text: // Ceres Solver - A fast non-linear least squares minimizer
  // Copyright 2014 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)
  
  #include "ceres/internal/port.h"
  
  #include <algorithm>
  #include <ctime>
  #include <set>
  #include <vector>
  
  #include "ceres/block_random_access_dense_matrix.h"
  #include "ceres/block_random_access_matrix.h"
  #include "ceres/block_random_access_sparse_matrix.h"
  #include "ceres/block_sparse_matrix.h"
  #include "ceres/block_structure.h"
  #include "ceres/cxsparse.h"
  #include "ceres/detect_structure.h"
  #include "ceres/internal/eigen.h"
  #include "ceres/internal/scoped_ptr.h"
  #include "ceres/lapack.h"
  #include "ceres/linear_solver.h"
  #include "ceres/schur_complement_solver.h"
  #include "ceres/suitesparse.h"
  #include "ceres/triplet_sparse_matrix.h"
  #include "ceres/types.h"
  #include "ceres/wall_time.h"
  #include "Eigen/Dense"
  #include "Eigen/SparseCore"
  
  namespace ceres {
  namespace internal {
  
  LinearSolver::Summary SchurComplementSolver::SolveImpl(
      BlockSparseMatrix* A,
      const double* b,
      const LinearSolver::PerSolveOptions& per_solve_options,
      double* x) {
    EventLogger event_logger("SchurComplementSolver::Solve");
  
    if (eliminator_.get() == NULL) {
      InitStorage(A->block_structure());
      DetectStructure(*A->block_structure(),
                      options_.elimination_groups[0],
                      &options_.row_block_size,
                      &options_.e_block_size,
                      &options_.f_block_size);
      eliminator_.reset(CHECK_NOTNULL(SchurEliminatorBase::Create(options_)));
      eliminator_->Init(options_.elimination_groups[0], A->block_structure());
    };
    fill(x, x + A->num_cols(), 0.0);
    event_logger.AddEvent("Setup");
  
    eliminator_->Eliminate(A, b, per_solve_options.D, lhs_.get(), rhs_.get());
    event_logger.AddEvent("Eliminate");
  
    double* reduced_solution = x + A->num_cols() - lhs_->num_cols();
    const LinearSolver::Summary summary =
        SolveReducedLinearSystem(reduced_solution);
    event_logger.AddEvent("ReducedSolve");
  
    if (summary.termination_type == LINEAR_SOLVER_SUCCESS) {
      eliminator_->BackSubstitute(A, b, per_solve_options.D, reduced_solution, x);
      event_logger.AddEvent("BackSubstitute");
    }
  
    return summary;
  }
  
  // Initialize a BlockRandomAccessDenseMatrix to store the Schur
  // complement.
  void DenseSchurComplementSolver::InitStorage(
      const CompressedRowBlockStructure* bs) {
    const int num_eliminate_blocks = options().elimination_groups[0];
    const int num_col_blocks = bs->cols.size();
  
    vector<int> blocks(num_col_blocks - num_eliminate_blocks, 0);
    for (int i = num_eliminate_blocks, j = 0;
         i < num_col_blocks;
         ++i, ++j) {
      blocks[j] = bs->cols[i].size;
    }
  
    set_lhs(new BlockRandomAccessDenseMatrix(blocks));
    set_rhs(new double[lhs()->num_rows()]);
  }
  
  // Solve the system Sx = r, assuming that the matrix S is stored in a
  // BlockRandomAccessDenseMatrix. The linear system is solved using
  // Eigen's Cholesky factorization.
  LinearSolver::Summary
  DenseSchurComplementSolver::SolveReducedLinearSystem(double* solution) {
    LinearSolver::Summary summary;
    summary.num_iterations = 0;
    summary.termination_type = LINEAR_SOLVER_SUCCESS;
    summary.message = "Success.";
  
    const BlockRandomAccessDenseMatrix* m =
        down_cast<const BlockRandomAccessDenseMatrix*>(lhs());
    const int num_rows = m->num_rows();
  
    // The case where there are no f blocks, and the system is block
    // diagonal.
    if (num_rows == 0) {
      return summary;
    }
  
    summary.num_iterations = 1;
  
    if (options().dense_linear_algebra_library_type == EIGEN) {
      Eigen::LLT<Matrix, Eigen::Upper> llt =
          ConstMatrixRef(m->values(), num_rows, num_rows)
          .selfadjointView<Eigen::Upper>()
          .llt();
      if (llt.info() != Eigen::Success) {
        summary.termination_type = LINEAR_SOLVER_FAILURE;
        summary.message =
            "Eigen failure. Unable to perform dense Cholesky factorization.";
        return summary;
      }
  
      VectorRef(solution, num_rows) = llt.solve(ConstVectorRef(rhs(), num_rows));
    } else {
      VectorRef(solution, num_rows) = ConstVectorRef(rhs(), num_rows);
      summary.termination_type =
          LAPACK::SolveInPlaceUsingCholesky(num_rows,
                                            m->values(),
                                            solution,
                                            &summary.message);
    }
  
    return summary;
  }
  
  SparseSchurComplementSolver::SparseSchurComplementSolver(
      const LinearSolver::Options& options)
      : SchurComplementSolver(options),
        factor_(NULL),
        cxsparse_factor_(NULL) {
  }
  
  SparseSchurComplementSolver::~SparseSchurComplementSolver() {
    if (factor_ != NULL) {
      ss_.Free(factor_);
      factor_ = NULL;
    }
  
    if (cxsparse_factor_ != NULL) {
      cxsparse_.Free(cxsparse_factor_);
      cxsparse_factor_ = NULL;
    }
  }
  
  // Determine the non-zero blocks in the Schur Complement matrix, and
  // initialize a BlockRandomAccessSparseMatrix object.
  void SparseSchurComplementSolver::InitStorage(
      const CompressedRowBlockStructure* bs) {
    const int num_eliminate_blocks = options().elimination_groups[0];
    const int num_col_blocks = bs->cols.size();
    const int num_row_blocks = bs->rows.size();
  
    blocks_.resize(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;
    }
  
    set<pair<int, int> > block_pairs;
    for (int i = 0; i < blocks_.size(); ++i) {
      block_pairs.insert(make_pair(i, i));
    }
  
    int r = 0;
    while (r < num_row_blocks) {
      int e_block_id = bs->rows[r].cells.front().block_id;
      if (e_block_id >= num_eliminate_blocks) {
        break;
      }
      vector<int> f_blocks;
  
      // Add to the chunk until the first block in the row is
      // different than the one in the first row for the chunk.
      for (; r < num_row_blocks; ++r) {
        const CompressedRow& row = bs->rows[r];
        if (row.cells.front().block_id != e_block_id) {
          break;
        }
  
        // Iterate over the blocks in the row, ignoring the first
        // block since it is the one to be eliminated.
        for (int c = 1; c < row.cells.size(); ++c) {
          const Cell& cell = row.cells[c];
          f_blocks.push_back(cell.block_id - num_eliminate_blocks);
        }
      }
  
      sort(f_blocks.begin(), f_blocks.end());
      f_blocks.erase(unique(f_blocks.begin(), f_blocks.end()), f_blocks.end());
      for (int i = 0; i < f_blocks.size(); ++i) {
        for (int j = i + 1; j < f_blocks.size(); ++j) {
          block_pairs.insert(make_pair(f_blocks[i], f_blocks[j]));
        }
      }
    }
  
    // Remaing rows do not contribute to the chunks and directly go
    // into the schur complement via an outer product.
    for (; r < num_row_blocks; ++r) {
      const CompressedRow& row = bs->rows[r];
      CHECK_GE(row.cells.front().block_id, num_eliminate_blocks);
      for (int i = 0; i < row.cells.size(); ++i) {
        int r_block1_id = row.cells[i].block_id - num_eliminate_blocks;
        for (int j = 0; j < row.cells.size(); ++j) {
          int r_block2_id = row.cells[j].block_id - num_eliminate_blocks;
          if (r_block1_id <= r_block2_id) {
            block_pairs.insert(make_pair(r_block1_id, r_block2_id));
          }
        }
      }
    }
  
    set_lhs(new BlockRandomAccessSparseMatrix(blocks_, block_pairs));
    set_rhs(new double[lhs()->num_rows()]);
  }
  
  LinearSolver::Summary
  SparseSchurComplementSolver::SolveReducedLinearSystem(double* solution) {
    switch (options().sparse_linear_algebra_library_type) {
      case SUITE_SPARSE:
        return SolveReducedLinearSystemUsingSuiteSparse(solution);
      case CX_SPARSE:
        return SolveReducedLinearSystemUsingCXSparse(solution);
      case EIGEN_SPARSE:
        return SolveReducedLinearSystemUsingEigen(solution);
      default:
        LOG(FATAL) << "Unknown sparse linear algebra library : "
                   << options().sparse_linear_algebra_library_type;
    }
  
    return LinearSolver::Summary();
  }
  
  // Solve the system Sx = r, assuming that the matrix S is stored in a
  // BlockRandomAccessSparseMatrix.  The linear system is solved using
  // CHOLMOD's sparse cholesky factorization routines.
  LinearSolver::Summary
  SparseSchurComplementSolver::SolveReducedLinearSystemUsingSuiteSparse(
      double* solution) {
  #ifdef CERES_NO_SUITESPARSE
  
    LinearSolver::Summary summary;
    summary.num_iterations = 0;
    summary.termination_type = LINEAR_SOLVER_FATAL_ERROR;
    summary.message = "Ceres was not built with SuiteSparse support. "
        "Therefore, SPARSE_SCHUR cannot be used with SUITE_SPARSE";
    return summary;
  
  #else
  
    LinearSolver::Summary summary;
    summary.num_iterations = 0;
    summary.termination_type = LINEAR_SOLVER_SUCCESS;
    summary.message = "Success.";
  
    TripletSparseMatrix* tsm =
        const_cast<TripletSparseMatrix*>(
            down_cast<const BlockRandomAccessSparseMatrix*>(lhs())->matrix());
    const int num_rows = tsm->num_rows();
  
    // The case where there are no f blocks, and the system is block
    // diagonal.
    if (num_rows == 0) {
      return summary;
    }
  
    summary.num_iterations = 1;
    cholmod_sparse* cholmod_lhs = NULL;
    if (options().use_postordering) {
      // If we are going to do a full symbolic analysis of the schur
      // complement matrix from scratch and not rely on the
      // pre-ordering, then the fastest path in cholmod_factorize is the
      // one corresponding to upper triangular matrices.
  
      // Create a upper triangular symmetric matrix.
      cholmod_lhs = ss_.CreateSparseMatrix(tsm);
      cholmod_lhs->stype = 1;
  
      if (factor_ == NULL) {
        factor_ = ss_.BlockAnalyzeCholesky(cholmod_lhs,
                                           blocks_,
                                           blocks_,
                                           &summary.message);
      }
    } else {
      // If we are going to use the natural ordering (i.e. rely on the
      // pre-ordering computed by solver_impl.cc), then the fastest
      // path in cholmod_factorize is the one corresponding to lower
      // triangular matrices.
  
      // Create a upper triangular symmetric matrix.
      cholmod_lhs = ss_.CreateSparseMatrixTranspose(tsm);
      cholmod_lhs->stype = -1;
  
      if (factor_ == NULL) {
        factor_ = ss_.AnalyzeCholeskyWithNaturalOrdering(cholmod_lhs,
                                                         &summary.message);
      }
    }
  
    if (factor_ == NULL) {
      ss_.Free(cholmod_lhs);
      summary.termination_type = LINEAR_SOLVER_FATAL_ERROR;
      // No need to set message as it has already been set by the
      // symbolic analysis routines above.
      return summary;
    }
  
    summary.termination_type =
      ss_.Cholesky(cholmod_lhs, factor_, &summary.message);
  
    ss_.Free(cholmod_lhs);
  
    if (summary.termination_type != LINEAR_SOLVER_SUCCESS) {
      // No need to set message as it has already been set by the
      // numeric factorization routine above.
      return summary;
    }
  
    cholmod_dense*  cholmod_rhs =
        ss_.CreateDenseVector(const_cast<double*>(rhs()), num_rows, num_rows);
    cholmod_dense* cholmod_solution = ss_.Solve(factor_,
                                                cholmod_rhs,
                                                &summary.message);
    ss_.Free(cholmod_rhs);
  
    if (cholmod_solution == NULL) {
      summary.message =
          "SuiteSparse failure. Unable to perform triangular solve.";
      summary.termination_type = LINEAR_SOLVER_FAILURE;
      return summary;
    }
  
    VectorRef(solution, num_rows)
        = VectorRef(static_cast<double*>(cholmod_solution->x), num_rows);
    ss_.Free(cholmod_solution);
    return summary;
  #endif  // CERES_NO_SUITESPARSE
  }
  
  // Solve the system Sx = r, assuming that the matrix S is stored in a
  // BlockRandomAccessSparseMatrix.  The linear system is solved using
  // CXSparse's sparse cholesky factorization routines.
  LinearSolver::Summary
  SparseSchurComplementSolver::SolveReducedLinearSystemUsingCXSparse(
      double* solution) {
  #ifdef CERES_NO_CXSPARSE
  
    LinearSolver::Summary summary;
    summary.num_iterations = 0;
    summary.termination_type = LINEAR_SOLVER_FATAL_ERROR;
    summary.message = "Ceres was not built with CXSparse support. "
        "Therefore, SPARSE_SCHUR cannot be used with CX_SPARSE";
    return summary;
  
  #else
  
    LinearSolver::Summary summary;
    summary.num_iterations = 0;
    summary.termination_type = LINEAR_SOLVER_SUCCESS;
    summary.message = "Success.";
  
    // Extract the TripletSparseMatrix that is used for actually storing S.
    TripletSparseMatrix* tsm =
        const_cast<TripletSparseMatrix*>(
            down_cast<const BlockRandomAccessSparseMatrix*>(lhs())->matrix());
    const int num_rows = tsm->num_rows();
  
    // The case where there are no f blocks, and the system is block
    // diagonal.
    if (num_rows == 0) {
      return summary;
    }
  
    cs_di* lhs = CHECK_NOTNULL(cxsparse_.CreateSparseMatrix(tsm));
    VectorRef(solution, num_rows) = ConstVectorRef(rhs(), num_rows);
  
    // Compute symbolic factorization if not available.
    if (cxsparse_factor_ == NULL) {
      cxsparse_factor_ = cxsparse_.BlockAnalyzeCholesky(lhs, blocks_, blocks_);
    }
  
    if (cxsparse_factor_ == NULL) {
      summary.termination_type = LINEAR_SOLVER_FATAL_ERROR;
      summary.message =
          "CXSparse failure. Unable to find symbolic factorization.";
    } else if (!cxsparse_.SolveCholesky(lhs, cxsparse_factor_, solution)) {
      summary.termination_type = LINEAR_SOLVER_FAILURE;
      summary.message = "CXSparse::SolveCholesky failed.";
    }
  
    cxsparse_.Free(lhs);
    return summary;
  #endif  // CERES_NO_CXPARSE
  }
  
  // Solve the system Sx = r, assuming that the matrix S is stored in a
  // BlockRandomAccessSparseMatrix.  The linear system is solved using
  // Eigen's sparse cholesky factorization routines.
  LinearSolver::Summary
  SparseSchurComplementSolver::SolveReducedLinearSystemUsingEigen(
      double* solution) {
  #ifndef CERES_USE_EIGEN_SPARSE
  
    LinearSolver::Summary summary;
    summary.num_iterations = 0;
    summary.termination_type = LINEAR_SOLVER_FATAL_ERROR;
    summary.message =
        "SPARSE_SCHUR cannot be used with EIGEN_SPARSE. "
        "Ceres was not built with support for "
        "Eigen's SimplicialLDLT decomposition. "
        "This requires enabling building with -DEIGENSPARSE=ON.";
    return summary;
  
  #else
    EventLogger event_logger("SchurComplementSolver::EigenSolve");
    LinearSolver::Summary summary;
    summary.num_iterations = 0;
    summary.termination_type = LINEAR_SOLVER_SUCCESS;
    summary.message = "Success.";
  
    // Extract the TripletSparseMatrix that is used for actually storing S.
    TripletSparseMatrix* tsm =
        const_cast<TripletSparseMatrix*>(
            down_cast<const BlockRandomAccessSparseMatrix*>(lhs())->matrix());
    const int num_rows = tsm->num_rows();
  
    // The case where there are no f blocks, and the system is block
    // diagonal.
    if (num_rows == 0) {
      return summary;
    }
  
    // This is an upper triangular matrix.
    CompressedRowSparseMatrix crsm(*tsm);
    // Map this to a column major, lower triangular matrix.
    Eigen::MappedSparseMatrix<double, Eigen::ColMajor> eigen_lhs(
        crsm.num_rows(),
        crsm.num_rows(),
        crsm.num_nonzeros(),
        crsm.mutable_rows(),
        crsm.mutable_cols(),
        crsm.mutable_values());
    event_logger.AddEvent("ToCompressedRowSparseMatrix");
  
    // Compute symbolic factorization if one does not exist.
    if (simplicial_ldlt_.get() == NULL) {
      simplicial_ldlt_.reset(new SimplicialLDLT);
      // This ordering is quite bad. The scalar ordering produced by the
      // AMD algorithm is quite bad and can be an order of magnitude
      // worse than the one computed using the block version of the
      // algorithm.
      simplicial_ldlt_->analyzePattern(eigen_lhs);
      event_logger.AddEvent("Analysis");
      if (simplicial_ldlt_->info() != Eigen::Success) {
        summary.termination_type = LINEAR_SOLVER_FATAL_ERROR;
        summary.message =
            "Eigen failure. Unable to find symbolic factorization.";
        return summary;
      }
    }
  
    simplicial_ldlt_->factorize(eigen_lhs);
    event_logger.AddEvent("Factorize");
    if (simplicial_ldlt_->info() != Eigen::Success) {
      summary.termination_type = LINEAR_SOLVER_FAILURE;
      summary.message = "Eigen failure. Unable to find numeric factoriztion.";
      return summary;
    }
  
    VectorRef(solution, num_rows) =
        simplicial_ldlt_->solve(ConstVectorRef(rhs(), num_rows));
    event_logger.AddEvent("Solve");
    if (simplicial_ldlt_->info() != Eigen::Success) {
      summary.termination_type = LINEAR_SOLVER_FAILURE;
      summary.message = "Eigen failure. Unable to do triangular solve.";
    }
  
    return summary;
  #endif  // CERES_USE_EIGEN_SPARSE
  }
  
  }  // namespace internal
  }  // namespace ceres
  