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+// Ceres Solver - A fast non-linear least squares minimizer
+// Copyright 2010, 2011, 2012 Google Inc. All rights reserved.
+// http://code.google.com/p/ceres-solver/
+//
+// 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/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;
+class BlockSparseMatrixBase;
+
+// 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.
+  ImplicitSchurComplement(int num_eliminate_blocks, bool preconditioner);
+  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 BlockSparseMatrixBase& 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();
+
+  int num_eliminate_blocks_;
+  bool preconditioner_;
+
+  scoped_ptr<PartitionedMatrixView> 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_