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-// Ceres Solver - A fast non-linear least squares minimizer
-// Copyright 2015 Google Inc. All rights reserved.
-// http://ceres-solver.org/
-//
-// Redistribution and use in source and binary forms, with or without
-// modification, are permitted provided that the following conditions are met:
-//
-// * Redistributions of source code must retain the above copyright notice,
-// this list of conditions and the following disclaimer.
-// * Redistributions in binary form must reproduce the above copyright notice,
-// this list of conditions and the following disclaimer in the documentation
-// and/or other materials provided with the distribution.
-// * Neither the name of Google Inc. nor the names of its contributors may be
-// used to endorse or promote products derived from this software without
-// specific prior written permission.
-//
-// THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
-// AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
-// IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
-// ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE
-// LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
-// CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
-// SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
-// INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
-// CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
-// ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
-// POSSIBILITY OF SUCH DAMAGE.
-//
-// Author: sameeragarwal@google.com (Sameer Agarwal)
-//
-// An iterative solver for solving the Schur complement/reduced camera
-// linear system that arise in SfM problems.
-
-#ifndef CERES_INTERNAL_IMPLICIT_SCHUR_COMPLEMENT_H_
-#define CERES_INTERNAL_IMPLICIT_SCHUR_COMPLEMENT_H_
-
-#include "ceres/linear_operator.h"
-#include "ceres/linear_solver.h"
-#include "ceres/partitioned_matrix_view.h"
-#include "ceres/internal/eigen.h"
-#include "ceres/internal/scoped_ptr.h"
-#include "ceres/types.h"
-
-namespace ceres {
-namespace internal {
-
-class BlockSparseMatrix;
-
-// This class implements various linear algebraic operations related
-// to the Schur complement without explicitly forming it.
-//
-//
-// Given a reactangular linear system Ax = b, where
-//
-// A = [E F]
-//
-// The normal equations are given by
-//
-// A'Ax = A'b
-//
-// |E'E E'F||y| = |E'b|
-// |F'E F'F||z| |F'b|
-//
-// and the Schur complement system is given by
-//
-// [F'F - F'E (E'E)^-1 E'F] z = F'b - F'E (E'E)^-1 E'b
-//
-// Now if we wish to solve Ax = b in the least squares sense, one way
-// is to form this Schur complement system and solve it using
-// Preconditioned Conjugate Gradients.
-//
-// The key operation in a conjugate gradient solver is the evaluation of the
-// matrix vector product with the Schur complement
-//
-// S = F'F - F'E (E'E)^-1 E'F
-//
-// It is straightforward to see that matrix vector products with S can
-// be evaluated without storing S in memory. Instead, given (E'E)^-1
-// (which for our purposes is an easily inverted block diagonal
-// matrix), it can be done in terms of matrix vector products with E,
-// F and (E'E)^-1. This class implements this functionality and other
-// auxilliary bits needed to implement a CG solver on the Schur
-// complement using the PartitionedMatrixView object.
-//
-// THREAD SAFETY: This class is nqot thread safe. In particular, the
-// RightMultiply (and the LeftMultiply) methods are not thread safe as
-// they depend on mutable arrays used for the temporaries needed to
-// compute the product y += Sx;
-class ImplicitSchurComplement : public LinearOperator {
- public:
- // num_eliminate_blocks is the number of E blocks in the matrix
- // A.
- //
- // preconditioner indicates whether the inverse of the matrix F'F
- // should be computed or not as a preconditioner for the Schur
- // Complement.
- //
- // TODO(sameeragarwal): Get rid of the two bools below and replace
- // them with enums.
- explicit ImplicitSchurComplement(const LinearSolver::Options& options);
- virtual ~ImplicitSchurComplement();
-
- // Initialize the Schur complement for a linear least squares
- // problem of the form
- //
- // |A | x = |b|
- // |diag(D)| |0|
- //
- // If D is null, then it is treated as a zero dimensional matrix. It
- // is important that the matrix A have a BlockStructure object
- // associated with it and has a block structure that is compatible
- // with the SchurComplement solver.
- void Init(const BlockSparseMatrix& A, const double* D, const double* b);
-
- // y += Sx, where S is the Schur complement.
- virtual void RightMultiply(const double* x, double* y) const;
-
- // The Schur complement is a symmetric positive definite matrix,
- // thus the left and right multiply operators are the same.
- virtual void LeftMultiply(const double* x, double* y) const {
- RightMultiply(x, y);
- }
-
- // y = (E'E)^-1 (E'b - E'F x). Given an estimate of the solution to
- // the Schur complement system, this method computes the value of
- // the e_block variables that were eliminated to form the Schur
- // complement.
- void BackSubstitute(const double* x, double* y);
-
- virtual int num_rows() const { return A_->num_cols_f(); }
- virtual int num_cols() const { return A_->num_cols_f(); }
- const Vector& rhs() const { return rhs_; }
-
- const BlockSparseMatrix* block_diagonal_EtE_inverse() const {
- return block_diagonal_EtE_inverse_.get();
- }
-
- const BlockSparseMatrix* block_diagonal_FtF_inverse() const {
- return block_diagonal_FtF_inverse_.get();
- }
-
- private:
- void AddDiagonalAndInvert(const double* D, BlockSparseMatrix* matrix);
- void UpdateRhs();
-
- const LinearSolver::Options& options_;
-
- scoped_ptr<PartitionedMatrixViewBase> A_;
- const double* D_;
- const double* b_;
-
- scoped_ptr<BlockSparseMatrix> block_diagonal_EtE_inverse_;
- scoped_ptr<BlockSparseMatrix> block_diagonal_FtF_inverse_;
-
- Vector rhs_;
-
- // Temporary storage vectors used to implement RightMultiply.
- mutable Vector tmp_rows_;
- mutable Vector tmp_e_cols_;
- mutable Vector tmp_e_cols_2_;
- mutable Vector tmp_f_cols_;
-};
-
-} // namespace internal
-} // namespace ceres
-
-#endif // CERES_INTERNAL_IMPLICIT_SCHUR_COMPLEMENT_H_