Matrix-free solvers

Iterative solvers such as ConjugateGradient and BiCGSTAB can be used in a matrix free context. To this end, user must provide a wrapper class inheriting EigenBase<> and implementing the following methods:

• Index rows() and Index cols(): returns number of rows and columns respectively
• operator* with your type and an Eigen dense column vector (its actual implementation goes in a specialization of the internal::generic_product_impl class)

Eigen::internal::traits<> must also be specialized for the wrapper type.

Here is a complete example wrapping an Eigen::SparseMatrix:

#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 inherit its traits:
template <>
struct traits<MatrixReplacement> : public Eigen::internal::traits<Eigen::SparseMatrix<double> > {};
}  // namespace internal
}  // namespace Eigen

// 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 mp_mat->rows(); }
  Index cols() const { return mp_mat->cols(); }

  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() : mp_mat(0) {}

  void attachMyMatrix(const SparseMatrix<double>& mat) { mp_mat = &mat; }
  const SparseMatrix<double> my_matrix() const { return *mp_mat; }

 private:
  const SparseMatrix<double>* mp_mat;
};

// 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.
    eigen_assert(alpha == Scalar(1) && "scaling is not implemented");
    EIGEN_ONLY_USED_FOR_DEBUG(alpha);

    // Here we could simply call dst.noalias() += lhs.my_matrix() * rhs,
    // but let's do something fancier (and less efficient):
    for (Index i = 0; i < lhs.cols(); ++i) dst += rhs(i) * lhs.my_matrix().col(i);
  }
};

}  // namespace internal
}  // namespace Eigen

int main() {
  int n = 10;
  Eigen::SparseMatrix<double> S = Eigen::MatrixXd::Random(n, n).sparseView(0.5, 1);
  S = S.transpose() * S;

  MatrixReplacement A;
  A.attachMyMatrix(S);

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

  // 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;
  }

  {
    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;
  }

  {
    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;
  }

  {
    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;
  }

  {
    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;
  }
}

Output:

CG:       #iterations: 19, estimated error: 1.0666e-18
BiCGSTAB: #iterations: 20, estimated error: 1.56382e-24
GMRES:    #iterations: 10, estimated error: 0
DGMRES:   #iterations: 20, estimated error: 5.63173e-29
MINRES:   #iterations: 19, estimated error: 5.04856e-18