MINRES solver

unsupported/Eigen/IterativeSolvers

@@ -35,6 +35,7 @@
 #include "src/IterativeSolvers/GMRES.h"
 #include "src/IterativeSolvers/IncompleteCholesky.h"
 //#include "src/IterativeSolvers/SSORPreconditioner.h"
+#include "src/IterativeSolvers/MINRES.h"
 
 //@}
 

unsupported/Eigen/src/IterativeSolvers/MINRES.h

@@ -0,0 +1,301 @@
+// 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 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, int& iters,
+                    typename Dest::RealScalar& tol_error)
+        {
+            typedef typename Dest::RealScalar RealScalar;
+            typedef typename Dest::Scalar Scalar;
+            typedef Matrix<Scalar,Dynamic,1> VectorType;
+
+            // initialize
+            const int maxIters(iters);  // initialize maxIters to iters
+            const int N(mat.cols());    // the size of the matrix
+            const RealScalar rhsNorm2(rhs.squaredNorm());
+            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));
+            assert(beta_new2 >= 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
+                assert(beta_new2 >= 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;
+                residualNorm2 *= s*s;
+                
+                if ( residualNorm2 < threshold2){
+                    break;
+                }
+                
+                eta=-s*eta; // update eta
+                iters++; // increment iteration number (for output purposes)
+            }
+            tol_error = std::sqrt(residualNorm2 / rhsNorm2); // return error. Note that this is the estimated error. The real error |Ax-b|/|b| may be slightly larger 
+        }
+        
+    }
+    
+    template< typename _MatrixType, int _UpLo=Lower,
+    typename _Preconditioner = IdentityPreconditioner>
+//    typename _Preconditioner = IdentityPreconditioner<typename _MatrixType::Scalar> > // preconditioner must be positive definite
+    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
+     *               or Upper. 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. Here is a step by
+     * step execution example starting with a random guess and printing the evolution
+     * of the estimated error:
+     * * \code
+     * x = VectorXd::Random(n);
+     * mr.setMaxIterations(1);
+     * int i = 0;
+     * do {
+     *   x = mr.solveWithGuess(b,x);
+     *   std::cout << i << " : " << mr.error() << std::endl;
+     *   ++i;
+     * } while (mr.info()!=Success && i<100);
+     * \endcode
+     * Note that such a step by step excution is slightly slower.
+     *
+     * \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::mp_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::Index Index;
+        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.
+         */
+        MINRES(const MatrixType& A) : Base(A) {}
+        
+        /** Destructor. */
+        ~MINRES(){}
+		
+        /** \returns the solution x of \f$ A x = b \f$ using the current decomposition of A
+         * \a x0 as an initial solution.
+         *
+         * \sa compute()
+         */
+        template<typename Rhs,typename Guess>
+        inline const internal::solve_retval_with_guess<MINRES, Rhs, Guess>
+        solveWithGuess(const MatrixBase<Rhs>& b, const Guess& x0) const
+        {
+            eigen_assert(m_isInitialized && "MINRES is not initialized.");
+            eigen_assert(Base::rows()==b.rows()
+                         && "MINRES::solve(): invalid number of rows of the right hand side matrix b");
+            return internal::solve_retval_with_guess
+            <MINRES, Rhs, Guess>(*this, b.derived(), x0);
+        }
+        
+        /** \internal */
+        template<typename Rhs,typename Dest>
+        void _solveWithGuess(const Rhs& b, Dest& x) const
+        {
+            m_iterations = Base::maxIterations();
+            m_error = Base::m_tolerance;
+            
+            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(mp_matrix->template selfadjointView<UpLo>(), 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(const Rhs& b, Dest& x) const
+        {
+            x.setZero();
+            _solveWithGuess(b,x);
+        }
+        
+    protected:
+        
+    };
+    
+    namespace internal {
+        
+        template<typename _MatrixType, int _UpLo, typename _Preconditioner, typename Rhs>
+        struct solve_retval<MINRES<_MatrixType,_UpLo,_Preconditioner>, Rhs>
+        : solve_retval_base<MINRES<_MatrixType,_UpLo,_Preconditioner>, Rhs>
+        {
+            typedef MINRES<_MatrixType,_UpLo,_Preconditioner> Dec;
+            EIGEN_MAKE_SOLVE_HELPERS(Dec,Rhs)
+            
+            template<typename Dest> void evalTo(Dest& dst) const
+            {
+                dec()._solve(rhs(),dst);
+            }
+        };
+        
+    } // end namespace internal
+    
+} // end namespace Eigen
+
+#endif // EIGEN_MINRES_H
+

unsupported/test/CMakeLists.txt

@@ -85,4 +85,4 @@ ei_add_test(polynomialutils)
 ei_add_test(kronecker_product)
 ei_add_test(splines)
 ei_add_test(gmres)
-
+ei_add_test(minres)

unsupported/test/minres.cpp

@@ -0,0 +1,33 @@
+// This file is part of Eigen, a lightweight C++ template library
+// for linear algebra.
+//
+// Copyright (C) 2011 Gael Guennebaud <g.gael@free.fr>
+// Copyright (C) 2012 Giacomo Po <gpo@ucla.edu>
+//
+// 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/.
+
+#include "../../test/sparse_solver.h"
+#include <Eigen/IterativeSolvers>
+
+template<typename T> void test_minres_T()
+{
+//  MINRES<SparseMatrix<T>, Lower, DiagonalPreconditioner<T> > minres_colmajor_diag;
+  MINRES<SparseMatrix<T>, Lower, IdentityPreconditioner    > minres_colmajor_I;
+//  MINRES<SparseMatrix<T>, Lower, IncompleteLUT<T> >           minres_colmajor_ilut;
+  //minres<SparseMatrix<T>, SSORPreconditioner<T> >     minres_colmajor_ssor;
+
+//  CALL_SUBTEST( check_sparse_square_solving(minres_colmajor_diag)  );
+  CALL_SUBTEST( check_sparse_spd_solving(minres_colmajor_I)     );
+ // CALL_SUBTEST( check_sparse_square_solving(minres_colmajor_ilut)     );
+  //CALL_SUBTEST( check_sparse_square_solving(minres_colmajor_ssor)     );
+}
+
+void test_minres()
+{
+  for(int i = 0; i < g_repeat; i++) {
+    CALL_SUBTEST_1(test_minres_T<double>());
+  //  CALL_SUBTEST_2(test_minres_T<std::complex<double> >());
+  }
+}