Re-use Eigen::SimplicialLLT's symbolic decomposition

问题

I am struggling a bit with the API of the Eigen Library, namely the SimplicialLLT class for Cholesky factorization of sparse matrices. I have three matrices that I need to factor and later use to solve many equation systems (changing only the right side) - therefore I would like to factor these matrices only once and then just re-use them. Moreover, they all have the same sparcity pattern, so I would like to do the symbolic decomposition only once and then use it for the numerical decomposition for all three matrices. According to the documentation, this is exactly what the SimplicialLLT::analyzePattern and SimplicialLLT::factor methods are for. However, I can't seem to find a way to keep all three factors in the memory. This is my code:

I have these member variables in my class I would like to fill with the factors:

Eigen::SimplicialLLT<Eigen::SparseMatrix<double>> choleskyA;
Eigen::SimplicialLLT<Eigen::SparseMatrix<double>> choleskyB;
Eigen::SimplicialLLT<Eigen::SparseMatrix<double>> choleskyC;

Then I create the three sparse matrices A, B and C and want to factor them:

choleskyA.analyzePattern(A);
choleskyA.factorize(A);
choleskyB.analyzePattern(B); // this has already been done!
choleskyB.factorize(B);
choleskyC.analyzePattern(C); // this has already been done!
choleskyC.factorize(C);

And later I can use them for solutions over and over again, changing just the b vectors of right sides:

xA = choleskyA.solve(bA);
xB = choleskyB.solve(bB);
xC = choleskyC.solve(bC);

This works (I think), but the second and third call to analyzePattern are redundant. What I would like to do is something like:

choleskyA.analyzePattern(A);
choleskyA.factorize(A);
choleskyB = choleskyA.factorize(B);
choleskyC = choleskyA.factorize(C);

But that is not an option with the current API (we use Eigen 3.2.3, but if I see correctly there is no change in this regard in 3.3.2). The problem here is that the subsequent calls to factorize on the same instance of SimplicialLLT will overwrite the previously computed factor and at the same time, I can't find a way to make a copy of it to keep. I took a look at the sources but I have to admit that didn't help much as I can't see any simple way to copy the underlying data structures. It seems to me like a rather common usage, so I feel like I am missing something obvious, please help.

What I have tried:

I tried using simply choleskyB = choleskyA hoping that the default copy constructor will get things done, but I have found out that the base classes are designed to be non-copyable.

I can get the L and U matrices (there's a getter for them) from choleskyA, make a copy of them and store only those and then basically copy-paste the content of SimplicialCholeskyBase::_solve_impl() (copy-pasted below) to write the method for solving myself using the previously stored L and U directly.

template<typename Rhs,typename Dest>
void _solve_impl(const MatrixBase<Rhs> &b, MatrixBase<Dest> &dest) const
{
  eigen_assert(m_factorizationIsOk && "The decomposition is not in a valid state for solving, you must first call either compute() or symbolic()/numeric()");
  eigen_assert(m_matrix.rows()==b.rows());

  if(m_info!=Success)
    return;

  if(m_P.size()>0)
    dest = m_P * b;
  else
    dest = b;

  if(m_matrix.nonZeros()>0) // otherwise L==I
    derived().matrixL().solveInPlace(dest);

  if(m_diag.size()>0)
    dest = m_diag.asDiagonal().inverse() * dest;

  if (m_matrix.nonZeros()>0) // otherwise U==I
    derived().matrixU().solveInPlace(dest);

  if(m_P.size()>0)
    dest = m_Pinv * dest;
}

...but that's quite an ugly solution plus I would probably screw it up since I don't have that good understanding of the process (I don't need the m_diag from the above code since I am doing LLT, right? that would be relevant only if I was using LDLT?). I hope this is not what I need to do...

A final note - adding the necessary getters/setters to the Eigen classes and compiling "my own" Eigen is not an option (well, not a good one) as this code will (hopefully) be further redistributed as open source, so it would be troublesome.

回答1:

This is a quite unusual pattern. In practice the symbolic factorization is very cheap compared to the numerical factorization, so I'm not sure it's worth bothering much. The cleanest solution to address this pattern would be to let SimplicialL?LT to be copiable.

来源：https://stackoverflow.com/questions/41899666/re-use-eigensimplicialllts-symbolic-decomposition