internal/ceres/householder_vector.h

// Ceres Solver - A fast non-linear least squares minimizer
// Copyright 2015 Google Inc. All rights reserved.
// http://code.google.com/p/ceres-solver/
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// Author: vitus@google.com (Michael Vitus)

#ifndef CERES_PUBLIC_HOUSEHOLDER_VECTOR_H_
#define CERES_PUBLIC_HOUSEHOLDER_VECTOR_H_

#include "Eigen/Core"
#include "glog/logging.h"

namespace ceres {
namespace 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.
template <typename Scalar>
void ComputeHouseholderVector(const Eigen::Matrix<Scalar, Eigen::Dynamic, 1>& x,
                              Eigen::Matrix<Scalar, Eigen::Dynamic, 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;
}

}  // namespace internal
}  // namespace ceres

#endif  // CERES_PUBLIC_HOUSEHOLDER_VECTOR_H_