// 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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// modification, are permitted provided that the following conditions are met:

//

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//   this list of conditions and the following disclaimer.

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//   and/or other materials provided with the distribution.

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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_NOTNULL(beta);

  CHECK_NOTNULL(v);

  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_