internal/ceres/iterative_refiner.h

// Ceres Solver - A fast non-linear least squares minimizer
// Copyright 2018 Google Inc. All rights reserved.
// http://ceres-solver.org/
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// Author: sameeragarwal@google.com (Sameer Agarwal)

#ifndef CERES_INTERNAL_ITERATIVE_REFINER_H_
#define CERES_INTERNAL_ITERATIVE_REFINER_H_

// This include must come before any #ifndef check on Ceres compile options.
#include "ceres/internal/port.h"
#include "ceres/internal/eigen.h"

namespace ceres {
namespace internal {

class SparseMatrix;
class SparseCholesky;

// Iterative refinement
// (https://en.wikipedia.org/wiki/Iterative_refinement) is the process
// of improving the solution to a linear system, by using the
// following iteration.
//
// r_i = b - Ax_i
// Ad_i = r_i
// x_{i+1} = x_i + d_i
//
// IterativeRefiner implements this process for Symmetric Positive
// Definite linear systems.
//
// The above iterative loop is run until max_num_iterations is reached
// or the following convergence criterion is satisfied:
//
//    |b - Ax|
// ------------- < 5e-15
// |A| |x| + |b|
//
// All norms in the above expression are max-norms. The above
// expression is what is recommended and used by Hogg & Scott in "A
// fast and robust mixed-precision solver for the solution of sparse
// symmetric linear systems".
//
// For example usage, please see sparse_normal_cholesky_solver.cc
class IterativeRefiner {
 public:
  struct Summary {
    bool converged = false;
    int num_iterations = -1;
    double lhs_max_norm = -1;
    double rhs_max_norm = -1;
    double solution_max_norm = -1;
    double residual_max_norm = -1;
  };

  // max_num_iterations is the maximum number of refinement iterations
  // to perform.
  IterativeRefiner(int max_num_iterations);

  // Needed for mocking.
  virtual ~IterativeRefiner();

  // Given an initial estimate of the solution of lhs * x = rhs, use
  // iterative refinement to improve it.
  //
  // sparse_cholesky is assumed to contain an already computed
  // factorization (or approximation thereof) of lhs.
  //
  // solution is expected to contain a approximation to the solution
  // to lhs * x = rhs. It can be zero.
  //
  // This method is virtual to facilitate mocking.
  //
  // TODO(sameeragarwal): Consider dropping the Summary object, and
  // simplifying the internal implementation to improve efficiency,
  // since we do not seem to be using the output at all.
  virtual Summary Refine(const SparseMatrix& lhs,
                         const double* rhs,
                         SparseCholesky* sparse_cholesky,
                         double* solution);

 private:
  void Allocate(int num_cols);

  int max_num_iterations_;
  Vector residual_;
  Vector correction_;
  Vector lhs_x_solution_;
};

}  // namespace internal
}  // namespace ceres

#endif  // CERES_INTERNAL_ITERATIVE_REFINER_H_