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add a conjugate gradient solver
This commit is contained in:
Gael Guennebaud
429
unsupported/Eigen/src/IterativeSolvers/ConjugateGradient.h
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429
unsupported/Eigen/src/IterativeSolvers/ConjugateGradient.h
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// This file is part of Eigen, a lightweight C++ template library
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// for linear algebra.
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//
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// Copyright (C) 2011 Gael Guennebaud <gael.guennebaud@inria.fr>
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//
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// Eigen is free software; you can redistribute it and/or
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// modify it under the terms of the GNU Lesser General Public
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// License as published by the Free Software Foundation; either
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// version 3 of the License, or (at your option) any later version.
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//
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// Alternatively, you can redistribute it and/or
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// modify it under the terms of the GNU General Public License as
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// published by the Free Software Foundation; either version 2 of
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// the License, or (at your option) any later version.
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//
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// Eigen is distributed in the hope that it will be useful, but WITHOUT ANY
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// WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
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// FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License or the
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// GNU General Public License for more details.
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//
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// You should have received a copy of the GNU Lesser General Public
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// License and a copy of the GNU General Public License along with
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// Eigen. If not, see <http://www.gnu.org/licenses/>.
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#ifndef EIGEN_CONJUGATE_GRADIENT_H
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#define EIGEN_CONJUGATE_GRADIENT_H
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namespace internal {
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/** \internal Low-level conjugate gradient algorithm
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* \param mat The matrix A
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* \param rhs The right hand side vector b
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* \param x On input and initial solution, on output the computed solution.
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* \param precond A preconditioner being able to efficiently solve for an
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* approximation of Ax=b (regardless of b)
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* \param iters On input the max number of iteration, on output the number of performed iterations.
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* \param tol_error On input the tolerance error, on output an estimation of the relative error.
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*/
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template<typename MatrixType, typename Rhs, typename Dest, typename Preconditioner>
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void conjugate_gradient(const MatrixType& mat, const Rhs& rhs, Dest& x,
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const Preconditioner& precond, int& iters,
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typename Dest::RealScalar& tol_error)
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{
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using std::sqrt;
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using std::abs;
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typedef typename Dest::RealScalar RealScalar;
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typedef typename Dest::Scalar Scalar;
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typedef Dest VectorType;
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RealScalar tol = tol_error;
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int maxIters = iters;
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int n = mat.cols();
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VectorType residual = rhs - mat * x; //initial residual
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VectorType p(n);
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p = precond.solve(residual); //initial search direction
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VectorType z(n), tmp(n);
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RealScalar absNew = internal::real(residual.dot(p)); // the square of the absolute value of r scaled by invM
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RealScalar absInit = absNew; // the initial absolute value
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int i = 0;
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while ((i < maxIters) && (absNew > tol*tol*absInit))
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{
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tmp.noalias() = mat * p; // the bottleneck of the algorithm
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Scalar alpha = absNew / p.dot(tmp); // the amount we travel on dir
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x += alpha * p; // update solution
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residual -= alpha * tmp; // update residue
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z = precond.solve(residual); // approximately solve for "A z = residual"
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RealScalar absOld = absNew;
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absNew = internal::real(residual.dot(z)); // update the absolute value of r
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RealScalar beta = absNew / absOld; // calculate the Gram-Schmidit value used to create the new search direction
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p = z + beta * p; // update search direction
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i++;
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}
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tol_error = sqrt(abs(absNew / absInit));
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iters = i;
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}
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}
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/** \brief A preconditioner based on the digonal entries
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*
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* This class allows to approximately solve for A.x = b problems assuming A is a diagonal matrix.
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* In other words, this preconditioner neglects all off diagonal entries and, in Eigen's language, solves for:
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* \code
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* A.diagonal().asDiagonal() . x = b
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* \endcode
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*
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* \tparam _Scalar the type of the scalar.
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*
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* This preconditioner is suitable for both selfadjoint and general problems.
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* The diagonal entries are pre-inverted and stored into a dense vector.
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*
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* \note A variant that has yet to be implemented would attempt to preserve the norm of each column.
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*
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*/
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template <typename _Scalar>
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class DiagonalPreconditioner
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{
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typedef _Scalar Scalar;
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typedef Matrix<Scalar,Dynamic,1> Vector;
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typedef typename Vector::Index Index;
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public:
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typedef Matrix<Scalar,Dynamic,Dynamic> MatrixType;
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DiagonalPreconditioner() : m_isInitialized(false) {}
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template<typename MatrixType>
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DiagonalPreconditioner(const MatrixType& mat) : m_invdiag(mat.cols())
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{
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compute(mat);
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}
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Index rows() const { return m_invdiag.size(); }
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Index cols() const { return m_invdiag.size(); }
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template<typename MatrixType>
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DiagonalPreconditioner& compute(const MatrixType& mat)
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{
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m_invdiag.resize(mat.cols());
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for(int j=0; j<mat.outerSize(); ++j)
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{
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typename MatrixType::InnerIterator it(mat,j);
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while(it && it.index()!=j) ++it;
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if(it.index()==j)
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m_invdiag(j) = Scalar(1)/it.value();
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else
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m_invdiag(j) = 0;
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}
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m_isInitialized = true;
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return *this;
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}
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template<typename Rhs, typename Dest>
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void _solve(const Rhs& b, Dest& x) const
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{
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x = m_invdiag.array() * b.array() ;
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}
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template<typename Rhs> inline const internal::solve_retval<DiagonalPreconditioner, Rhs>
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solve(const MatrixBase<Rhs>& b) const
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{
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eigen_assert(m_isInitialized && "DiagonalPreconditioner is not initialized.");
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eigen_assert(m_invdiag.size()==b.rows()
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&& "DiagonalPreconditioner::solve(): invalid number of rows of the right hand side matrix b");
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return internal::solve_retval<DiagonalPreconditioner, Rhs>(*this, b.derived());
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}
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protected:
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Vector m_invdiag;
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bool m_isInitialized;
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};
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namespace internal {
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template<typename _MatrixType, typename Rhs>
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struct solve_retval<DiagonalPreconditioner<_MatrixType>, Rhs>
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: solve_retval_base<DiagonalPreconditioner<_MatrixType>, Rhs>
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{
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typedef DiagonalPreconditioner<_MatrixType> Dec;
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EIGEN_MAKE_SOLVE_HELPERS(Dec,Rhs)
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template<typename Dest> void evalTo(Dest& dst) const
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{
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dec()._solve(rhs(),dst);
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}
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};
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template<typename CG, typename Rhs, typename Guess>
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class conjugate_gradient_solve_retval_with_guess;
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}
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/** \brief A conjugate gradient solver for sparse self-adjoint problems
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*
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* This class allows to solve for A.x = b sparse linear problems using a conjugate gradient algorithm.
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* The sparse matrix A must be selfadjoint. The vectors x and b can be either dense or sparse.
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*
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* \tparam _MatrixType the type of the sparse matrix A, can be a dense or a sparse matrix.
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* \tparam _UpLo the triangular part that will be used for the computations. It can be Lower
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* or Upper. Default is Lower.
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* \tparam _Preconditioner the type of the preconditioner. Default is DiagonalPreconditioner
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*
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* The maximal number of iterations and tolerance value can be controlled via the setMaxIterations()
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* and setTolerance() methods. The default are 1000 max iterations and NumTraits<Scalar>::epsilon()
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* for the tolerance.
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*
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* This class can be used as the direct solver classes. Here is a typical usage example:
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* \code
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* int n = 10000;
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* VectorXd x(n), b(n);
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* SparseMatrix<double> A(n,n);
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* // fill A and b
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* ConjugateGradient<SparseMatrix<double> > cg;
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* cg(A);
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* x = cg.solve(b);
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* std::cout << "#iterations: " << cg.iterations() << std::endl;
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* std::cout << "estimated error: " << cg.error() << std::endl;
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* // update b, and solve again
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* x = cg.solve(b);
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* \endcode
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*
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* By default the iterations start with x=0 as an initial guess of the solution.
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* One can control the start using the solveWithGuess() method. Here is a step by
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* step execution example starting with a random guess and printing the evolution
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* of the estimated error:
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* * \code
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* x = VectorXd::Random(n);
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* cg.setMaxIterations(1);
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* int i = 0;
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* do {
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* x = cg.solveWithGuess(b,x);
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* std::cout << i << " : " << cg.error() << std::endl;
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* ++i;
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* } while (cg.info()!=Success && i<100);
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* \endcode
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* Note that such a step by step excution is slightly slower.
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*
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*/
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template< typename _MatrixType, int _UpLo=Lower,
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typename _Preconditioner = DiagonalPreconditioner<typename _MatrixType::Scalar> >
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class ConjugateGradient
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{
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public:
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typedef _MatrixType MatrixType;
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typedef typename MatrixType::Scalar Scalar;
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typedef typename MatrixType::Index Index;
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typedef typename MatrixType::RealScalar RealScalar;
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typedef _Preconditioner Preconditioner;
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enum {
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UpLo = _UpLo
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};
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public:
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/** Default constructor. */
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ConjugateGradient()
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: mp_matrix(0)
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{
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init();
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}
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/** Initialize the solver with matrix \a A for further \c Ax=b solving.
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*
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* \warning this class stores a reference to the matrix A as well as some
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* precomputed values that depend on it. Therefore, if \a A is changed
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* this class becomes invalid. Call compute() to update it with the new
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* matrix A, or modify a copy of A.
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*/
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ConjugateGradient(const MatrixType& A)
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{
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init();
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compute(A);
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}
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~ConjugateGradient() {}
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/** Initializes the iterative solver with the matrix \a A for further solving \c Ax=b problems.
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*
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* \warning this class stores a reference to the matrix A as well as some
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* precomputed values that depend on it. Therefore, if \a A is changed
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* this class becomes invalid. Call compute() to update it with the new
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* matrix A, or modify a copy of A.
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*/
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ConjugateGradient& compute(const MatrixType& A)
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{
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mp_matrix = &A;
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m_preconditioner.compute(A);
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m_isInitialized = true;
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return *this;
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}
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/** \internal */
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Index rows() const { return mp_matrix->rows(); }
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/** \internal */
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Index cols() const { return mp_matrix->cols(); }
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/** \returns the tolerance threshold used by the stopping criteria */
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RealScalar tolerance() const { return m_tolerance; }
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/** Sets the tolerance threshold used by the stopping criteria */
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ConjugateGradient& setTolerance(RealScalar tolerance)
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{
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m_tolerance = tolerance;
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return *this;
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}
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/** \returns the max number of iterations */
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int maxIterations() const { return m_maxIterations; }
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/** Sets the max number of iterations */
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ConjugateGradient& setMaxIterations(int maxIters)
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{
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m_maxIterations = maxIters;
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return *this;
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}
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/** \returns the number of iterations performed during the last solve */
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int iterations() const
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{
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eigen_assert(m_isInitialized && "ConjugateGradient is not initialized.");
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return m_iterations;
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}
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/** \returns the tolerance error reached during the last solve */
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RealScalar error() const
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{
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eigen_assert(m_isInitialized && "ConjugateGradient is not initialized.");
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return m_error;
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}
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/** \returns the solution x of \f$ A x = b \f$ using the current decomposition of A.
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*
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* \sa compute()
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*/
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template<typename Rhs> inline const internal::solve_retval<ConjugateGradient, Rhs>
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solve(const MatrixBase<Rhs>& b) const
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{
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eigen_assert(m_isInitialized && "ConjugateGradient is not initialized.");
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eigen_assert(rows()==b.rows()
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&& "ConjugateGradient::solve(): invalid number of rows of the right hand side matrix b");
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return internal::solve_retval<ConjugateGradient, Rhs>(*this, b.derived());
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}
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/** \returns the solution x of \f$ A x = b \f$ using the current decomposition of A
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* \a x0 as an initial solution.
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*
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* \sa compute()
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*/
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template<typename Rhs,typename Guess>
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inline const internal::conjugate_gradient_solve_retval_with_guess<ConjugateGradient, Rhs, Guess>
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solveWithGuess(const MatrixBase<Rhs>& b, const Guess& x0) const
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{
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eigen_assert(m_isInitialized && "ConjugateGradient is not initialized.");
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eigen_assert(rows()==b.rows()
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&& "ConjugateGradient::solve(): invalid number of rows of the right hand side matrix b");
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return internal::conjugate_gradient_solve_retval_with_guess
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<ConjugateGradient, Rhs, Guess>(*this, b.derived(), x0);
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}
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/** \returns Success if the iterations converged, and NoConvergence otherwise. */
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ComputationInfo info() const
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{
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eigen_assert(m_isInitialized && "ConjugateGradient is not initialized.");
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return m_info;
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}
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/** \internal */
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template<typename Rhs,typename Dest>
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void _solve(const Rhs& b, Dest& x) const
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{
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m_iterations = m_maxIterations;
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m_error = m_tolerance;
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internal::conjugate_gradient(mp_matrix->template selfadjointView<UpLo>(), b, x,
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m_preconditioner, m_iterations, m_error);
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m_isInitialized = true;
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m_info = m_error <= m_tolerance ? Success : NoConvergence;
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}
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protected:
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void init()
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{
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m_isInitialized = false;
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m_maxIterations = 1000;
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m_tolerance = NumTraits<Scalar>::epsilon();
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}
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const MatrixType* mp_matrix;
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Preconditioner m_preconditioner;
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int m_maxIterations;
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RealScalar m_tolerance;
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mutable RealScalar m_error;
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mutable int m_iterations;
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mutable ComputationInfo m_info;
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mutable bool m_isInitialized;
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};
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namespace internal {
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template<typename _MatrixType, int _UpLo, typename _Preconditioner, typename Rhs>
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struct solve_retval<ConjugateGradient<_MatrixType,_UpLo,_Preconditioner>, Rhs>
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: solve_retval_base<ConjugateGradient<_MatrixType,_UpLo,_Preconditioner>, Rhs>
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{
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typedef ConjugateGradient<_MatrixType,_UpLo,_Preconditioner> Dec;
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EIGEN_MAKE_SOLVE_HELPERS(Dec,Rhs)
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template<typename Dest> void evalTo(Dest& dst) const
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{
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dst.setZero();
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dec()._solve(rhs(),dst);
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}
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};
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template<typename CG, typename Rhs, typename Guess>
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class conjugate_gradient_solve_retval_with_guess
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: solve_retval_base<CG, Rhs>
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{
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typedef Eigen::internal::solve_retval_base<CG,Rhs> Base;
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using Base::dec;
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using Base::rhs;
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conjugate_gradient_solve_retval_with_guess(const CG& cg, const Rhs& rhs, const Guess guess)
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: Base(cg, rhs), m_guess(guess)
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{}
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template<typename Dest> void evalTo(Dest& dst) const
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{
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dst = m_guess;
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dec()._solve(rhs(), dst);
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}
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protected:
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const Guess& m_guess;
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};
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}
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#endif // EIGEN_CONJUGATE_GRADIENT_H
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