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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)
//
// Solve dense rectangular systems Ax = b using the QR factorization.
#ifndef CERES_INTERNAL_DENSE_QR_SOLVER_H_
#define CERES_INTERNAL_DENSE_QR_SOLVER_H_
#include "ceres/linear_solver.h"
#include "ceres/internal/eigen.h"
namespace ceres {
namespace internal {
class DenseSparseMatrix;
// This class implements the LinearSolver interface for solving
// rectangular/unsymmetric (well constrained) linear systems of the
// form
//
// Ax = b
//
// Since there does not usually exist a solution that satisfies these
// equations, the solver instead solves the linear least squares
// problem
//
// min_x |Ax - b|^2
//
// The solution strategy is based on computing the QR decomposition of
// A, i.e.
//
// A = QR
//
// Where Q is an orthonormal matrix and R is an upper triangular
// matrix. Then
//
// Ax = b
// QRx = b
// Q'QRx = Q'b
// Rx = Q'b
// x = R^{-1} Q'b
//
// If the PerSolveOptions struct has a non-null array D, then the
// augmented/regularized linear system
//
// [ A ]x = [b]
// [ diag(D) ] [0]
//
// is solved.
//
// This class uses the dense QR factorization routines from the Eigen
// library. This solver always returns a solution, it is the user's
// responsibility to judge if the solution is good enough for their
// purposes.
class DenseQRSolver: public DenseSparseMatrixSolver {
public:
explicit DenseQRSolver(const LinearSolver::Options& options);
private:
LinearSolver::Summary SolveImpl(
DenseSparseMatrix* A,
const double* b,
const LinearSolver::PerSolveOptions& per_solve_options,
double* x) final;
LinearSolver::Summary SolveUsingEigen(
DenseSparseMatrix* A,
const double* b,
const LinearSolver::PerSolveOptions& per_solve_options,
double* x);
LinearSolver::Summary SolveUsingLAPACK(
DenseSparseMatrix* A,
const double* b,
const LinearSolver::PerSolveOptions& per_solve_options,
double* x);
const LinearSolver::Options options_;
ColMajorMatrix lhs_;
Vector rhs_;
Vector work_;
};
} // namespace internal
} // namespace ceres
#endif // CERES_INTERNAL_DENSE_QR_SOLVER_H_
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