsRowMajor)
                       &&  (!NumTraits<Scalar>::IsComplex)
     };
     typedef typename internal::conditional<TransposeInput,Transpose<const ActualMatrixType>, ActualMatrixType const&>::type RowMajorWrapper;
     EIGEN_STATIC_ASSERT(EIGEN_IMPLIES(MatrixWrapper::MatrixFree,UpLo==(Lower|Upper)),MATRIX_FREE_CONJUGATE_GRADIENT_IS_COMPATIBLE_WITH_UPPER_UNION_LOWER_MODE_ONLY);
     typedef typename internal::conditional<UpLo==(Lower|Upper),
                                            RowMajorWrapper,
                                            typename MatrixWrapper::template ConstSelfAdjointViewReturnType<UpLo>::Type
                                           >::type SelfAdjointWrapper;
 
     m_iterations = Base::maxIterations();
     m_error = Base::m_tolerance;
 
     RowMajorWrapper row_mat(matrix());
     internal::conjugate_gradient(SelfAdjointWrapper(row_mat), b, x, Base::m_preconditioner, m_iterations, m_error);
     m_info = m_error <= Base::m_tolerance ? Success : NoConvergence;
   }
 
 protected:
 
 };
 
 } // end namespace Eigen
 
 #endif // EIGEN_CONJUGATE_GRADIENT_H