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diff --git a/internal/ceres/implicit_schur_complement.cc b/internal/ceres/implicit_schur_complement.cc
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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)
+
+#include "ceres/implicit_schur_complement.h"
+
+#include "Eigen/Dense"
+#include "ceres/block_sparse_matrix.h"
+#include "ceres/block_structure.h"
+#include "ceres/internal/eigen.h"
+#include "ceres/linear_solver.h"
+#include "ceres/types.h"
+#include "glog/logging.h"
+
+namespace ceres {
+namespace internal {
+
+ImplicitSchurComplement::ImplicitSchurComplement(
+ const LinearSolver::Options& options)
+ : options_(options),
+ D_(NULL),
+ b_(NULL) {
+}
+
+ImplicitSchurComplement::~ImplicitSchurComplement() {
+}
+
+void ImplicitSchurComplement::Init(const BlockSparseMatrix& A,
+ const double* D,
+ const double* b) {
+ // Since initialization is reasonably heavy, perhaps we can save on
+ // constructing a new object everytime.
+ if (A_ == NULL) {
+ A_.reset(PartitionedMatrixViewBase::Create(options_, A));
+ }
+
+ D_ = D;
+ b_ = b;
+
+ // Initialize temporary storage and compute the block diagonals of
+ // E'E and F'E.
+ if (block_diagonal_EtE_inverse_ == NULL) {
+ block_diagonal_EtE_inverse_.reset(A_->CreateBlockDiagonalEtE());
+ if (options_.preconditioner_type == JACOBI) {
+ block_diagonal_FtF_inverse_.reset(A_->CreateBlockDiagonalFtF());
+ }
+ rhs_.resize(A_->num_cols_f());
+ rhs_.setZero();
+ tmp_rows_.resize(A_->num_rows());
+ tmp_e_cols_.resize(A_->num_cols_e());
+ tmp_e_cols_2_.resize(A_->num_cols_e());
+ tmp_f_cols_.resize(A_->num_cols_f());
+ } else {
+ A_->UpdateBlockDiagonalEtE(block_diagonal_EtE_inverse_.get());
+ if (options_.preconditioner_type == JACOBI) {
+ A_->UpdateBlockDiagonalFtF(block_diagonal_FtF_inverse_.get());
+ }
+ }
+
+ // The block diagonals of the augmented linear system contain
+ // contributions from the diagonal D if it is non-null. Add that to
+ // the block diagonals and invert them.
+ AddDiagonalAndInvert(D_, block_diagonal_EtE_inverse_.get());
+ if (options_.preconditioner_type == JACOBI) {
+ AddDiagonalAndInvert((D_ == NULL) ? NULL : D_ + A_->num_cols_e(),
+ block_diagonal_FtF_inverse_.get());
+ }
+
+ // Compute the RHS of the Schur complement system.
+ UpdateRhs();
+}
+
+// Evaluate the product
+//
+// Sx = [F'F - F'E (E'E)^-1 E'F]x
+//
+// By breaking it down into individual matrix vector products
+// involving the matrices E and F. This is implemented using a
+// PartitionedMatrixView of the input matrix A.
+void ImplicitSchurComplement::RightMultiply(const double* x, double* y) const {
+ // y1 = F x
+ tmp_rows_.setZero();
+ A_->RightMultiplyF(x, tmp_rows_.data());
+
+ // y2 = E' y1
+ tmp_e_cols_.setZero();
+ A_->LeftMultiplyE(tmp_rows_.data(), tmp_e_cols_.data());
+
+ // y3 = -(E'E)^-1 y2
+ tmp_e_cols_2_.setZero();
+ block_diagonal_EtE_inverse_->RightMultiply(tmp_e_cols_.data(),
+ tmp_e_cols_2_.data());
+ tmp_e_cols_2_ *= -1.0;
+
+ // y1 = y1 + E y3
+ A_->RightMultiplyE(tmp_e_cols_2_.data(), tmp_rows_.data());
+
+ // y5 = D * x
+ if (D_ != NULL) {
+ ConstVectorRef Dref(D_ + A_->num_cols_e(), num_cols());
+ VectorRef(y, num_cols()) =
+ (Dref.array().square() *
+ ConstVectorRef(x, num_cols()).array()).matrix();
+ } else {
+ VectorRef(y, num_cols()).setZero();
+ }
+
+ // y = y5 + F' y1
+ A_->LeftMultiplyF(tmp_rows_.data(), y);
+}
+
+// Given a block diagonal matrix and an optional array of diagonal
+// entries D, add them to the diagonal of the matrix and compute the
+// inverse of each diagonal block.
+void ImplicitSchurComplement::AddDiagonalAndInvert(
+ const double* D,
+ BlockSparseMatrix* block_diagonal) {
+ const CompressedRowBlockStructure* block_diagonal_structure =
+ block_diagonal->block_structure();
+ for (int r = 0; r < block_diagonal_structure->rows.size(); ++r) {
+ const int row_block_pos = block_diagonal_structure->rows[r].block.position;
+ const int row_block_size = block_diagonal_structure->rows[r].block.size;
+ const Cell& cell = block_diagonal_structure->rows[r].cells[0];
+ MatrixRef m(block_diagonal->mutable_values() + cell.position,
+ row_block_size, row_block_size);
+
+ if (D != NULL) {
+ ConstVectorRef d(D + row_block_pos, row_block_size);
+ m += d.array().square().matrix().asDiagonal();
+ }
+
+ m = m
+ .selfadjointView<Eigen::Upper>()
+ .llt()
+ .solve(Matrix::Identity(row_block_size, row_block_size));
+ }
+}
+
+// Similar to RightMultiply, use the block structure of the matrix A
+// to compute y = (E'E)^-1 (E'b - E'F x).
+void ImplicitSchurComplement::BackSubstitute(const double* x, double* y) {
+ const int num_cols_e = A_->num_cols_e();
+ const int num_cols_f = A_->num_cols_f();
+ const int num_cols = A_->num_cols();
+ const int num_rows = A_->num_rows();
+
+ // y1 = F x
+ tmp_rows_.setZero();
+ A_->RightMultiplyF(x, tmp_rows_.data());
+
+ // y2 = b - y1
+ tmp_rows_ = ConstVectorRef(b_, num_rows) - tmp_rows_;
+
+ // y3 = E' y2
+ tmp_e_cols_.setZero();
+ A_->LeftMultiplyE(tmp_rows_.data(), tmp_e_cols_.data());
+
+ // y = (E'E)^-1 y3
+ VectorRef(y, num_cols).setZero();
+ block_diagonal_EtE_inverse_->RightMultiply(tmp_e_cols_.data(), y);
+
+ // The full solution vector y has two blocks. The first block of
+ // variables corresponds to the eliminated variables, which we just
+ // computed via back substitution. The second block of variables
+ // corresponds to the Schur complement system, so we just copy those
+ // values from the solution to the Schur complement.
+ VectorRef(y + num_cols_e, num_cols_f) = ConstVectorRef(x, num_cols_f);
+}
+
+// Compute the RHS of the Schur complement system.
+//
+// rhs = F'b - F'E (E'E)^-1 E'b
+//
+// Like BackSubstitute, we use the block structure of A to implement
+// this using a series of matrix vector products.
+void ImplicitSchurComplement::UpdateRhs() {
+ // y1 = E'b
+ tmp_e_cols_.setZero();
+ A_->LeftMultiplyE(b_, tmp_e_cols_.data());
+
+ // y2 = (E'E)^-1 y1
+ Vector y2 = Vector::Zero(A_->num_cols_e());
+ block_diagonal_EtE_inverse_->RightMultiply(tmp_e_cols_.data(), y2.data());
+
+ // y3 = E y2
+ tmp_rows_.setZero();
+ A_->RightMultiplyE(y2.data(), tmp_rows_.data());
+
+ // y3 = b - y3
+ tmp_rows_ = ConstVectorRef(b_, A_->num_rows()) - tmp_rows_;
+
+ // rhs = F' y3
+ rhs_.setZero();
+ A_->LeftMultiplyF(tmp_rows_.data(), rhs_.data());
+}
+
+} // namespace internal
+} // namespace ceres