include/ceres/internal/householder_vector.h

// Ceres Solver - A fast non-linear least squares minimizer
// Copyright 2023 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: vitus@google.com (Michael Vitus)

#ifndef CERES_PUBLIC_INTERNAL_HOUSEHOLDER_VECTOR_H_
#define CERES_PUBLIC_INTERNAL_HOUSEHOLDER_VECTOR_H_

#include "Eigen/Core"
#include "absl/log/check.h"

namespace ceres::internal {

// Algorithm 5.1.1 from 'Matrix Computations' by Golub et al. (Johns Hopkins
// Studies in Mathematical Sciences) but using the nth element of the input
// vector as pivot instead of first. This computes the vector v with v(n) = 1
// and beta such that H = I - beta * v * v^T is orthogonal and
// H * x = ||x||_2 * e_n.
//
// NOTE: Some versions of MSVC have trouble deducing the type of v if
// you do not specify all the template arguments explicitly.
template <typename XVectorType, typename Scalar, int N>
void ComputeHouseholderVector(const XVectorType& x,
                              Eigen::Matrix<Scalar, N, 1>* v,
                              Scalar* beta) {
  CHECK(beta != nullptr);
  CHECK(v != nullptr);
  CHECK_GT(x.rows(), 1);
  CHECK_EQ(x.rows(), v->rows());

  Scalar sigma = x.head(x.rows() - 1).squaredNorm();
  *v = x;
  (*v)(v->rows() - 1) = Scalar(1.0);

  *beta = Scalar(0.0);
  const Scalar& x_pivot = x(x.rows() - 1);

  if (sigma <= Scalar(std::numeric_limits<double>::epsilon())) {
    if (x_pivot < Scalar(0.0)) {
      *beta = Scalar(2.0);
    }
    return;
  }

  const Scalar mu = sqrt(x_pivot * x_pivot + sigma);
  Scalar v_pivot = Scalar(1.0);

  if (x_pivot <= Scalar(0.0)) {
    v_pivot = x_pivot - mu;
  } else {
    v_pivot = -sigma / (x_pivot + mu);
  }

  *beta = Scalar(2.0) * v_pivot * v_pivot / (sigma + v_pivot * v_pivot);

  v->head(v->rows() - 1) /= v_pivot;
}

template <typename XVectorType, typename Derived>
typename Derived::PlainObject ApplyHouseholderVector(
    const XVectorType& y,
    const Eigen::MatrixBase<Derived>& v,
    const typename Derived::Scalar& beta) {
  return (y - v * (beta * (v.transpose() * y)));
}

}  // namespace ceres::internal

#endif  // CERES_PUBLIC_INTERNAL_HOUSEHOLDER_VECTOR_H_