Updated fuzzy comparisons to use L2 norm as all my experiments

tends to show L2 norm works very well here.
(the legacy implementation is still available via a preprocessor token
 to allow further experiments if needed...)

Eigen/src/Core/Fuzzy.h

@@ -26,6 +26,78 @@
 #ifndef EIGEN_FUZZY_H
 #define EIGEN_FUZZY_H
 
+#ifndef EIGEN_LEGACY_COMPARES
+
+/** \returns \c true if \c *this is approximately equal to \a other, within the precision
+  * determined by \a prec.
+  *
+  * \note The fuzzy compares are done multiplicatively. Two vectors \f$ v \f$ and \f$ w \f$
+  * are considered to be approximately equal within precision \f$ p \f$ if
+  * \f[ \Vert v - w \Vert \leqslant p\,\min(\Vert v\Vert, \Vert w\Vert). \f]
+  * For matrices, the comparison is done using the Hilbert-Schmidt norm (aka Frobenius norm
+  * L2 norm).
+  *
+  * \note Because of the multiplicativeness of this comparison, one can't use this function
+  * to check whether \c *this is approximately equal to the zero matrix or vector.
+  * Indeed, \c isApprox(zero) returns false unless \c *this itself is exactly the zero matrix
+  * or vector. If you want to test whether \c *this is zero, use ei_isMuchSmallerThan(const
+  * RealScalar&, RealScalar) instead.
+  *
+  * \sa ei_isMuchSmallerThan(const RealScalar&, RealScalar) const
+  */
+template<typename Derived>
+template<typename OtherDerived>
+bool MatrixBase<Derived>::isApprox(
+  const MatrixBase<OtherDerived>& other,
+  typename NumTraits<Scalar>::Real prec
+) const
+{
+  const typename ei_nested<Derived,2>::type nested(derived());
+  const typename ei_nested<OtherDerived,2>::type otherNested(other.derived());
+  return (nested - otherNested).cwiseAbs2().sum() <= prec * prec * std::min(nested.cwiseAbs2().sum(), otherNested.cwiseAbs2().sum());
+}
+
+/** \returns \c true if the norm of \c *this is much smaller than \a other,
+  * within the precision determined by \a prec.
+  *
+  * \note The fuzzy compares are done multiplicatively. A vector \f$ v \f$ is
+  * considered to be much smaller than \f$ x \f$ within precision \f$ p \f$ if
+  * \f[ \Vert v \Vert \leqslant p\,\vert x\vert. \f]
+  * For matrices, the comparison is done using the Hilbert-Schmidt norm.
+  *
+  * \sa isApprox(), isMuchSmallerThan(const MatrixBase<OtherDerived>&, RealScalar) const
+  */
+template<typename Derived>
+bool MatrixBase<Derived>::isMuchSmallerThan(
+  const typename NumTraits<Scalar>::Real& other,
+  typename NumTraits<Scalar>::Real prec
+) const
+{
+  return cwiseAbs2().sum() <= prec * prec * other * other * cols() * rows();
+}
+
+/** \returns \c true if the norm of \c *this is much smaller than the norm of \a other,
+  * within the precision determined by \a prec.
+  *
+  * \note The fuzzy compares are done multiplicatively. A vector \f$ v \f$ is
+  * considered to be much smaller than a vector \f$ w \f$ within precision \f$ p \f$ if
+  * \f[ \Vert v \Vert \leqslant p\,\Vert w\Vert. \f]
+  * For matrices, the comparison is done using the Hilbert-Schmidt norm.
+  *
+  * \sa isApprox(), isMuchSmallerThan(const RealScalar&, RealScalar) const
+  */
+template<typename Derived>
+template<typename OtherDerived>
+bool MatrixBase<Derived>::isMuchSmallerThan(
+  const MatrixBase<OtherDerived>& other,
+  typename NumTraits<Scalar>::Real prec
+) const
+{
+  return this->cwiseAbs2().sum() <= prec * prec * other.cwiseAbs2().sum();
+}
+
+#else
+
 template<typename Derived, typename OtherDerived=Derived, bool IsVector=Derived::IsVectorAtCompileTime>
 struct ei_fuzzy_selector;
 
@@ -154,4 +226,6 @@ struct ei_fuzzy_selector<Derived,OtherDerived,false>
   }
 };
 
+#endif
+
 #endif // EIGEN_FUZZY_H