gmres_householder.hpp 5.63 KB
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#pragma once
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#include <algorithm>
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#include <boost/numeric/mtl/concept/collection.hpp>
#include <boost/numeric/mtl/vector/dense_vector.hpp>
#include <boost/numeric/mtl/matrix/dense2D.hpp>
#include <boost/numeric/mtl/matrix/multi_vector.hpp>
#include <boost/numeric/mtl/operation/givens.hpp>
#include <boost/numeric/mtl/operation/two_norm.hpp>
#include <boost/numeric/mtl/utility/exception.hpp>
#include <boost/numeric/mtl/utility/irange.hpp>

#include "details.hpp"

namespace itl
{
  /// Generalized Minimal Residual method (without restart) using householder othogonalization.
  /** It computes at most kmax_in iterations (or size(x) depending on what is smaller)
      regardless on whether the termination criterion is reached or not.   **/
  template <typename Matrix, typename Vector, typename LeftPreconditioner, typename Iteration>
  int gmres_householder_full(const Matrix& A, Vector& x, const Vector& b,
                             LeftPreconditioner& L, Iteration& iter)
  {
    using mtl::irange;
    using std::abs;
    using math::reciprocal;
    using mtl::iall;
    using mtl::imax;
    using mtl::signum;
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    using mtl::vec::dot;
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    using mtl::conj;
    typedef typename mtl::Collection<Vector>::value_type Scalar;
    typedef typename mtl::Collection<Vector>::size_type  Size;

    if (size(b) == 0) throw mtl::logic_error("empty rhs vector");

    const Scalar zero= math::zero(Scalar()), dbl_tol= 1.e-16;
    Scalar       rho, bnrm2, temp, beta;
    Size         k, kmax(std::min(size(x), Size(iter.max_iterations() - iter.iterations())));
    Vector       w(b - A *x), r(solve(L,w));
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    mtl::mat::multi_vector<Vector>   V(Vector(resource(x), zero), kmax+1);
    mtl::vec::dense_vector<Scalar>   sn(kmax, zero), cs(kmax, zero), s(kmax+1, zero), y(kmax, zero);  // replicated in distributed solvers
    mtl::mat::dense2D<Scalar>        H(kmax, kmax);
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    bnrm2 = two_norm(b);
    if (bnrm2 < dbl_tol)
      bnrm2 = 1.0;

    temp = two_norm(r);				// norm of preconditioned residual
    rho = temp * reciprocal(bnrm2);
    if (iter.finished(rho))			// initial guess is good enough solution
      return iter;

    // u = r + sign(r(0))*||r||*e0
    beta = signum(r[0])*temp;
    w = r;
    w[0] += beta;
    w *= reciprocal(two_norm(w));

    V.vector(0) = w;
    H = zero;
    s[0] = -beta;

    // GMRES iteration
    for (k= 0; k < kmax && !iter.finished(rho); ++k, ++iter)
    {

      w = (-2.0 * V.vector(k)[k])*V.vector(k);
      w[k] += 1.0;
      // v := P_0*...*P_{k-2}*(P_{k-1} * e_k)
      for (Size i= k; i > 0; i--)
      {
        temp = 2.0 * dot(V.vector(i-1), w);
        w -= temp * V.vector(i-1);
      }

      temp = two_norm(w);
      if (temp == zero)
        return iter.fail(2, "GMRES: breakdown");

      // Explicitly normalize v to reduce the effects of round-off.
      w *= reciprocal(temp);
      w = solve(L, Vector(A*w));

      // P_{k-1}*...*P_0*Av
      for (Size i = 0; i <= k; i++)
      {
        temp = 2.0 * dot(V.vector(i), w);
        w -= temp * V.vector(i);
      }

      temp = two_norm(w);
      if (temp == zero)
        return iter.fail(3, "GMRES: breakdown");

      irange range_to_end(k+1,imax);
      set_to_zero(V.vector(k+1));
      V.vector(k+1)[range_to_end] = w[range_to_end];
      beta = two_norm(V.vector(k+1));
      if (beta != 0.0)
      {
        beta *= signum(w[k+1]);
        V.vector(k+1)[k+1] += beta;
        V.vector(k+1) *= reciprocal(two_norm(V.vector(k+1)));

        w[k+1] = -beta;
      }

      for (Size i= 0; i < k; i++)
      {
        temp   =  conj(cs[i])*w[i] + conj(sn[i])*w[i+1];
        w[i+1] = -sn[i]*w[i] + cs[i]*w[i+1];
        w[i]   =  temp;
      }

      details::rotmat(w[k], w[k+1], cs[k], sn[k]);

      s[k+1] = -sn[k]*s[k];
      s[k]   = conj(cs[k])*s[k];
      w[k]   = cs[k]*w[k] + sn[k]*w[k+1];
      w[k+1] = 0.0;

      irange range(num_rows(H));
      H[iall][k] = w[range];

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      using std::abs;
      rho = abs(s[k+1]) / bnrm2;
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    }

    // reduce k, to get regular matrix
    //     while (k > 0 && std::abs(s[k-1]) <= iter.atol()) k--;

    // iteration is finished -> compute x: solve H*y=s as far as rank of H allows
    irange range(k);
    for (; !range.empty(); --range)
    {
      try
      {
        y[range] = upper_trisolve(H[range][range], s[range]);
      }
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      catch (mtl::matrix_singular const&)
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      {
        continue;    // if singular then try with sub-matrix
      }
      break;
    }

    if (range.finish() < k)
      std::cerr << "GMRES orhogonalized with " << k << " vectors but matrix singular, can only use "
                << range.finish() << " vectors!\n";
    if (range.empty())
      return iter.fail(3, "GMRES did not find any direction to correct x");

    kmax = k-1;

    w = V.vector(kmax) * (-2.0 * y[kmax] * conj(V.vector(kmax)[kmax]));
    w[kmax] += y[kmax];
    for (Size i= kmax; i > 0; i--)
    {
      w[i-1] += y[i-1];
      temp = 2.0 * dot(V.vector(i-1), w);
      w -= temp * V.vector(i-1);
    }
    x += w;

    r = b - A*x;
    return iter.terminate(r);
  }

  /// Generalized Minimal Residual method with restart
  template <typename Matrix, typename Vector, typename LeftPreconditioner,
            typename Iteration>
  int gmres_householder(const Matrix& A, Vector& x, const Vector& b,
                        LeftPreconditioner& L,
                        Iteration& iter, typename mtl::Collection<Vector>::size_type restart)
  {
    do
    {
      Iteration inner(iter);
      inner.set_max_iterations(std::min(int(iter.iterations()+restart), iter.max_iterations()));
      inner.suppress_resume(true);
      gmres_householder_full(A, x, b, L, inner);
      iter.update_progress(inner);
    }
    while (!iter.finished());

    return iter;
  }

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} // end namespace itl