unsupported/Eigen/src/IterativeSolvers/MINRES.h - eigen/fuchsia - Git at Google

 // This file is part of Eigen, a lightweight C++ template library
 // for linear algebra.
 //
 // Copyright (C) 2012 Giacomo Po <gpo@ucla.edu>
 // 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_MINRES_H_
 #define EIGEN_MINRES_H_


 namespace Eigen {

     namespace internal {

         /** \internal Low-level MINRES 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 right 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 minres(const MatrixType& mat, const Rhs& rhs, Dest& x,
                     const Preconditioner& precond, Index& iters,
                     typename Dest::RealScalar& tol_error)
         {
             using std::sqrt;
             typedef typename Dest::RealScalar RealScalar;
             typedef typename Dest::Scalar Scalar;
             typedef Matrix<Scalar,Dynamic,1> VectorType;

             // Check for zero rhs
             const RealScalar rhsNorm2(rhs.squaredNorm());
             if(rhsNorm2 == 0)
             {
                 x.setZero();
                 iters = 0;
                 tol_error = 0;
                 return;
             }

             // initialize
             const Index maxIters(iters);  // initialize maxIters to iters
             const Index N(mat.cols());    // the size of the matrix
             const RealScalar threshold2(tol_error*tol_error*rhsNorm2); // convergence threshold (compared to residualNorm2)

             // Initialize preconditioned Lanczos
             VectorType v_old(N); // will be initialized inside loop
             VectorType v( VectorType::Zero(N) ); //initialize v
             VectorType v_new(rhs-mat*x); //initialize v_new
             RealScalar residualNorm2(v_new.squaredNorm());
             VectorType w(N); // will be initialized inside loop
             VectorType w_new(precond.solve(v_new)); // initialize w_new
 //            RealScalar beta; // will be initialized inside loop
             RealScalar beta_new2(v_new.dot(w_new));
             eigen_assert(beta_new2 >= 0.0 && "PRECONDITIONER IS NOT POSITIVE DEFINITE");
             RealScalar beta_new(sqrt(beta_new2));
             const RealScalar beta_one(beta_new);
             v_new /= beta_new;
             w_new /= beta_new;
             // Initialize other variables
             RealScalar c(1.0); // the cosine of the Givens rotation
             RealScalar c_old(1.0);
             RealScalar s(0.0); // the sine of the Givens rotation
             RealScalar s_old(0.0); // the sine of the Givens rotation
             VectorType p_oold(N); // will be initialized in loop
             VectorType p_old(VectorType::Zero(N)); // initialize p_old=0
             VectorType p(p_old); // initialize p=0
             RealScalar eta(1.0);

             iters = 0; // reset iters
             while ( iters < maxIters )
             {
                 // Preconditioned Lanczos
                 /* Note that there are 4 variants on the Lanczos algorithm. These are
                  * described in Paige, C. C. (1972). Computational variants of
                  * the Lanczos method for the eigenproblem. IMA Journal of Applied
                  * Mathematics, 10(3), 373–381. The current implementation corresponds
                  * to the case A(2,7) in the paper. It also corresponds to
                  * algorithm 6.14 in Y. Saad, Iterative Methods for Sparse Linear
                  * Systems, 2003 p.173. For the preconditioned version see
                  * A. Greenbaum, Iterative Methods for Solving Linear Systems, SIAM (1987).
                  */
                 const RealScalar beta(beta_new);
                 v_old = v; // update: at first time step, this makes v_old = 0 so value of beta doesn't matter
 //                const VectorType v_old(v); // NOT SURE IF CREATING v_old EVERY ITERATION IS EFFICIENT
                 v = v_new; // update
                 w = w_new; // update
 //                const VectorType w(w_new); // NOT SURE IF CREATING w EVERY ITERATION IS EFFICIENT
                 v_new.noalias() = mat*w - beta*v_old; // compute v_new
                 const RealScalar alpha = v_new.dot(w);
                 v_new -= alpha*v; // overwrite v_new
                 w_new = precond.solve(v_new); // overwrite w_new
                 beta_new2 = v_new.dot(w_new); // compute beta_new
                 eigen_assert(beta_new2 >= 0.0 && "PRECONDITIONER IS NOT POSITIVE DEFINITE");
                 beta_new = sqrt(beta_new2); // compute beta_new
                 v_new /= beta_new; // overwrite v_new for next iteration
                 w_new /= beta_new; // overwrite w_new for next iteration

                 // Givens rotation
                 const RealScalar r2 =s*alpha+c*c_old*beta; // s, s_old, c and c_old are still from previous iteration
                 const RealScalar r3 =s_old*beta; // s, s_old, c and c_old are still from previous iteration
                 const RealScalar r1_hat=c*alpha-c_old*s*beta;
                 const RealScalar r1 =sqrt( std::pow(r1_hat,2) + std::pow(beta_new,2) );
                 c_old = c; // store for next iteration
                 s_old = s; // store for next iteration
                 c=r1_hat/r1; // new cosine
                 s=beta_new/r1; // new sine

                 // Update solution
                 p_oold = p_old;
 //                const VectorType p_oold(p_old); // NOT SURE IF CREATING p_oold EVERY ITERATION IS EFFICIENT
                 p_old = p;
                 p.noalias()=(w-r2*p_old-r3*p_oold) /r1; // IS NOALIAS REQUIRED?
                 x += beta_one*c*eta*p;

                 /* Update the squared residual. Note that this is the estimated residual.
                 The real residual |Ax-b|^2 may be slightly larger */
                 residualNorm2 *= s*s;

                 if ( residualNorm2 < threshold2)
                 {
                     break;
                 }

                 eta=-s*eta; // update eta
                 iters++; // increment iteration number (for output purposes)
             }

             /* Compute error. Note that this is the estimated error. The real
              error |Ax-b|/|b| may be slightly larger */
             tol_error = std::sqrt(residualNorm2 / rhsNorm2);
         }

     }

     template< typename _MatrixType, int _UpLo=Lower,
     typename _Preconditioner = IdentityPreconditioner>
     class MINRES;

     namespace internal {

         template< typename _MatrixType, int _UpLo, typename _Preconditioner>
         struct traits<MINRES<_MatrixType,_UpLo,_Preconditioner> >
         {
             typedef _MatrixType MatrixType;
             typedef _Preconditioner Preconditioner;
         };

     }

     /** \ingroup IterativeLinearSolvers_Module
      * \brief A minimal residual solver for sparse symmetric problems
      *
      * This class allows to solve for A.x = b sparse linear problems using the MINRES algorithm
      * of Paige and Saunders (1975). The sparse matrix A must be symmetric (possibly indefinite).
      * The vectors x and b can be either dense or sparse.
      *
      * \tparam _MatrixType the type of the sparse 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,
      *               Upper, or Lower|Upper in which the full matrix entries will be considered. Default is Lower.
      * \tparam _Preconditioner the type of the preconditioner. Default is DiagonalPreconditioner
      *
      * 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.
      *
      * 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
      * MINRES<SparseMatrix<double> > mr;
      * mr.compute(A);
      * x = mr.solve(b);
      * std::cout << "#iterations:     " << mr.iterations() << std::endl;
      * std::cout << "estimated error: " << mr.error()      << std::endl;
      * // update b, and solve again
      * x = mr.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.
      *
      * MINRES can also be used in a matrix-free context, see the following \link MatrixfreeSolverExample example \endlink.
      *
      * \sa class ConjugateGradient, BiCGSTAB, SimplicialCholesky, DiagonalPreconditioner, IdentityPreconditioner
      */
     template< typename _MatrixType, int _UpLo, typename _Preconditioner>
     class MINRES : public IterativeSolverBase<MINRES<_MatrixType,_UpLo,_Preconditioner> >
     {

         typedef IterativeSolverBase<MINRES> Base;
         using Base::matrix;
         using Base::m_error;
         using Base::m_iterations;
         using Base::m_info;
         using Base::m_isInitialized;
     public:
         using Base::_solve_impl;
         typedef _MatrixType MatrixType;
         typedef typename MatrixType::Scalar Scalar;
         typedef typename MatrixType::RealScalar RealScalar;
         typedef _Preconditioner Preconditioner;

         enum {UpLo = _UpLo};

     public:

         /** Default constructor. */
         MINRES() : 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 MINRES(const EigenBase<MatrixDerived>& A) : Base(A.derived()) {}

         /** Destructor. */
         ~MINRES(){}

         /** \internal */
         template<typename Rhs,typename Dest>
         void _solve_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());
             for(int j=0; j<b.cols(); ++j)
             {
                 m_iterations = Base::maxIterations();
                 m_error = Base::m_tolerance;

                 typename Dest::ColXpr xj(x,j);
                 internal::minres(SelfAdjointWrapper(row_mat), b.col(j), xj,
                                  Base::m_preconditioner, m_iterations, m_error);
             }

             m_isInitialized = true;
             m_info = m_error <= Base::m_tolerance ? Success : NoConvergence;
         }

         /** \internal */
         template<typename Rhs,typename Dest>
         void _solve_impl(const Rhs& b, MatrixBase<Dest> &x) const
         {
             x.setZero();
             _solve_with_guess_impl(b,x.derived());
         }

     protected:

     };

 } // end namespace Eigen

 #endif // EIGEN_MINRES_H
	// This file is part of Eigen, a lightweight C++ template library
	// for linear algebra.
	//
	// Copyright (C) 2012 Giacomo Po <gpo@ucla.edu>
	// 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_MINRES_H_
	#define EIGEN_MINRES_H_


	namespace Eigen {

	namespace internal {

	/** \internal Low-level MINRES 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 right 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 minres(const MatrixType& mat, const Rhs& rhs, Dest& x,
	const Preconditioner& precond, Index& iters,
	typename Dest::RealScalar& tol_error)
	{
	using std::sqrt;
	typedef typename Dest::RealScalar RealScalar;
	typedef typename Dest::Scalar Scalar;
	typedef Matrix<Scalar,Dynamic,1> VectorType;

	// Check for zero rhs
	const RealScalar rhsNorm2(rhs.squaredNorm());
	if(rhsNorm2 == 0)
	{
	x.setZero();
	iters = 0;
	tol_error = 0;
	return;
	}

	// initialize
	const Index maxIters(iters); // initialize maxIters to iters
	const Index N(mat.cols()); // the size of the matrix
	const RealScalar threshold2(tol_errortol_errorrhsNorm2); // convergence threshold (compared to residualNorm2)

	// Initialize preconditioned Lanczos
	VectorType v_old(N); // will be initialized inside loop
	VectorType v( VectorType::Zero(N) ); //initialize v
	VectorType v_new(rhs-mat*x); //initialize v_new
	RealScalar residualNorm2(v_new.squaredNorm());
	VectorType w(N); // will be initialized inside loop
	VectorType w_new(precond.solve(v_new)); // initialize w_new
	// RealScalar beta; // will be initialized inside loop
	RealScalar beta_new2(v_new.dot(w_new));
	eigen_assert(beta_new2 >= 0.0 && "PRECONDITIONER IS NOT POSITIVE DEFINITE");
	RealScalar beta_new(sqrt(beta_new2));
	const RealScalar beta_one(beta_new);
	v_new /= beta_new;
	w_new /= beta_new;
	// Initialize other variables
	RealScalar c(1.0); // the cosine of the Givens rotation
	RealScalar c_old(1.0);
	RealScalar s(0.0); // the sine of the Givens rotation
	RealScalar s_old(0.0); // the sine of the Givens rotation
	VectorType p_oold(N); // will be initialized in loop
	VectorType p_old(VectorType::Zero(N)); // initialize p_old=0
	VectorType p(p_old); // initialize p=0
	RealScalar eta(1.0);

	iters = 0; // reset iters
	while ( iters < maxIters )
	{
	// Preconditioned Lanczos
	/* Note that there are 4 variants on the Lanczos algorithm. These are
	* described in Paige, C. C. (1972). Computational variants of
	* the Lanczos method for the eigenproblem. IMA Journal of Applied
	* Mathematics, 10(3), 373–381. The current implementation corresponds
	* to the case A(2,7) in the paper. It also corresponds to
	* algorithm 6.14 in Y. Saad, Iterative Methods for Sparse Linear
	* Systems, 2003 p.173. For the preconditioned version see
	* A. Greenbaum, Iterative Methods for Solving Linear Systems, SIAM (1987).
	*/
	const RealScalar beta(beta_new);
	v_old = v; // update: at first time step, this makes v_old = 0 so value of beta doesn't matter
	// const VectorType v_old(v); // NOT SURE IF CREATING v_old EVERY ITERATION IS EFFICIENT
	v = v_new; // update
	w = w_new; // update
	// const VectorType w(w_new); // NOT SURE IF CREATING w EVERY ITERATION IS EFFICIENT
	v_new.noalias() = matw - betav_old; // compute v_new
	const RealScalar alpha = v_new.dot(w);
	v_new -= alpha*v; // overwrite v_new
	w_new = precond.solve(v_new); // overwrite w_new
	beta_new2 = v_new.dot(w_new); // compute beta_new
	eigen_assert(beta_new2 >= 0.0 && "PRECONDITIONER IS NOT POSITIVE DEFINITE");
	beta_new = sqrt(beta_new2); // compute beta_new
	v_new /= beta_new; // overwrite v_new for next iteration
	w_new /= beta_new; // overwrite w_new for next iteration

	// Givens rotation
	const RealScalar r2 =salpha+cc_old*beta; // s, s_old, c and c_old are still from previous iteration
	const RealScalar r3 =s_old*beta; // s, s_old, c and c_old are still from previous iteration
	const RealScalar r1_hat=calpha-c_olds*beta;
	const RealScalar r1 =sqrt( std::pow(r1_hat,2) + std::pow(beta_new,2) );
	c_old = c; // store for next iteration
	s_old = s; // store for next iteration
	c=r1_hat/r1; // new cosine
	s=beta_new/r1; // new sine

	// Update solution
	p_oold = p_old;
	// const VectorType p_oold(p_old); // NOT SURE IF CREATING p_oold EVERY ITERATION IS EFFICIENT
	p_old = p;
	p.noalias()=(w-r2p_old-r3p_oold) /r1; // IS NOALIAS REQUIRED?
	x += beta_oneceta*p;

	/* Update the squared residual. Note that this is the estimated residual.
	The real residual \|Ax-b\|^2 may be slightly larger */
	residualNorm2 = ss;

	if ( residualNorm2 < threshold2)
	{
	break;
	}

	eta=-s*eta; // update eta
	iters++; // increment iteration number (for output purposes)
	}

	/* Compute error. Note that this is the estimated error. The real
	error \|Ax-b\|/\|b\| may be slightly larger */
	tol_error = std::sqrt(residualNorm2 / rhsNorm2);
	}

	}

	template< typename _MatrixType, int _UpLo=Lower,
	typename _Preconditioner = IdentityPreconditioner>
	class MINRES;

	namespace internal {

	template< typename _MatrixType, int _UpLo, typename _Preconditioner>
	struct traits<MINRES<_MatrixType,_UpLo,_Preconditioner> >
	{
	typedef _MatrixType MatrixType;
	typedef _Preconditioner Preconditioner;
	};

	}

	/** \ingroup IterativeLinearSolvers_Module
	* \brief A minimal residual solver for sparse symmetric problems
	*
	* This class allows to solve for A.x = b sparse linear problems using the MINRES algorithm
	* of Paige and Saunders (1975). The sparse matrix A must be symmetric (possibly indefinite).
	* The vectors x and b can be either dense or sparse.
	*
	* \tparam _MatrixType the type of the sparse 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,
	* Upper, or Lower\|Upper in which the full matrix entries will be considered. Default is Lower.
	* \tparam _Preconditioner the type of the preconditioner. Default is DiagonalPreconditioner
	*
	* 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.
	*
	* 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
	* MINRES<SparseMatrix<double> > mr;
	* mr.compute(A);
	* x = mr.solve(b);
	* std::cout << "#iterations: " << mr.iterations() << std::endl;
	* std::cout << "estimated error: " << mr.error() << std::endl;
	* // update b, and solve again
	* x = mr.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.
	*
	* MINRES can also be used in a matrix-free context, see the following \link MatrixfreeSolverExample example \endlink.
	*
	* \sa class ConjugateGradient, BiCGSTAB, SimplicialCholesky, DiagonalPreconditioner, IdentityPreconditioner
	*/
	template< typename _MatrixType, int _UpLo, typename _Preconditioner>
	class MINRES : public IterativeSolverBase<MINRES<_MatrixType,_UpLo,_Preconditioner> >
	{

	typedef IterativeSolverBase<MINRES> Base;
	using Base::matrix;
	using Base::m_error;
	using Base::m_iterations;
	using Base::m_info;
	using Base::m_isInitialized;
	public:
	using Base::_solve_impl;
	typedef _MatrixType MatrixType;
	typedef typename MatrixType::Scalar Scalar;
	typedef typename MatrixType::RealScalar RealScalar;
	typedef _Preconditioner Preconditioner;

	enum {UpLo = _UpLo};

	public:

	/** Default constructor. */
	MINRES() : 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 MINRES(const EigenBase<MatrixDerived>& A) : Base(A.derived()) {}

	/** Destructor. */
	~MINRES(){}

	/** \internal */
	template<typename Rhs,typename Dest>
	void _solve_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());
	for(int j=0; j<b.cols(); ++j)
	{
	m_iterations = Base::maxIterations();
	m_error = Base::m_tolerance;

	typename Dest::ColXpr xj(x,j);
	internal::minres(SelfAdjointWrapper(row_mat), b.col(j), xj,
	Base::m_preconditioner, m_iterations, m_error);
	}

	m_isInitialized = true;
	m_info = m_error <= Base::m_tolerance ? Success : NoConvergence;
	}

	/** \internal */
	template<typename Rhs,typename Dest>
	void _solve_impl(const Rhs& b, MatrixBase<Dest> &x) const
	{
	x.setZero();
	_solve_with_guess_impl(b,x.derived());
	}

	protected:

	};

	} // end namespace Eigen

	#endif // EIGEN_MINRES_H