10ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong// Ceres Solver - A fast non-linear least squares minimizer 20ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong// Copyright 2010, 2011, 2012 Google Inc. All rights reserved. 30ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong// http://code.google.com/p/ceres-solver/ 40ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong// 50ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong// Redistribution and use in source and binary forms, with or without 60ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong// modification, are permitted provided that the following conditions are met: 70ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong// 80ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong// * Redistributions of source code must retain the above copyright notice, 90ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong// this list of conditions and the following disclaimer. 100ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong// * Redistributions in binary form must reproduce the above copyright notice, 110ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong// this list of conditions and the following disclaimer in the documentation 120ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong// and/or other materials provided with the distribution. 130ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong// * Neither the name of Google Inc. nor the names of its contributors may be 140ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong// used to endorse or promote products derived from this software without 150ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong// specific prior written permission. 160ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong// 170ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong// THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" 180ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong// AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE 190ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong// IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE 200ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong// ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE 210ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong// LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR 220ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong// CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF 230ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong// SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS 240ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong// INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN 250ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong// CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) 260ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong// ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE 270ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong// POSSIBILITY OF SUCH DAMAGE. 280ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong// 290ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong// Author: sameeragarwal@google.com (Sameer Agarwal) 300ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong 310ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong#include "ceres/schur_eliminator.h" 320ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong 330ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong#include "Eigen/Dense" 340ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong#include "ceres/block_random_access_dense_matrix.h" 350ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong#include "ceres/block_sparse_matrix.h" 360ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong#include "ceres/casts.h" 370ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong#include "ceres/detect_structure.h" 380ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong#include "ceres/internal/eigen.h" 390ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong#include "ceres/internal/scoped_ptr.h" 400ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong#include "ceres/linear_least_squares_problems.h" 410ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong#include "ceres/test_util.h" 420ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong#include "ceres/triplet_sparse_matrix.h" 430ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong#include "ceres/types.h" 440ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong#include "glog/logging.h" 450ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong#include "gtest/gtest.h" 460ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong 470ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong// TODO(sameeragarwal): Reduce the size of these tests and redo the 480ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong// parameterization to be more efficient. 490ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong 500ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kongnamespace ceres { 510ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kongnamespace internal { 520ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong 530ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kongclass SchurEliminatorTest : public ::testing::Test { 540ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong protected: 550ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong void SetUpFromId(int id) { 560ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong scoped_ptr<LinearLeastSquaresProblem> 570ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong problem(CreateLinearLeastSquaresProblemFromId(id)); 580ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong CHECK_NOTNULL(problem.get()); 590ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong SetupHelper(problem.get()); 600ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong } 610ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong 620ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong void SetupHelper(LinearLeastSquaresProblem* problem) { 630ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong A.reset(down_cast<BlockSparseMatrix*>(problem->A.release())); 640ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong b.reset(problem->b.release()); 650ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong D.reset(problem->D.release()); 660ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong 670ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong num_eliminate_blocks = problem->num_eliminate_blocks; 680ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong num_eliminate_cols = 0; 690ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong const CompressedRowBlockStructure* bs = A->block_structure(); 700ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong 710ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong for (int i = 0; i < num_eliminate_blocks; ++i) { 720ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong num_eliminate_cols += bs->cols[i].size; 730ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong } 740ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong } 750ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong 760ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong // Compute the golden values for the reduced linear system and the 770ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong // solution to the linear least squares problem using dense linear 780ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong // algebra. 790ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong void ComputeReferenceSolution(const Vector& D) { 800ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong Matrix J; 810ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong A->ToDenseMatrix(&J); 820ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong VectorRef f(b.get(), J.rows()); 830ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong 840ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong Matrix H = (D.cwiseProduct(D)).asDiagonal(); 850ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong H.noalias() += J.transpose() * J; 860ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong 870ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong const Vector g = J.transpose() * f; 880ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong const int schur_size = J.cols() - num_eliminate_cols; 890ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong 900ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong lhs_expected.resize(schur_size, schur_size); 910ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong lhs_expected.setZero(); 920ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong 930ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong rhs_expected.resize(schur_size); 940ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong rhs_expected.setZero(); 950ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong 960ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong sol_expected.resize(J.cols()); 970ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong sol_expected.setZero(); 980ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong 990ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong Matrix P = H.block(0, 0, num_eliminate_cols, num_eliminate_cols); 1000ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong Matrix Q = H.block(0, 1010ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong num_eliminate_cols, 1020ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong num_eliminate_cols, 1030ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong schur_size); 1040ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong Matrix R = H.block(num_eliminate_cols, 1050ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong num_eliminate_cols, 1060ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong schur_size, 1070ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong schur_size); 1080ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong int row = 0; 1090ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong const CompressedRowBlockStructure* bs = A->block_structure(); 1100ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong for (int i = 0; i < num_eliminate_blocks; ++i) { 1110ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong const int block_size = bs->cols[i].size; 1120ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong P.block(row, row, block_size, block_size) = 1130ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong P 1140ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong .block(row, row, block_size, block_size) 115399f7d09e0c45af54b77b4ab9508d6f23759b927Scott Ettinger .llt() 1160ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong .solve(Matrix::Identity(block_size, block_size)); 1170ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong row += block_size; 1180ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong } 1190ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong 1200ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong lhs_expected 1210ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong .triangularView<Eigen::Upper>() = R - Q.transpose() * P * Q; 1220ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong rhs_expected = 1230ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong g.tail(schur_size) - Q.transpose() * P * g.head(num_eliminate_cols); 124399f7d09e0c45af54b77b4ab9508d6f23759b927Scott Ettinger sol_expected = H.llt().solve(g); 1250ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong } 1260ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong 1270ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong void EliminateSolveAndCompare(const VectorRef& diagonal, 1280ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong bool use_static_structure, 1290ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong const double relative_tolerance) { 1300ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong const CompressedRowBlockStructure* bs = A->block_structure(); 1310ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong const int num_col_blocks = bs->cols.size(); 1320ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong vector<int> blocks(num_col_blocks - num_eliminate_blocks, 0); 1330ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong for (int i = num_eliminate_blocks; i < num_col_blocks; ++i) { 1340ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong blocks[i - num_eliminate_blocks] = bs->cols[i].size; 1350ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong } 1360ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong 1370ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong BlockRandomAccessDenseMatrix lhs(blocks); 1380ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong 1390ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong const int num_cols = A->num_cols(); 1400ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong const int schur_size = lhs.num_rows(); 1410ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong 1420ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong Vector rhs(schur_size); 1430ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong 1440ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong LinearSolver::Options options; 1450ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong options.elimination_groups.push_back(num_eliminate_blocks); 1460ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong if (use_static_structure) { 1470ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong DetectStructure(*bs, 1480ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong num_eliminate_blocks, 1490ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong &options.row_block_size, 1500ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong &options.e_block_size, 1510ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong &options.f_block_size); 1520ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong } 1530ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong 1540ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong scoped_ptr<SchurEliminatorBase> eliminator; 1550ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong eliminator.reset(SchurEliminatorBase::Create(options)); 1560ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong eliminator->Init(num_eliminate_blocks, A->block_structure()); 1570ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong eliminator->Eliminate(A.get(), b.get(), diagonal.data(), &lhs, rhs.data()); 1580ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong 1590ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong MatrixRef lhs_ref(lhs.mutable_values(), lhs.num_rows(), lhs.num_cols()); 1600ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong Vector reduced_sol = 1610ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong lhs_ref 1620ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong .selfadjointView<Eigen::Upper>() 163399f7d09e0c45af54b77b4ab9508d6f23759b927Scott Ettinger .llt() 1640ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong .solve(rhs); 1650ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong 1660ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong // Solution to the linear least squares problem. 1670ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong Vector sol(num_cols); 1680ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong sol.setZero(); 1690ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong sol.tail(schur_size) = reduced_sol; 1700ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong eliminator->BackSubstitute(A.get(), 1710ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong b.get(), 1720ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong diagonal.data(), 1730ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong reduced_sol.data(), 1740ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong sol.data()); 1750ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong 1760ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong Matrix delta = (lhs_ref - lhs_expected).selfadjointView<Eigen::Upper>(); 1770ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong double diff = delta.norm(); 1780ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong EXPECT_NEAR(diff / lhs_expected.norm(), 0.0, relative_tolerance); 1790ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong EXPECT_NEAR((rhs - rhs_expected).norm() / rhs_expected.norm(), 0.0, 1800ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong relative_tolerance); 1810ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong EXPECT_NEAR((sol - sol_expected).norm() / sol_expected.norm(), 0.0, 1820ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong relative_tolerance); 1830ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong } 1840ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong 1850ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong scoped_ptr<BlockSparseMatrix> A; 1860ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong scoped_array<double> b; 1870ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong scoped_array<double> D; 1880ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong int num_eliminate_blocks; 1890ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong int num_eliminate_cols; 1900ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong 1910ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong Matrix lhs_expected; 1920ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong Vector rhs_expected; 1930ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong Vector sol_expected; 1940ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong}; 1950ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong 1960ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus KongTEST_F(SchurEliminatorTest, ScalarProblem) { 1970ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong SetUpFromId(2); 1980ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong Vector zero(A->num_cols()); 1990ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong zero.setZero(); 2000ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong 2010ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong ComputeReferenceSolution(VectorRef(zero.data(), A->num_cols())); 2020ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong EliminateSolveAndCompare(VectorRef(zero.data(), A->num_cols()), true, 1e-14); 2030ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong EliminateSolveAndCompare(VectorRef(zero.data(), A->num_cols()), false, 1e-14); 2040ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong 2050ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong ComputeReferenceSolution(VectorRef(D.get(), A->num_cols())); 2060ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong EliminateSolveAndCompare(VectorRef(D.get(), A->num_cols()), true, 1e-14); 2070ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong EliminateSolveAndCompare(VectorRef(D.get(), A->num_cols()), false, 1e-14); 2080ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong} 2090ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong 2100ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong} // namespace internal 2110ae28bd5885b5daa526898fcf7c323dc2c3e1963Angus Kong} // namespace ceres 212