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// 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
namespace Eigen {
namespace internal {
template<typename ExpressionType, typename Scalar>
inline void stable_norm_kernel(const ExpressionType& bl, Scalar& ssq, Scalar& scale, Scalar& invScale)
{
Scalar max = bl.cwiseAbs().maxCoeff();
if (max>scale)
{
ssq = ssq * abs2(scale/max);
scale = max;
invScale = Scalar(1)/scale;
}
// TODO if the max is much much smaller than the current scale,
// then we can neglect this sub vector
ssq += (bl*invScale).squaredNorm();
}
}
/** \returns the \em l2 norm of \c *this avoiding underflow and overflow.
* This version use a blockwise two passes algorithm:
* 1 - find the absolute largest coefficient \c s
* 2 - compute \f$ s \Vert \frac{*this}{s} \Vert \f$ in a standard way
*
* For architecture/scalar types supporting vectorization, this version
* is faster than blueNorm(). Otherwise the blueNorm() is much faster.
*
* \sa norm(), blueNorm(), hypotNorm()
*/
template<typename Derived>
inline typename NumTraits<typename internal::traits<Derived>::Scalar>::Real
MatrixBase<Derived>::stableNorm() const
{
using std::min;
const Index blockSize = 4096;
RealScalar scale(0);
RealScalar invScale(1);
RealScalar ssq(0); // sum of square
enum {
Alignment = (int(Flags)&DirectAccessBit) || (int(Flags)&AlignedBit) ? 1 : 0
};
Index n = size();
Index bi = internal::first_aligned(derived());
if (bi>0)
internal::stable_norm_kernel(this->head(bi), ssq, scale, invScale);
for (; bi<n; bi+=blockSize)
internal::stable_norm_kernel(this->segment(bi,(min)(blockSize, n - bi)).template forceAlignedAccessIf<Alignment>(), ssq, scale, invScale);
return scale * internal::sqrt(ssq);
}
/** \returns the \em l2 norm of \c *this using the Blue's algorithm.
* 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
{
using std::pow;
using std::min;
using std::max;
static Index nmax = -1;
static RealScalar b1, b2, s1m, s2m, overfl, rbig, relerr;
if(nmax <= 0)
{
int nbig, ibeta, it, iemin, iemax, iexp;
RealScalar abig, eps;
// This program calculates the machine-dependent constants
// bl, b2, slm, s2m, relerr overfl, nmax
// 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
nbig = (std::numeric_limits<Index>::max)(); // largest integer
ibeta = std::numeric_limits<RealScalar>::radix; // base for floating-point numbers
it = std::numeric_limits<RealScalar>::digits; // number of base-beta digits in mantissa
iemin = std::numeric_limits<RealScalar>::min_exponent; // minimum exponent
iemax = std::numeric_limits<RealScalar>::max_exponent; // maximum exponent
rbig = (std::numeric_limits<RealScalar>::max)(); // largest floating-point number
iexp = -((1-iemin)/2);
b1 = RealScalar(pow(RealScalar(ibeta),RealScalar(iexp))); // lower boundary of midrange
iexp = (iemax + 1 - it)/2;
b2 = RealScalar(pow(RealScalar(ibeta),RealScalar(iexp))); // upper boundary of midrange
iexp = (2-iemin)/2;
s1m = RealScalar(pow(RealScalar(ibeta),RealScalar(iexp))); // scaling factor for lower range
iexp = - ((iemax+it)/2);
s2m = RealScalar(pow(RealScalar(ibeta),RealScalar(iexp))); // scaling factor for upper range
overfl = rbig*s2m; // overflow boundary for abig
eps = RealScalar(pow(double(ibeta), 1-it));
relerr = internal::sqrt(eps); // tolerance for neglecting asml
abig = RealScalar(1.0/eps - 1.0);
if (RealScalar(nbig)>abig) nmax = int(abig); // largest safe n
else nmax = nbig;
}
Index n = size();
RealScalar ab2 = b2 / RealScalar(n);
RealScalar asml = RealScalar(0);
RealScalar amed = RealScalar(0);
RealScalar abig = RealScalar(0);
for(Index j=0; j<n; ++j)
{
RealScalar ax = internal::abs(coeff(j));
if(ax > ab2) abig += internal::abs2(ax*s2m);
else if(ax < b1) asml += internal::abs2(ax*s1m);
else amed += internal::abs2(ax);
}
if(abig > RealScalar(0))
{
abig = internal::sqrt(abig);
if(abig > overfl)
{
eigen_assert(false && "overflow");
return rbig;
}
if(amed > RealScalar(0))
{
abig = abig/s2m;
amed = internal::sqrt(amed);
}
else
return abig/s2m;
}
else if(asml > RealScalar(0))
{
if (amed > RealScalar(0))
{
abig = internal::sqrt(amed);
amed = internal::sqrt(asml) / s1m;
}
else
return internal::sqrt(asml)/s1m;
}
else
return internal::sqrt(amed);
asml = (min)(abig, amed);
abig = (max)(abig, amed);
if(asml <= abig*relerr)
return abig;
else
return abig * internal::sqrt(RealScalar(1) + internal::abs2(asml/abig));
}
/** \returns the \em l2 norm of \c *this avoiding undeflow and overflow.
* This version use a concatenation of hypot() calls, and it is very slow.
*
* \sa norm(), stableNorm()
*/
template<typename Derived>
inline typename NumTraits<typename internal::traits<Derived>::Scalar>::Real
MatrixBase<Derived>::hypotNorm() const
{
return this->cwiseAbs().redux(internal::scalar_hypot_op<RealScalar>());
}
} // end namespace Eigen
#endif // EIGEN_STABLENORM_H
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