Simple LDL^T solve interface

[roryhubbard]

What?

Simplify the interface to solving a linear system via LDL^T factorization (https://en.wikipedia.org/wiki/Cholesky_decomposition).
I overloaded the preexisting "ldl_solve" function to accept just "A" and "b" and return the solution to the linear system "Ax=b".

Why?

Allow user to write less when using the "ldl_solve" function. Currently, one must write code like this:

casadi::SX D, LT;
std::vector<casadi_int> p;
ldl(A, D, LT, p, false);
auto x = ldl_solve(b, D, LT, p);

The Eigen library has a more concise interface (https://eigen.tuxfamily.org/dox/group__TutorialLinearAlgebra.html):

auto x = A.ldlt().solve(b)

This pull request will allow a user to call "ldl_solve" like this:

auto x = ldl_solve(A, b);

[jgillis]

@jaeandersson what's your opinion?
Should we instead expose this through solve(A,b,'ldl')?

[jaeandersson]

@jaeandersson what's your opinion? Should we instead expose this through solve(A,b,'ldl')?

Yes, you're right. I think we should reserve ldl_solve for the lower level routine, that is separate steps for factorization and solution. qr_solve works the same way. It does make sense to have an overload of solve(A,b,'ldl') as you suggest. It probably already works by the way, but only for casadi::DM, not for casadi::SX. You could easily add an implementation for SX here:

  template<typename Scalar>
  Matrix<Scalar> Matrix<Scalar>::
  solve(const Matrix<Scalar>& a, const Matrix<Scalar>& b,
           const std::string& lsolver, const Dict& dict) {
    casadi_error("'solve' with plugin not defined for " + type_name());
    return Matrix<Scalar>();
  }

Just make a if-else to cover ldl and qr:

  template<typename Scalar>
  Matrix<Scalar> Matrix<Scalar>::
  solve(const Matrix<Scalar>& a, const Matrix<Scalar>& b,
           const std::string& lsolver, const Dict& dict) {
    if (lsolver == "qr") {
       ...
    } else if (lsolver == "ldl") {
       ...
    } else {
        casadi_error("'solve' with plugin not defined for " + type_name());
        return Matrix<Scalar>();
    }
  }

[roryhubbard]

Okay sorry for the delay. Is it correct to use qr_sparse over qr? And how do we feel about the hard coded false values for qr_sparse, qr_solve, and ldl? I'm not sure if it would be better to set any of those to true as the default value instead.

Also, is Sparsity::is_symmetric only checking for structural symmetry as oppose to numerical symmetry? It is currently possible to call solve(A, b, 'ldl') solve with a non symmetric square A and it will return a false solution.