Implemented a modifier conjugate gradient algorithm to support stable
constraining. The algorithm is described in the paper "Large Steps in Cloth Simulation" (Baraff/Witkin 1998). The same method was (incorrectly) implemented in the old cloth solver. It is based on restricting the degrees of freedom (ndof) of vertices using a block matrix and a vector of target velocity deltas. See chapter 5 of the paper for details.
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@ -109,6 +109,7 @@ set(SRC
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intern/image_gen.c
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intern/implicit.c
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intern/implicit_eigen.cpp
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intern/ConstrainedConjugateGradient.h # XXX move this to a better place
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intern/ipo.c
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intern/key.c
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intern/lamp.c
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@ -0,0 +1,294 @@
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#ifndef EIGEN_CONSTRAINEDCG_H
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#define EIGEN_CONSTRAINEDCG_H
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#include <Eigen/Core>
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namespace Eigen {
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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 FilterMatrixType, typename Preconditioner>
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EIGEN_DONT_INLINE
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void constrained_conjugate_gradient(const MatrixType& mat, const Rhs& rhs, Dest& x,
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const FilterMatrixType &filter,
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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 Matrix<Scalar,Dynamic,1> 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 = filter * (rhs - mat * x); //initial residual
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RealScalar rhsNorm2 = (filter * rhs).squaredNorm();
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if(rhsNorm2 == 0)
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{
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/* XXX TODO set constrained result here */
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x.setZero();
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iters = 0;
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tol_error = 0;
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return;
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}
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RealScalar threshold = tol*tol*rhsNorm2;
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RealScalar residualNorm2 = residual.squaredNorm();
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if (residualNorm2 < threshold)
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{
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iters = 0;
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tol_error = sqrt(residualNorm2 / rhsNorm2);
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return;
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}
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VectorType p(n);
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p = filter * precond.solve(residual); //initial search direction
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VectorType z(n), tmp(n);
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RealScalar absNew = numext::real(residual.dot(p)); // the square of the absolute value of r scaled by invM
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int i = 0;
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while(i < maxIters)
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{
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tmp.noalias() = filter * (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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residualNorm2 = residual.squaredNorm();
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if(residualNorm2 < threshold)
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break;
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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 = numext::real(residual.dot(z)); // update the absolute value of r
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RealScalar beta = absNew / absOld; // calculate the Gram-Schmidt value used to create the new search direction
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p = filter * (z + beta * p); // update search direction
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i++;
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}
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tol_error = sqrt(residualNorm2 / rhsNorm2);
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iters = i;
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}
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}
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#if 0 /* unused */
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template<typename MatrixType>
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struct MatrixFilter
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{
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MatrixFilter() :
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m_cmat(NULL)
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{
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}
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MatrixFilter(const MatrixType &cmat) :
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m_cmat(&cmat)
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{
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}
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void setMatrix(const MatrixType &cmat) { m_cmat = &cmat; }
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template <typename VectorType>
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void apply(VectorType v) const
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{
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v = (*m_cmat) * v;
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}
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protected:
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const MatrixType *m_cmat;
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};
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#endif
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template< typename _MatrixType, int _UpLo=Lower,
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typename _FilterMatrixType = _MatrixType,
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typename _Preconditioner = DiagonalPreconditioner<typename _MatrixType::Scalar> >
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class ConstrainedConjugateGradient;
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namespace internal {
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template< typename _MatrixType, int _UpLo, typename _FilterMatrixType, typename _Preconditioner>
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struct traits<ConstrainedConjugateGradient<_MatrixType,_UpLo,_FilterMatrixType,_Preconditioner> >
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{
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typedef _MatrixType MatrixType;
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typedef _FilterMatrixType FilterMatrixType;
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typedef _Preconditioner Preconditioner;
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};
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}
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/** \ingroup IterativeLinearSolvers_Module
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* \brief A conjugate gradient solver for sparse self-adjoint problems with additional constraints
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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 defaults are the size of the problem for the maximal number of iterations
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* and NumTraits<Scalar>::epsilon() 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.compute(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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* \sa class SimplicialCholesky, DiagonalPreconditioner, IdentityPreconditioner
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*/
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template< typename _MatrixType, int _UpLo, typename _FilterMatrixType, typename _Preconditioner>
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class ConstrainedConjugateGradient : public IterativeSolverBase<ConstrainedConjugateGradient<_MatrixType,_UpLo,_FilterMatrixType,_Preconditioner> >
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{
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typedef IterativeSolverBase<ConstrainedConjugateGradient> Base;
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using Base::mp_matrix;
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using Base::m_error;
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using Base::m_iterations;
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using Base::m_info;
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using Base::m_isInitialized;
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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 _FilterMatrixType FilterMatrixType;
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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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ConstrainedConjugateGradient() : Base() {}
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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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* This constructor is a shortcut for the default constructor followed
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* by a call to compute().
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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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ConstrainedConjugateGradient(const MatrixType& A) : Base(A) {}
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~ConstrainedConjugateGradient() {}
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FilterMatrixType &filter() { return m_filter; }
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const FilterMatrixType &filter() const { return m_filter; }
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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::solve_retval_with_guess<ConstrainedConjugateGradient, 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(Base::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_with_guess
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<ConstrainedConjugateGradient, Rhs, Guess>(*this, b.derived(), x0);
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}
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/** \internal */
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template<typename Rhs,typename Dest>
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void _solveWithGuess(const Rhs& b, Dest& x) const
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{
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m_iterations = Base::maxIterations();
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m_error = Base::m_tolerance;
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for(int j=0; j<b.cols(); ++j)
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{
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m_iterations = Base::maxIterations();
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m_error = Base::m_tolerance;
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typename Dest::ColXpr xj(x,j);
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internal::constrained_conjugate_gradient(mp_matrix->template selfadjointView<UpLo>(), b.col(j), xj, m_filter,
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Base::m_preconditioner, m_iterations, m_error);
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}
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m_isInitialized = true;
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m_info = m_error <= Base::m_tolerance ? Success : NoConvergence;
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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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x.setOnes();
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_solveWithGuess(b,x);
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}
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protected:
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FilterMatrixType m_filter;
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};
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namespace internal {
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template<typename _MatrixType, int _UpLo, typename _Filter, typename _Preconditioner, typename Rhs>
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struct solve_retval<ConstrainedConjugateGradient<_MatrixType,_UpLo,_Filter,_Preconditioner>, Rhs>
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: solve_retval_base<ConstrainedConjugateGradient<_MatrixType,_UpLo,_Filter,_Preconditioner>, Rhs>
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{
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typedef ConstrainedConjugateGradient<_MatrixType,_UpLo,_Filter,_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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dec()._solve(rhs(),dst);
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}
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};
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} // end namespace internal
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} // end namespace Eigen
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#endif // EIGEN_CONSTRAINEDCG_H
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