StableNorm.h

// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2009 Gael Guennebaud <gael.guennebaud@inria.fr>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
 
#ifndef EIGEN_STABLENORM_H
#define EIGEN_STABLENORM_H
 
// IWYU pragma: private
#include "./InternalHeaderCheck.h"
 
namespace Eigen {
 
namespace internal {
 
template <typename ExpressionType, typename Scalar>
inline void stable_norm_kernel(const ExpressionType& bl, Scalar& ssq, Scalar& scale, Scalar& invScale) {
  Scalar maxCoeff = bl.cwiseAbs().maxCoeff();
 
  if (maxCoeff > scale) {
    ssq = ssq * numext::abs2(scale / maxCoeff);
    Scalar tmp = Scalar(1) / maxCoeff;
    if (tmp > NumTraits<Scalar>::highest()) {
      invScale = NumTraits<Scalar>::highest();
      scale = Scalar(1) / invScale;
    } else if (maxCoeff > NumTraits<Scalar>::highest())  // we got a INF
    {
      invScale = Scalar(1);
      scale = maxCoeff;
    } else {
      scale = maxCoeff;
      invScale = tmp;
    }
  } else if (maxCoeff != maxCoeff)  // we got a NaN
  {
    scale = maxCoeff;
  }
 
  // TODO if the maxCoeff is much much smaller than the current scale,
  // then we can neglect this sub vector
  if (scale > Scalar(0))  // if scale==0, then bl is 0
    ssq += (bl * invScale).squaredNorm();
}
 
template <typename VectorType, typename RealScalar>
void stable_norm_impl_inner_step(const VectorType& vec, RealScalar& ssq, RealScalar& scale, RealScalar& invScale) {
  const Index blockSize = 4096;
 
  Index n = vec.size();
  Index blockEnd = numext::round_down(n, blockSize);
  for (Index i = 0; i < blockEnd; i += blockSize) {
    internal::stable_norm_kernel(vec.template segment<blockSize>(i), ssq, scale, invScale);
  }
  if (n > blockEnd) {
    internal::stable_norm_kernel(vec.tail(n - blockEnd), ssq, scale, invScale);
  }
}
 
template <typename VectorType>
typename VectorType::RealScalar stable_norm_impl(const VectorType& vec,
                                                 std::enable_if_t<VectorType::IsVectorAtCompileTime>* = 0) {
  using std::abs;
  using std::sqrt;
 
  Index n = vec.size();
  if (EIGEN_PREDICT_FALSE(n == 1)) return abs(vec.coeff(0));
 
  typedef typename VectorType::RealScalar RealScalar;
  RealScalar scale(0);
  RealScalar invScale(1);
  RealScalar ssq(0);  // sum of squares
 
  stable_norm_impl_inner_step(vec, ssq, scale, invScale);
 
  return scale * sqrt(ssq);
}
 
template <typename MatrixType>
typename MatrixType::RealScalar stable_norm_impl(const MatrixType& mat,
                                                 std::enable_if_t<!MatrixType::IsVectorAtCompileTime>* = 0) {
  using std::sqrt;
 
  typedef typename MatrixType::RealScalar RealScalar;
  RealScalar scale(0);
  RealScalar invScale(1);
  RealScalar ssq(0);  // sum of squares
 
  for (Index j = 0; j < mat.outerSize(); ++j) stable_norm_impl_inner_step(mat.innerVector(j), ssq, scale, invScale);
  return scale * sqrt(ssq);
}
 
template <typename Derived>
inline typename NumTraits<typename traits<Derived>::Scalar>::Real blueNorm_impl(const EigenBase<Derived>& _vec) {
  typedef typename Derived::RealScalar RealScalar;
  using std::abs;
  using std::pow;
  using std::sqrt;
 
  // This program calculates the machine-dependent constants
  // bl, b2, slm, s2m, relerr overfl
  // from the "basic" machine-dependent numbers
  // nbig, ibeta, it, iemin, iemax, rbig.
  // The following define the basic machine-dependent constants.
  // For portability, the PORT subprograms "ilmaeh" and "rlmach"
  // are used. For any specific computer, each of the assignment
  // statements can be replaced
  static const int ibeta = std::numeric_limits<RealScalar>::radix;  // base for floating-point numbers
  static const int it = NumTraits<RealScalar>::digits();            // number of base-beta digits in mantissa
  static const int iemin = NumTraits<RealScalar>::min_exponent();   // minimum exponent
  static const int iemax = NumTraits<RealScalar>::max_exponent();   // maximum exponent
  static const RealScalar rbig = NumTraits<RealScalar>::highest();  // largest floating-point number
  static const RealScalar b1 =
      RealScalar(pow(RealScalar(ibeta), RealScalar(-((1 - iemin) / 2))));  // lower boundary of midrange
  static const RealScalar b2 =
      RealScalar(pow(RealScalar(ibeta), RealScalar((iemax + 1 - it) / 2)));  // upper boundary of midrange
  static const RealScalar s1m =
      RealScalar(pow(RealScalar(ibeta), RealScalar((2 - iemin) / 2)));  // scaling factor for lower range
  static const RealScalar s2m =
      RealScalar(pow(RealScalar(ibeta), RealScalar(-((iemax + it) / 2))));  // scaling factor for upper range
  static const RealScalar eps = RealScalar(pow(double(ibeta), 1 - it));
  static const RealScalar relerr = sqrt(eps);  // tolerance for neglecting asml
 
  const Derived& vec(_vec.derived());
  Index n = vec.size();
  RealScalar ab2 = b2 / RealScalar(n);
  RealScalar asml = RealScalar(0);
  RealScalar amed = RealScalar(0);
  RealScalar abig = RealScalar(0);
 
  for (Index j = 0; j < vec.outerSize(); ++j) {
    for (typename Derived::InnerIterator iter(vec, j); iter; ++iter) {
      RealScalar ax = abs(iter.value());
      if (ax > ab2)
        abig += numext::abs2(ax * s2m);
      else if (ax < b1)
        asml += numext::abs2(ax * s1m);
      else
        amed += numext::abs2(ax);
    }
  }
  if (amed != amed) return amed;  // we got a NaN
  if (abig > RealScalar(0)) {
    abig = sqrt(abig);
    if (abig > rbig)  // overflow, or *this contains INF values
      return abig;    // return INF
    if (amed > RealScalar(0)) {
      abig = abig / s2m;
      amed = sqrt(amed);
    } else
      return abig / s2m;
  } else if (asml > RealScalar(0)) {
    if (amed > RealScalar(0)) {
      abig = sqrt(amed);
      amed = sqrt(asml) / s1m;
    } else
      return sqrt(asml) / s1m;
  } else
    return sqrt(amed);
  asml = numext::mini(abig, amed);
  abig = numext::maxi(abig, amed);
  if (asml <= abig * relerr)
    return abig;
  else
    return abig * sqrt(RealScalar(1) + numext::abs2(asml / abig));
}
 
}  // end namespace internal

template <typename Derived>
inline typename NumTraits<typename internal::traits<Derived>::Scalar>::Real MatrixBase<Derived>::stableNorm() const {
  return internal::stable_norm_impl(derived());
}

 * A Portable Fortran Program to Find the Euclidean Norm of a Vector,
 * ACM TOMS, Vol 4, Issue 1, 1978.
 *
 * For architecture/scalar types without vectorization, this version
 * is much faster than stableNorm(). Otherwise the stableNorm() is faster.
 *
 * \sa norm(), stableNorm(), hypotNorm()
 */
template <typename Derived>
inline typename NumTraits<typename internal::traits<Derived>::Scalar>::Real MatrixBase<Derived>::blueNorm() const {
  return internal::blueNorm_impl(*this);
}

template <typename Derived>
inline typename NumTraits<typename internal::traits<Derived>::Scalar>::Real MatrixBase<Derived>::hypotNorm() const {
  if (size() == 1)
    return numext::abs(coeff(0, 0));
  else
    return this->cwiseAbs().redux(internal::scalar_hypot_op<RealScalar>());
}
 
}  // end namespace Eigen
 
#endif  // EIGEN_STABLENORM_H