Eigen/src/IterativeLinearSolvers/ConjugateGradient.h

// 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

// IWYU pragma: private
#include "./InternalHeaderCheck.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) {
  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 = numext::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 = numext::sqrt(residualNorm2 / rhsNorm2);
  iters = i;
}

}  // namespace internal

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;
};

}  // namespace internal

/** \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::m_error;
  using Base::m_info;
  using Base::m_isInitialized;
  using Base::m_iterations;
  using Base::matrix;

 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 std::conditional_t<TransposeInput, Transpose<const ActualMatrixType>, ActualMatrixType const&>
        RowMajorWrapper;
    EIGEN_STATIC_ASSERT(internal::check_implication(MatrixWrapper::MatrixFree, UpLo == (Lower | Upper)),
                        MATRIX_FREE_CONJUGATE_GRADIENT_IS_COMPATIBLE_WITH_UPPER_UNION_LOWER_MODE_ONLY);
    typedef std::conditional_t<UpLo == (Lower | Upper), RowMajorWrapper,
                               typename MatrixWrapper::template ConstSelfAdjointViewReturnType<UpLo>::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