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+// 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;
+ }
+ RealScalar threshold = tol*tol*rhsNorm2;
+ 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_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;
+
+ for(Index j=0; j<b.cols(); ++j)
+ {
+ m_iterations = Base::maxIterations();
+ m_error = Base::m_tolerance;
+
+ typename Dest::ColXpr xj(x,j);
+ RowMajorWrapper row_mat(matrix());
+ internal::conjugate_gradient(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 */
+ using Base::_solve_impl;
+ template<typename Rhs,typename Dest>
+ void _solve_impl(const MatrixBase<Rhs>& b, Dest& x) const
+ {
+ x.setZero();
+ _solve_with_guess_impl(b.derived(),x);
+ }
+
+protected:
+
+};
+
+} // end namespace Eigen
+
+#endif // EIGEN_CONJUGATE_GRADIENT_H