// 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
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// 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