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diff --git a/internal/ceres/implicit_schur_complement_test.cc b/internal/ceres/implicit_schur_complement_test.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 <cstddef>
+#include <memory>
+#include "Eigen/Dense"
+#include "ceres/block_random_access_dense_matrix.h"
+#include "ceres/block_sparse_matrix.h"
+#include "ceres/casts.h"
+#include "ceres/context_impl.h"
+#include "ceres/internal/eigen.h"
+#include "ceres/linear_least_squares_problems.h"
+#include "ceres/linear_solver.h"
+#include "ceres/schur_eliminator.h"
+#include "ceres/triplet_sparse_matrix.h"
+#include "ceres/types.h"
+#include "glog/logging.h"
+#include "gtest/gtest.h"
+
+namespace ceres {
+namespace internal {
+
+using testing::AssertionResult;
+
+const double kEpsilon = 1e-14;
+
+class ImplicitSchurComplementTest : public ::testing::Test {
+ protected :
+ virtual void SetUp() {
+ std::unique_ptr<LinearLeastSquaresProblem> problem(
+ CreateLinearLeastSquaresProblemFromId(2));
+
+ CHECK(problem != nullptr);
+ A_.reset(down_cast<BlockSparseMatrix*>(problem->A.release()));
+ b_.reset(problem->b.release());
+ D_.reset(problem->D.release());
+
+ num_cols_ = A_->num_cols();
+ num_rows_ = A_->num_rows();
+ num_eliminate_blocks_ = problem->num_eliminate_blocks;
+ }
+
+ void ReducedLinearSystemAndSolution(double* D,
+ Matrix* lhs,
+ Vector* rhs,
+ Vector* solution) {
+ const CompressedRowBlockStructure* bs = A_->block_structure();
+ const int num_col_blocks = bs->cols.size();
+ std::vector<int> blocks(num_col_blocks - num_eliminate_blocks_, 0);
+ for (int i = num_eliminate_blocks_; i < num_col_blocks; ++i) {
+ blocks[i - num_eliminate_blocks_] = bs->cols[i].size;
+ }
+
+ BlockRandomAccessDenseMatrix blhs(blocks);
+ const int num_schur_rows = blhs.num_rows();
+
+ LinearSolver::Options options;
+ options.elimination_groups.push_back(num_eliminate_blocks_);
+ options.type = DENSE_SCHUR;
+ ContextImpl context;
+ options.context = &context;
+
+ std::unique_ptr<SchurEliminatorBase> eliminator(
+ SchurEliminatorBase::Create(options));
+ CHECK(eliminator != nullptr);
+ const bool kFullRankETE = true;
+ eliminator->Init(num_eliminate_blocks_, kFullRankETE, bs);
+
+ lhs->resize(num_schur_rows, num_schur_rows);
+ rhs->resize(num_schur_rows);
+
+ eliminator->Eliminate(A_.get(), b_.get(), D, &blhs, rhs->data());
+
+ MatrixRef lhs_ref(blhs.mutable_values(), num_schur_rows, num_schur_rows);
+
+ // lhs_ref is an upper triangular matrix. Construct a full version
+ // of lhs_ref in lhs by transposing lhs_ref, choosing the strictly
+ // lower triangular part of the matrix and adding it to lhs_ref.
+ *lhs = lhs_ref;
+ lhs->triangularView<Eigen::StrictlyLower>() =
+ lhs_ref.triangularView<Eigen::StrictlyUpper>().transpose();
+
+ solution->resize(num_cols_);
+ solution->setZero();
+ VectorRef schur_solution(solution->data() + num_cols_ - num_schur_rows,
+ num_schur_rows);
+ schur_solution = lhs->selfadjointView<Eigen::Upper>().llt().solve(*rhs);
+ eliminator->BackSubstitute(A_.get(), b_.get(), D,
+ schur_solution.data(), solution->data());
+ }
+
+ AssertionResult TestImplicitSchurComplement(double* D) {
+ Matrix lhs;
+ Vector rhs;
+ Vector reference_solution;
+ ReducedLinearSystemAndSolution(D, &lhs, &rhs, &reference_solution);
+
+ LinearSolver::Options options;
+ options.elimination_groups.push_back(num_eliminate_blocks_);
+ options.preconditioner_type = JACOBI;
+ ContextImpl context;
+ options.context = &context;
+ ImplicitSchurComplement isc(options);
+ isc.Init(*A_, D, b_.get());
+
+ int num_sc_cols = lhs.cols();
+
+ for (int i = 0; i < num_sc_cols; ++i) {
+ Vector x(num_sc_cols);
+ x.setZero();
+ x(i) = 1.0;
+
+ Vector y(num_sc_cols);
+ y = lhs * x;
+
+ Vector z(num_sc_cols);
+ isc.RightMultiply(x.data(), z.data());
+
+ // The i^th column of the implicit schur complement is the same as
+ // the explicit schur complement.
+ if ((y - z).norm() > kEpsilon) {
+ return testing::AssertionFailure()
+ << "Explicit and Implicit SchurComplements differ in "
+ << "column " << i << ". explicit: " << y.transpose()
+ << " implicit: " << z.transpose();
+ }
+ }
+
+ // Compare the rhs of the reduced linear system
+ if ((isc.rhs() - rhs).norm() > kEpsilon) {
+ return testing::AssertionFailure()
+ << "Explicit and Implicit SchurComplements differ in "
+ << "rhs. explicit: " << rhs.transpose()
+ << " implicit: " << isc.rhs().transpose();
+ }
+
+ // Reference solution to the f_block.
+ const Vector reference_f_sol =
+ lhs.selfadjointView<Eigen::Upper>().llt().solve(rhs);
+
+ // Backsubstituted solution from the implicit schur solver using the
+ // reference solution to the f_block.
+ Vector sol(num_cols_);
+ isc.BackSubstitute(reference_f_sol.data(), sol.data());
+ if ((sol - reference_solution).norm() > kEpsilon) {
+ return testing::AssertionFailure()
+ << "Explicit and Implicit SchurComplements solutions differ. "
+ << "explicit: " << reference_solution.transpose()
+ << " implicit: " << sol.transpose();
+ }
+
+ return testing::AssertionSuccess();
+ }
+
+ int num_rows_;
+ int num_cols_;
+ int num_eliminate_blocks_;
+
+ std::unique_ptr<BlockSparseMatrix> A_;
+ std::unique_ptr<double[]> b_;
+ std::unique_ptr<double[]> D_;
+};
+
+// Verify that the Schur Complement matrix implied by the
+// ImplicitSchurComplement class matches the one explicitly computed
+// by the SchurComplement solver.
+//
+// We do this with and without regularization to check that the
+// support for the LM diagonal is correct.
+TEST_F(ImplicitSchurComplementTest, SchurMatrixValuesTest) {
+ EXPECT_TRUE(TestImplicitSchurComplement(NULL));
+ EXPECT_TRUE(TestImplicitSchurComplement(D_.get()));
+}
+
+} // namespace internal
+} // namespace ceres