// Ceres Solver - A fast non-linear least squares minimizer
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// http://ceres-solver.org/
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// Author: sameeragarwal@google.com (Sameer Agarwal)
//
// Solve dense rectangular systems Ax = b by forming the normal
// equations and solving them using the Cholesky factorization.

#ifndef CERES_INTERNAL_DENSE_NORMAL_CHOLESKY_SOLVER_H_
#define CERES_INTERNAL_DENSE_NORMAL_CHOLESKY_SOLVER_H_

#include "ceres/dense_cholesky.h"
#include "ceres/linear_solver.h"

namespace ceres {
namespace internal {

class DenseSparseMatrix;

// This class implements the LinearSolver interface for solving
// rectangular/unsymmetric (well constrained) linear systems of the
// form
//
//   Ax = b
//
// Since there does not usually exist a solution that satisfies these
// equations, the solver instead solves the linear least squares
// problem
//
//   min_x |Ax - b|^2
//
// Setting the gradient of the above optimization problem to zero
// gives us the normal equations
//
//   A'Ax = A'b
//
// A'A is a positive definite matrix (hopefully), and the resulting
// linear system can be solved using Cholesky factorization.
//
// If the PerSolveOptions struct has a non-null array D, then the
// augmented/regularized linear system
//
//   [    A    ]x = [b]
//   [ diag(D) ]    [0]
//
// is solved.
//
// This class uses the LDLT factorization routines from the Eigen
// library. This solver always returns a solution, it is the user's
// responsibility to judge if the solution is good enough for their
// purposes.
class DenseNormalCholeskySolver : public DenseSparseMatrixSolver {
 public:
  explicit DenseNormalCholeskySolver(const LinearSolver::Options& options);

 private:
  LinearSolver::Summary SolveImpl(
      DenseSparseMatrix* A,
      const double* b,
      const LinearSolver::PerSolveOptions& per_solve_options,
      double* x) final;

  const LinearSolver::Options options_;
  std::unique_ptr<DenseCholesky> cholesky_;
};

}  // namespace internal
}  // namespace ceres

#endif  // CERES_INTERNAL_DENSE_NORMAL_CHOLESKY_SOLVER_H_