eigen/doc/examples/matrixfree.cpp

#include <iostream>
#include <Eigen/Core>
#include <Eigen/Dense>
#include <Eigen/IterativeLinearSolvers>
#include <unsupported/Eigen/IterativeSolvers>

class MatrixReplacement;
using Eigen::SparseMatrix;

namespace Eigen {
namespace internal {
  // MatrixReplacement looks-like a SparseMatrix, so let's inherits its traits:
  template<>
  struct traits<MatrixReplacement> :  public Eigen::internal::traits<Eigen::SparseMatrix<double> >
  {};
}
}

// Example of a matrix-free wrapper from a user type to Eigen's compatible type
// For the sake of simplicity, this example simply wrap a Eigen::SparseMatrix.
class MatrixReplacement : public Eigen::EigenBase<MatrixReplacement> {
public:
  // Required typedefs, constants, and method:
  typedef double Scalar;
  typedef double RealScalar;
  typedef int StorageIndex;
  enum {
    ColsAtCompileTime = Eigen::Dynamic,
    MaxColsAtCompileTime = Eigen::Dynamic,
    IsRowMajor = false
  };

  Index rows() const { return _n; }
  Index cols() const { return _n; }

  template<typename Rhs>
  Eigen::Product<MatrixReplacement,Rhs,Eigen::AliasFreeProduct> operator*(const Eigen::MatrixBase<Rhs>& x) const {
    return Eigen::Product<MatrixReplacement,Rhs,Eigen::AliasFreeProduct>(*this, x.derived());
  }

  // Custom API:
  MatrixReplacement(int n){ _n = n; }

private:
  int _n;
};


// Implementation of MatrixReplacement * Eigen::DenseVector though a specialization of internal::generic_product_impl:
namespace Eigen {
namespace internal {

  template<typename Rhs>
  struct generic_product_impl<MatrixReplacement, Rhs, SparseShape, DenseShape, GemvProduct> // GEMV stands for matrix-vector
  : generic_product_impl_base<MatrixReplacement,Rhs,generic_product_impl<MatrixReplacement,Rhs> >
  {
    typedef typename Product<MatrixReplacement,Rhs>::Scalar Scalar;

    template<typename Dest>
    static void scaleAndAddTo(Dest& dst, const MatrixReplacement& lhs, const Rhs& rhs, const Scalar& alpha)
    {
      // This method should implement "dst += alpha * lhs * rhs" inplace,
      // however, for iterative solvers, alpha is always equal to 1, so let's not bother about it.
      assert(alpha==Scalar(1) && "scaling is not implemented");
      EIGEN_ONLY_USED_FOR_DEBUG(alpha);

      // LLS:
      // the matrix comes from 1D Poisson Eq. -u_{xx} = f
      // FD scheme is: 2 U_{i} - U_{i-1}-U_{i+1} = f_i, i=0,...,n-1
      // BC condition: U_{-1}=0, U_{n}=0
      int n = lhs.cols();
      Scalar *dst_ptr = dst.data();
      const Scalar *rhs_ptr = rhs.data();

      for(int i=0;i<n;i++) {
        Scalar v=rhs_ptr[i];
        Scalar vp,vm;
        if(i==n-1) {
          vp = 0.0;
        } else {
          vp = rhs_ptr[i+1];
        }
        if(i==0) {
          vm = 0.0;
        } else {
          vm = rhs_ptr[i-1];
        }
        dst_ptr[i]=2.0*v-vp-vm;
      }
    }
  };

}
}

int main()
{
  int n = 4;

  MatrixReplacement A(n);

  Eigen::VectorXd b(n), x;
  b.setOnes();

  // Solve Ax = b using various iterative solver with matrix-free version:
  {
    Eigen::ConjugateGradient<MatrixReplacement, Eigen::Lower|Eigen::Upper, Eigen::IdentityPreconditioner> cg;
    cg.compute(A);
    x = cg.solve(b);
    std::cout << "CG:       #iterations: " << cg.iterations() << ", estimated error: " << cg.error() << std::endl;
    for(int i=0;i<n;i++) std::cout << "x["<<i<<"] = "<<x[i]<<"\n";
  }

  {
    Eigen::BiCGSTAB<MatrixReplacement, Eigen::IdentityPreconditioner> bicg;
    bicg.compute(A);
    x = bicg.solve(b);
    std::cout << "BiCGSTAB: #iterations: " << bicg.iterations() << ", estimated error: " << bicg.error() << std::endl;
    for(int i=0;i<n;i++) std::cout << "x["<<i<<"] = "<<x[i]<<"\n";
  }

  {
    Eigen::GMRES<MatrixReplacement, Eigen::IdentityPreconditioner> gmres;
    gmres.compute(A);
    x = gmres.solve(b);
    std::cout << "GMRES:    #iterations: " << gmres.iterations() << ", estimated error: " << gmres.error() << std::endl;
    for(int i=0;i<n;i++) std::cout << "x["<<i<<"] = "<<x[i]<<"\n";
  }

  {
    Eigen::DGMRES<MatrixReplacement, Eigen::IdentityPreconditioner> gmres;
    gmres.compute(A);
    x = gmres.solve(b);
    std::cout << "DGMRES:   #iterations: " << gmres.iterations() << ", estimated error: " << gmres.error() << std::endl;
    for(int i=0;i<n;i++) std::cout << "x["<<i<<"] = "<<x[i]<<"\n";
  }

  {
    Eigen::MINRES<MatrixReplacement, Eigen::Lower|Eigen::Upper, Eigen::IdentityPreconditioner> minres;
    minres.compute(A);
    x = minres.solve(b);
    std::cout << "MINRES:   #iterations: " << minres.iterations() << ", estimated error: " << minres.error() << std::endl;
    for(int i=0;i<n;i++) std::cout << "x["<<i<<"] = "<<x[i]<<"\n";
  }
}