internal/ceres/schur_eliminator_test.cc - ceres-solver - Git at Google

 // Ceres Solver - A fast non-linear least squares minimizer
 // Copyright 2023 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/schur_eliminator.h"

 #include <algorithm>
 #include <memory>
 #include <random>
 #include <vector>

 #include "Eigen/Dense"
 #include "ceres/block_random_access_dense_matrix.h"
 #include "ceres/block_sparse_matrix.h"
 #include "ceres/block_structure.h"
 #include "ceres/casts.h"
 #include "ceres/context_impl.h"
 #include "ceres/detect_structure.h"
 #include "ceres/internal/eigen.h"
 #include "ceres/linear_least_squares_problems.h"
 #include "ceres/test_util.h"
 #include "ceres/triplet_sparse_matrix.h"
 #include "ceres/types.h"
 #include "gtest/gtest.h"

 // TODO(sameeragarwal): Reduce the size of these tests and redo the
 // parameterization to be more efficient.

 namespace ceres::internal {

 class SchurEliminatorTest : public ::testing::Test {
  protected:
   void SetUpFromId(int id) {
     auto problem = CreateLinearLeastSquaresProblemFromId(id);
     ASSERT_TRUE(problem != nullptr);
     SetupHelper(problem.get());
   }

   void SetupHelper(LinearLeastSquaresProblem* problem) {
     A.reset(down_cast<BlockSparseMatrix*>(problem->A.release()));
     b = std::move(problem->b);
     D = std::move(problem->D);

     num_eliminate_blocks = problem->num_eliminate_blocks;
     num_eliminate_cols = 0;
     const CompressedRowBlockStructure* bs = A->block_structure();

     for (int i = 0; i < num_eliminate_blocks; ++i) {
       num_eliminate_cols += bs->cols[i].size;
     }
   }

   // Compute the golden values for the reduced linear system and the
   // solution to the linear least squares problem using dense linear
   // algebra.
   void ComputeReferenceSolution(const Vector& D) {
     Matrix J;
     A->ToDenseMatrix(&J);
     VectorRef f(b.get(), J.rows());

     Matrix H = (D.cwiseProduct(D)).asDiagonal();
     H.noalias() += J.transpose() * J;

     const Vector g = J.transpose() * f;
     const int schur_size = J.cols() - num_eliminate_cols;

     lhs_expected.resize(schur_size, schur_size);
     lhs_expected.setZero();

     rhs_expected.resize(schur_size);
     rhs_expected.setZero();

     sol_expected.resize(J.cols());
     sol_expected.setZero();

     Matrix P = H.block(0, 0, num_eliminate_cols, num_eliminate_cols);
     Matrix Q = H.block(0, num_eliminate_cols, num_eliminate_cols, schur_size);
     Matrix R =
         H.block(num_eliminate_cols, num_eliminate_cols, schur_size, schur_size);
     int row = 0;
     const CompressedRowBlockStructure* bs = A->block_structure();
     for (int i = 0; i < num_eliminate_blocks; ++i) {
       const int block_size = bs->cols[i].size;
       P.block(row, row, block_size, block_size) =
           P.block(row, row, block_size, block_size)
               .llt()
               .solve(Matrix::Identity(block_size, block_size));
       row += block_size;
     }

     lhs_expected.triangularView<Eigen::Upper>() = R - Q.transpose() * P * Q;
     rhs_expected =
         g.tail(schur_size) - Q.transpose() * P * g.head(num_eliminate_cols);
     sol_expected = H.llt().solve(g);
   }

   void EliminateSolveAndCompare(const VectorRef& diagonal,
                                 bool use_static_structure,
                                 const double relative_tolerance) {
     const CompressedRowBlockStructure* bs = A->block_structure();
     const int num_col_blocks = bs->cols.size();
     auto blocks = Tail(bs->cols, num_col_blocks - num_eliminate_blocks);
     BlockRandomAccessDenseMatrix lhs(blocks, &context_, 1);

     const int num_cols = A->num_cols();
     const int schur_size = lhs.num_rows();

     Vector rhs(schur_size);

     LinearSolver::Options options;
     options.context = &context_;
     options.elimination_groups.push_back(num_eliminate_blocks);
     if (use_static_structure) {
       DetectStructure(*bs,
                       num_eliminate_blocks,
                       &options.row_block_size,
                       &options.e_block_size,
                       &options.f_block_size);
     }

     std::unique_ptr<SchurEliminatorBase> eliminator =
         SchurEliminatorBase::Create(options);
     const bool kFullRankETE = true;
     eliminator->Init(num_eliminate_blocks, kFullRankETE, A->block_structure());
     eliminator->Eliminate(
         BlockSparseMatrixData(*A), b.get(), diagonal.data(), &lhs, rhs.data());

     MatrixRef lhs_ref(lhs.mutable_values(), lhs.num_rows(), lhs.num_cols());
     Vector reduced_sol =
         lhs_ref.selfadjointView<Eigen::Upper>().llt().solve(rhs);

     // Solution to the linear least squares problem.
     Vector sol(num_cols);
     sol.setZero();
     sol.tail(schur_size) = reduced_sol;
     eliminator->BackSubstitute(BlockSparseMatrixData(*A),
                                b.get(),
                                diagonal.data(),
                                reduced_sol.data(),
                                sol.data());

     Matrix delta = (lhs_ref - lhs_expected).selfadjointView<Eigen::Upper>();
     double diff = delta.norm();
     EXPECT_NEAR(diff / lhs_expected.norm(), 0.0, relative_tolerance);
     EXPECT_NEAR((rhs - rhs_expected).norm() / rhs_expected.norm(),
                 0.0,
                 relative_tolerance);
     EXPECT_NEAR((sol - sol_expected).norm() / sol_expected.norm(),
                 0.0,
                 relative_tolerance);
   }

   ContextImpl context_;

   std::unique_ptr<BlockSparseMatrix> A;
   std::unique_ptr<double[]> b;
   std::unique_ptr<double[]> D;
   int num_eliminate_blocks;
   int num_eliminate_cols;

   Matrix lhs_expected;
   Vector rhs_expected;
   Vector sol_expected;
 };

 TEST_F(SchurEliminatorTest, ScalarProblemNoRegularization) {
   SetUpFromId(2);
   Vector zero(A->num_cols());
   zero.setZero();

   ComputeReferenceSolution(VectorRef(zero.data(), A->num_cols()));
   EliminateSolveAndCompare(VectorRef(zero.data(), A->num_cols()), true, 1e-14);
   EliminateSolveAndCompare(VectorRef(zero.data(), A->num_cols()), false, 1e-14);
 }

 TEST_F(SchurEliminatorTest, ScalarProblemWithRegularization) {
   SetUpFromId(2);
   ComputeReferenceSolution(VectorRef(D.get(), A->num_cols()));
   EliminateSolveAndCompare(VectorRef(D.get(), A->num_cols()), true, 1e-14);
   EliminateSolveAndCompare(VectorRef(D.get(), A->num_cols()), false, 1e-14);
 }

 TEST_F(SchurEliminatorTest, VaryingFBlockSizeWithStaticStructure) {
   SetUpFromId(4);
   ComputeReferenceSolution(VectorRef(D.get(), A->num_cols()));
   EliminateSolveAndCompare(VectorRef(D.get(), A->num_cols()), true, 1e-14);
 }

 TEST_F(SchurEliminatorTest, VaryingFBlockSizeWithoutStaticStructure) {
   SetUpFromId(4);
   ComputeReferenceSolution(VectorRef(D.get(), A->num_cols()));
   EliminateSolveAndCompare(VectorRef(D.get(), A->num_cols()), false, 1e-14);
 }

 TEST(SchurEliminatorForOneFBlock, MatchesSchurEliminator) {
   constexpr int kRowBlockSize = 2;
   constexpr int kEBlockSize = 3;
   constexpr int kFBlockSize = 6;
   constexpr int num_e_blocks = 5;

   ContextImpl context;

   auto* bs = new CompressedRowBlockStructure;
   bs->cols.resize(num_e_blocks + 1);
   int col_pos = 0;
   for (int i = 0; i < num_e_blocks; ++i) {
     bs->cols[i].position = col_pos;
     bs->cols[i].size = kEBlockSize;
     col_pos += kEBlockSize;
   }
   bs->cols.back().position = col_pos;
   bs->cols.back().size = kFBlockSize;

   bs->rows.resize(2 * num_e_blocks + 1);
   int row_pos = 0;
   int cell_pos = 0;
   for (int i = 0; i < num_e_blocks; ++i) {
     {
       auto& row = bs->rows[2 * i];
       row.block.position = row_pos;
       row.block.size = kRowBlockSize;
       row_pos += kRowBlockSize;
       auto& cells = row.cells;
       cells.resize(2);
       cells[0].block_id = i;
       cells[0].position = cell_pos;
       cell_pos += kRowBlockSize * kEBlockSize;
       cells[1].block_id = num_e_blocks;
       cells[1].position = cell_pos;
       cell_pos += kRowBlockSize * kFBlockSize;
     }
     {
       auto& row = bs->rows[2 * i + 1];
       row.block.position = row_pos;
       row.block.size = kRowBlockSize;
       row_pos += kRowBlockSize;
       auto& cells = row.cells;
       cells.resize(1);
       cells[0].block_id = i;
       cells[0].position = cell_pos;
       cell_pos += kRowBlockSize * kEBlockSize;
     }
   }

   {
     auto& row = bs->rows.back();
     row.block.position = row_pos;
     row.block.size = kEBlockSize;
     row_pos += kRowBlockSize;
     auto& cells = row.cells;
     cells.resize(1);
     cells[0].block_id = num_e_blocks;
     cells[0].position = cell_pos;
     cell_pos += kEBlockSize * kEBlockSize;
   }

   BlockSparseMatrix matrix(bs);
   double* values = matrix.mutable_values();
   std::mt19937 prng;
   std::normal_distribution<> standard_normal;
   std::generate_n(values, matrix.num_nonzeros(), [&prng, &standard_normal] {
     return standard_normal(prng);
   });

   Vector b(matrix.num_rows());
   b.setRandom();

   Vector diagonal(matrix.num_cols());
   diagonal.setOnes();

   std::vector<Block> blocks;
   blocks.emplace_back(kFBlockSize, 0);
   BlockRandomAccessDenseMatrix actual_lhs(blocks, &context, 1);
   BlockRandomAccessDenseMatrix expected_lhs(blocks, &context, 1);
   Vector actual_rhs(kFBlockSize);
   Vector expected_rhs(kFBlockSize);

   Vector f_sol(kFBlockSize);
   f_sol.setRandom();
   Vector actual_e_sol(num_e_blocks * kEBlockSize);
   actual_e_sol.setZero();
   Vector expected_e_sol(num_e_blocks * kEBlockSize);
   expected_e_sol.setZero();

   {
     LinearSolver::Options linear_solver_options;
     linear_solver_options.e_block_size = kEBlockSize;
     linear_solver_options.row_block_size = kRowBlockSize;
     linear_solver_options.f_block_size = kFBlockSize;
     linear_solver_options.context = &context;
     std::unique_ptr<SchurEliminatorBase> eliminator(
         SchurEliminatorBase::Create(linear_solver_options));
     eliminator->Init(num_e_blocks, true, matrix.block_structure());
     eliminator->Eliminate(BlockSparseMatrixData(matrix),
                           b.data(),
                           diagonal.data(),
                           &expected_lhs,
                           expected_rhs.data());
     eliminator->BackSubstitute(BlockSparseMatrixData(matrix),
                                b.data(),
                                diagonal.data(),
                                f_sol.data(),
                                actual_e_sol.data());
   }

   {
     SchurEliminatorForOneFBlock<2, 3, 6> eliminator;
     eliminator.Init(num_e_blocks, true, matrix.block_structure());
     eliminator.Eliminate(BlockSparseMatrixData(matrix),
                          b.data(),
                          diagonal.data(),
                          &actual_lhs,
                          actual_rhs.data());
     eliminator.BackSubstitute(BlockSparseMatrixData(matrix),
                               b.data(),
                               diagonal.data(),
                               f_sol.data(),
                               expected_e_sol.data());
   }
   ConstMatrixRef actual_lhsref(
       actual_lhs.values(), actual_lhs.num_cols(), actual_lhs.num_cols());
   ConstMatrixRef expected_lhsref(
       expected_lhs.values(), actual_lhs.num_cols(), actual_lhs.num_cols());

   EXPECT_NEAR((actual_lhsref - expected_lhsref).norm() / expected_lhsref.norm(),
               0.0,
               1e-12)
       << "expected: \n"
       << expected_lhsref << "\nactual: \n"
       << actual_lhsref;

   EXPECT_NEAR(
       (actual_rhs - expected_rhs).norm() / expected_rhs.norm(), 0.0, 1e-12)
       << "expected: \n"
       << expected_rhs << "\nactual: \n"
       << actual_rhs;

   EXPECT_NEAR((actual_e_sol - expected_e_sol).norm() / expected_e_sol.norm(),
               0.0,
               1e-12)
       << "expected: \n"
       << expected_e_sol << "\nactual: \n"
       << actual_e_sol;
 }

 }  // namespace ceres::internal
	// Ceres Solver - A fast non-linear least squares minimizer
	// Copyright 2023 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/schur_eliminator.h"

	#include <algorithm>
	#include <memory>
	#include <random>
	#include <vector>

	#include "Eigen/Dense"
	#include "ceres/block_random_access_dense_matrix.h"
	#include "ceres/block_sparse_matrix.h"
	#include "ceres/block_structure.h"
	#include "ceres/casts.h"
	#include "ceres/context_impl.h"
	#include "ceres/detect_structure.h"
	#include "ceres/internal/eigen.h"
	#include "ceres/linear_least_squares_problems.h"
	#include "ceres/test_util.h"
	#include "ceres/triplet_sparse_matrix.h"
	#include "ceres/types.h"
	#include "gtest/gtest.h"

	// TODO(sameeragarwal): Reduce the size of these tests and redo the
	// parameterization to be more efficient.

	namespace ceres::internal {

	class SchurEliminatorTest : public ::testing::Test {
	protected:
	void SetUpFromId(int id) {
	auto problem = CreateLinearLeastSquaresProblemFromId(id);
	ASSERT_TRUE(problem != nullptr);
	SetupHelper(problem.get());
	}

	void SetupHelper(LinearLeastSquaresProblem* problem) {
	A.reset(down_cast<BlockSparseMatrix*>(problem->A.release()));
	b = std::move(problem->b);
	D = std::move(problem->D);

	num_eliminate_blocks = problem->num_eliminate_blocks;
	num_eliminate_cols = 0;
	const CompressedRowBlockStructure* bs = A->block_structure();

	for (int i = 0; i < num_eliminate_blocks; ++i) {
	num_eliminate_cols += bs->cols[i].size;
	}
	}

	// Compute the golden values for the reduced linear system and the
	// solution to the linear least squares problem using dense linear
	// algebra.
	void ComputeReferenceSolution(const Vector& D) {
	Matrix J;
	A->ToDenseMatrix(&J);
	VectorRef f(b.get(), J.rows());

	Matrix H = (D.cwiseProduct(D)).asDiagonal();
	H.noalias() += J.transpose() * J;

	const Vector g = J.transpose() * f;
	const int schur_size = J.cols() - num_eliminate_cols;

	lhs_expected.resize(schur_size, schur_size);
	lhs_expected.setZero();

	rhs_expected.resize(schur_size);
	rhs_expected.setZero();

	sol_expected.resize(J.cols());
	sol_expected.setZero();

	Matrix P = H.block(0, 0, num_eliminate_cols, num_eliminate_cols);
	Matrix Q = H.block(0, num_eliminate_cols, num_eliminate_cols, schur_size);
	Matrix R =
	H.block(num_eliminate_cols, num_eliminate_cols, schur_size, schur_size);
	int row = 0;
	const CompressedRowBlockStructure* bs = A->block_structure();
	for (int i = 0; i < num_eliminate_blocks; ++i) {
	const int block_size = bs->cols[i].size;
	P.block(row, row, block_size, block_size) =
	P.block(row, row, block_size, block_size)
	.llt()
	.solve(Matrix::Identity(block_size, block_size));
	row += block_size;
	}

	lhs_expected.triangularView<Eigen::Upper>() = R - Q.transpose() * P * Q;
	rhs_expected =
	g.tail(schur_size) - Q.transpose() * P * g.head(num_eliminate_cols);
	sol_expected = H.llt().solve(g);
	}

	void EliminateSolveAndCompare(const VectorRef& diagonal,
	bool use_static_structure,
	const double relative_tolerance) {
	const CompressedRowBlockStructure* bs = A->block_structure();
	const int num_col_blocks = bs->cols.size();
	auto blocks = Tail(bs->cols, num_col_blocks - num_eliminate_blocks);
	BlockRandomAccessDenseMatrix lhs(blocks, &context_, 1);

	const int num_cols = A->num_cols();
	const int schur_size = lhs.num_rows();

	Vector rhs(schur_size);

	LinearSolver::Options options;
	options.context = &context_;
	options.elimination_groups.push_back(num_eliminate_blocks);
	if (use_static_structure) {
	DetectStructure(*bs,
	num_eliminate_blocks,
	&options.row_block_size,
	&options.e_block_size,
	&options.f_block_size);
	}

	std::unique_ptr<SchurEliminatorBase> eliminator =
	SchurEliminatorBase::Create(options);
	const bool kFullRankETE = true;
	eliminator->Init(num_eliminate_blocks, kFullRankETE, A->block_structure());
	eliminator->Eliminate(
	BlockSparseMatrixData(*A), b.get(), diagonal.data(), &lhs, rhs.data());

	MatrixRef lhs_ref(lhs.mutable_values(), lhs.num_rows(), lhs.num_cols());
	Vector reduced_sol =
	lhs_ref.selfadjointView<Eigen::Upper>().llt().solve(rhs);

	// Solution to the linear least squares problem.
	Vector sol(num_cols);
	sol.setZero();
	sol.tail(schur_size) = reduced_sol;
	eliminator->BackSubstitute(BlockSparseMatrixData(*A),
	b.get(),
	diagonal.data(),
	reduced_sol.data(),
	sol.data());

	Matrix delta = (lhs_ref - lhs_expected).selfadjointView<Eigen::Upper>();
	double diff = delta.norm();
	EXPECT_NEAR(diff / lhs_expected.norm(), 0.0, relative_tolerance);
	EXPECT_NEAR((rhs - rhs_expected).norm() / rhs_expected.norm(),
	0.0,
	relative_tolerance);
	EXPECT_NEAR((sol - sol_expected).norm() / sol_expected.norm(),
	0.0,
	relative_tolerance);
	}

	ContextImpl context_;

	std::unique_ptr<BlockSparseMatrix> A;
	std::unique_ptr<double[]> b;
	std::unique_ptr<double[]> D;
	int num_eliminate_blocks;
	int num_eliminate_cols;

	Matrix lhs_expected;
	Vector rhs_expected;
	Vector sol_expected;
	};

	TEST_F(SchurEliminatorTest, ScalarProblemNoRegularization) {
	SetUpFromId(2);
	Vector zero(A->num_cols());
	zero.setZero();

	ComputeReferenceSolution(VectorRef(zero.data(), A->num_cols()));
	EliminateSolveAndCompare(VectorRef(zero.data(), A->num_cols()), true, 1e-14);
	EliminateSolveAndCompare(VectorRef(zero.data(), A->num_cols()), false, 1e-14);
	}

	TEST_F(SchurEliminatorTest, ScalarProblemWithRegularization) {
	SetUpFromId(2);
	ComputeReferenceSolution(VectorRef(D.get(), A->num_cols()));
	EliminateSolveAndCompare(VectorRef(D.get(), A->num_cols()), true, 1e-14);
	EliminateSolveAndCompare(VectorRef(D.get(), A->num_cols()), false, 1e-14);
	}

	TEST_F(SchurEliminatorTest, VaryingFBlockSizeWithStaticStructure) {
	SetUpFromId(4);
	ComputeReferenceSolution(VectorRef(D.get(), A->num_cols()));
	EliminateSolveAndCompare(VectorRef(D.get(), A->num_cols()), true, 1e-14);
	}

	TEST_F(SchurEliminatorTest, VaryingFBlockSizeWithoutStaticStructure) {
	SetUpFromId(4);
	ComputeReferenceSolution(VectorRef(D.get(), A->num_cols()));
	EliminateSolveAndCompare(VectorRef(D.get(), A->num_cols()), false, 1e-14);
	}

	TEST(SchurEliminatorForOneFBlock, MatchesSchurEliminator) {
	constexpr int kRowBlockSize = 2;
	constexpr int kEBlockSize = 3;
	constexpr int kFBlockSize = 6;
	constexpr int num_e_blocks = 5;

	ContextImpl context;

	auto* bs = new CompressedRowBlockStructure;
	bs->cols.resize(num_e_blocks + 1);
	int col_pos = 0;
	for (int i = 0; i < num_e_blocks; ++i) {
	bs->cols[i].position = col_pos;
	bs->cols[i].size = kEBlockSize;
	col_pos += kEBlockSize;
	}
	bs->cols.back().position = col_pos;
	bs->cols.back().size = kFBlockSize;

	bs->rows.resize(2 * num_e_blocks + 1);
	int row_pos = 0;
	int cell_pos = 0;
	for (int i = 0; i < num_e_blocks; ++i) {
	{
	auto& row = bs->rows[2 * i];
	row.block.position = row_pos;
	row.block.size = kRowBlockSize;
	row_pos += kRowBlockSize;
	auto& cells = row.cells;
	cells.resize(2);
	cells[0].block_id = i;
	cells[0].position = cell_pos;
	cell_pos += kRowBlockSize * kEBlockSize;
	cells[1].block_id = num_e_blocks;
	cells[1].position = cell_pos;
	cell_pos += kRowBlockSize * kFBlockSize;
	}
	{
	auto& row = bs->rows[2 * i + 1];
	row.block.position = row_pos;
	row.block.size = kRowBlockSize;
	row_pos += kRowBlockSize;
	auto& cells = row.cells;
	cells.resize(1);
	cells[0].block_id = i;
	cells[0].position = cell_pos;
	cell_pos += kRowBlockSize * kEBlockSize;
	}
	}

	{
	auto& row = bs->rows.back();
	row.block.position = row_pos;
	row.block.size = kEBlockSize;
	row_pos += kRowBlockSize;
	auto& cells = row.cells;
	cells.resize(1);
	cells[0].block_id = num_e_blocks;
	cells[0].position = cell_pos;
	cell_pos += kEBlockSize * kEBlockSize;
	}

	BlockSparseMatrix matrix(bs);
	double* values = matrix.mutable_values();
	std::mt19937 prng;
	std::normal_distribution<> standard_normal;
	std::generate_n(values, matrix.num_nonzeros(), [&prng, &standard_normal] {
	return standard_normal(prng);
	});

	Vector b(matrix.num_rows());
	b.setRandom();

	Vector diagonal(matrix.num_cols());
	diagonal.setOnes();

	std::vector<Block> blocks;
	blocks.emplace_back(kFBlockSize, 0);
	BlockRandomAccessDenseMatrix actual_lhs(blocks, &context, 1);
	BlockRandomAccessDenseMatrix expected_lhs(blocks, &context, 1);
	Vector actual_rhs(kFBlockSize);
	Vector expected_rhs(kFBlockSize);

	Vector f_sol(kFBlockSize);
	f_sol.setRandom();
	Vector actual_e_sol(num_e_blocks * kEBlockSize);
	actual_e_sol.setZero();
	Vector expected_e_sol(num_e_blocks * kEBlockSize);
	expected_e_sol.setZero();

	{
	LinearSolver::Options linear_solver_options;
	linear_solver_options.e_block_size = kEBlockSize;
	linear_solver_options.row_block_size = kRowBlockSize;
	linear_solver_options.f_block_size = kFBlockSize;
	linear_solver_options.context = &context;
	std::unique_ptr<SchurEliminatorBase> eliminator(
	SchurEliminatorBase::Create(linear_solver_options));
	eliminator->Init(num_e_blocks, true, matrix.block_structure());
	eliminator->Eliminate(BlockSparseMatrixData(matrix),
	b.data(),
	diagonal.data(),
	&expected_lhs,
	expected_rhs.data());
	eliminator->BackSubstitute(BlockSparseMatrixData(matrix),
	b.data(),
	diagonal.data(),
	f_sol.data(),
	actual_e_sol.data());
	}

	{
	SchurEliminatorForOneFBlock<2, 3, 6> eliminator;
	eliminator.Init(num_e_blocks, true, matrix.block_structure());
	eliminator.Eliminate(BlockSparseMatrixData(matrix),
	b.data(),
	diagonal.data(),
	&actual_lhs,
	actual_rhs.data());
	eliminator.BackSubstitute(BlockSparseMatrixData(matrix),
	b.data(),
	diagonal.data(),
	f_sol.data(),
	expected_e_sol.data());
	}
	ConstMatrixRef actual_lhsref(
	actual_lhs.values(), actual_lhs.num_cols(), actual_lhs.num_cols());
	ConstMatrixRef expected_lhsref(
	expected_lhs.values(), actual_lhs.num_cols(), actual_lhs.num_cols());

	EXPECT_NEAR((actual_lhsref - expected_lhsref).norm() / expected_lhsref.norm(),
	0.0,
	1e-12)
	<< "expected: \n"
	<< expected_lhsref << "\nactual: \n"
	<< actual_lhsref;

	EXPECT_NEAR(
	(actual_rhs - expected_rhs).norm() / expected_rhs.norm(), 0.0, 1e-12)
	<< "expected: \n"
	<< expected_rhs << "\nactual: \n"
	<< actual_rhs;

	EXPECT_NEAR((actual_e_sol - expected_e_sol).norm() / expected_e_sol.norm(),
	0.0,
	1e-12)
	<< "expected: \n"
	<< expected_e_sol << "\nactual: \n"
	<< actual_e_sol;
	}

	} // namespace ceres::internal