re-implement stableNorm using a homemade blocky and

vectorization friendly algorithm (slow if no vectorization)

Eigen/Core

@@ -161,6 +161,7 @@ namespace Eigen {
 #include "src/Core/CwiseNullaryOp.h"
 #include "src/Core/CwiseUnaryView.h"
 #include "src/Core/Dot.h"
+#include "src/Core/StableNorm.h"
 #include "src/Core/DiagonalProduct.h"
 #include "src/Core/SolveTriangular.h"
 #include "src/Core/MapBase.h"

Eigen/src/Core/Block.h

@@ -33,8 +33,8 @@
   * \param MatrixType the type of the object in which we are taking a block
   * \param BlockRows the number of rows of the block we are taking at compile time (optional)
   * \param BlockCols the number of columns of the block we are taking at compile time (optional)
-  * \param _PacketAccess allows to enforce aligned loads and stores if set to ForceAligned.
+  * \param _PacketAccess allows to enforce aligned loads and stores if set to \b ForceAligned.
-  *                      The default is AsRequested. This parameter is internaly used by Eigen
+  *                      The default is \b AsRequested. This parameter is internaly used by Eigen
   *                      in expressions such as \code mat.block() += other; \endcode and most of
   *                      the time this is the only way it is used.
   * \param _DirectAccessStatus \internal used for partial specialization

Eigen/src/Core/Dot.h

@@ -292,131 +292,6 @@ inline typename NumTraits<typename ei_traits<Derived>::Scalar>::Real MatrixBase<
   return ei_sqrt(squaredNorm());
 }
 
-/** \returns the \em l2 norm of \c *this using a numerically more stable
-  * algorithm.
-  *
-  * \sa norm(), dot(), squaredNorm(), blueNorm()
-  */
-template<typename Derived>
-inline typename NumTraits<typename ei_traits<Derived>::Scalar>::Real
-MatrixBase<Derived>::stableNorm() const
-{
-  return this->cwise().abs().redux(ei_scalar_hypot_op<RealScalar>());
-}
-
-/** \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.
-  *
-  * \sa norm(), dot(), squaredNorm(), stableNorm()
-  */
-template<typename Derived>
-inline typename NumTraits<typename ei_traits<Derived>::Scalar>::Real
-MatrixBase<Derived>::blueNorm() const
-{
-  static int nmax = -1;
-  static Scalar b1, b2, s1m, s2m, overfl, rbig, relerr;
-  int n;
-  Scalar ax, abig, amed, asml;
-
-  if(nmax <= 0)
-  {
-    int nbig, ibeta, it, iemin, iemax, iexp;
-    Scalar 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<int>::max();            // largest integer
-    ibeta = std::numeric_limits<Scalar>::radix; //NumTraits<Scalar>::Base;                    // base for floating-point numbers
-    it    = std::numeric_limits<Scalar>::digits; //NumTraits<Scalar>::Mantissa;                // number of base-beta digits in mantissa
-    iemin = std::numeric_limits<Scalar>::min_exponent;  // minimum exponent
-    iemax = std::numeric_limits<Scalar>::max_exponent;  // maximum exponent
-    rbig  = std::numeric_limits<Scalar>::max();         // largest floating-point number
-
-    // Check the basic machine-dependent constants.
-    if(iemin > 1 - 2*it || 1+it>iemax || (it==2 && ibeta<5)
-      || (it<=4 && ibeta <= 3 ) || it<2)
-    {
-      ei_assert(false && "the algorithm cannot be guaranteed on this computer");
-    }
-    iexp  = -((1-iemin)/2);
-    b1    = std::pow(ibeta, iexp);  // lower boundary of midrange
-    iexp  = (iemax + 1 - it)/2;
-    b2    = std::pow(ibeta,iexp);   // upper boundary of midrange
-
-    iexp  = (2-iemin)/2;
-    s1m   = std::pow(ibeta,iexp);   // scaling factor for lower range
-    iexp  = - ((iemax+it)/2);
-    s2m   = std::pow(ibeta,iexp);   // scaling factor for upper range
-
-    overfl  = rbig*s2m;          // overfow boundary for abig
-    eps     = std::pow(ibeta, 1-it);
-    relerr  = ei_sqrt(eps);      // tolerance for neglecting asml
-    abig    = 1.0/eps - 1.0;
-    if (Scalar(nbig)>abig)  nmax = abig;  // largest safe n
-    else                    nmax = nbig;
-  }
-  n = size();
-  if(n==0)
-    return 0;
-  asml = Scalar(0);
-  amed = Scalar(0);
-  abig = Scalar(0);
-  for(int j=0; j<n; ++j)
-  {
-    ax = ei_abs(coeff(j));
-    if(ax > b2)      abig += ei_abs2(ax*s2m);
-    else if(ax < b1) asml += ei_abs2(ax*s1m);
-    else             amed += ei_abs2(ax);
-  }
-  if(abig > Scalar(0))
-  {
-    abig = ei_sqrt(abig);
-    if(abig > overfl)
-    {
-      ei_assert(false && "overflow");
-      return rbig;
-    }
-    if(amed > Scalar(0))
-    {
-      abig = abig/s2m;
-      amed = ei_sqrt(amed);
-    }
-    else
-    {
-      return abig/s2m;
-    }
-
-  }
-  else if(asml > Scalar(0))
-  {
-    if (amed > Scalar(0))
-    {
-      abig = ei_sqrt(amed);
-      amed = ei_sqrt(asml) / s1m;
-    }
-    else
-    {
-      return ei_sqrt(asml)/s1m;
-    }
-  }
-  else
-  {
-    return ei_sqrt(amed);
-  }
-  asml = std::min(abig, amed);
-  abig = std::max(abig, amed);
-  if(asml <= abig*relerr)
-    return abig;
-  else
-    return abig * ei_sqrt(Scalar(1) + ei_abs2(asml/abig));
-}
-
 /** \returns an expression of the quotient of *this by its own norm.
   *
   * \only_for_vectors

Eigen/src/Core/Functors.h

@@ -124,9 +124,6 @@ template<typename Scalar> struct ei_scalar_hypot_op EIGEN_EMPTY_STRUCT {
 //   typedef typename NumTraits<Scalar>::Real result_type;
   EIGEN_STRONG_INLINE const Scalar operator() (const Scalar& _x, const Scalar& _y) const
   {
-//     typedef typename NumTraits<T>::Real RealScalar;
-//     RealScalar _x = ei_abs(x);
-//     RealScalar _y = ei_abs(y);
     Scalar p = std::max(_x, _y);
     Scalar q = std::min(_x, _y);
     Scalar qp = q/p;

Eigen/src/Core/MatrixBase.h

@@ -384,6 +384,7 @@ template<typename Derived> class MatrixBase
     RealScalar norm() const;
     RealScalar stableNorm() const;
     RealScalar blueNorm() const;
+    RealScalar hypotNorm() const;
     const PlainMatrixType normalized() const;
     void normalize();
 

Eigen/src/Core/StableNorm.h

@@ -0,0 +1,194 @@
+// This file is part of Eigen, a lightweight C++ template library
+// for linear algebra.
+//
+// Copyright (C) 2009 Gael Guennebaud <g.gael@free.fr>
+//
+// Eigen is free software; you can redistribute it and/or
+// modify it under the terms of the GNU Lesser General Public
+// License as published by the Free Software Foundation; either
+// version 3 of the License, or (at your option) any later version.
+//
+// Alternatively, you can redistribute it and/or
+// modify it under the terms of the GNU General Public License as
+// published by the Free Software Foundation; either version 2 of
+// the License, or (at your option) any later version.
+//
+// Eigen is distributed in the hope that it will be useful, but WITHOUT ANY
+// WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
+// FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License or the
+// GNU General Public License for more details.
+//
+// You should have received a copy of the GNU Lesser General Public
+// License and a copy of the GNU General Public License along with
+// Eigen. If not, see <http://www.gnu.org/licenses/>.
+
+#ifndef EIGEN_STABLENORM_H
+#define EIGEN_STABLENORM_H
+
+template<typename ExpressionType, typename Scalar>
+inline void ei_stable_norm_kernel(const ExpressionType& bl, Scalar& ssq, Scalar& scale, Scalar& invScale)
+{
+  Scalar max = bl.cwise().abs().maxCoeff();
+  if (max>scale)
+  {
+    ssq = ssq * ei_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 ei_traits<Derived>::Scalar>::Real
+MatrixBase<Derived>::stableNorm() const
+{
+  const int blockSize = 4096;
+  RealScalar scale = 0;
+  RealScalar invScale;
+  RealScalar ssq = 0; // sum of square
+  enum {
+    Alignment = (int(Flags)&DirectAccessBit) || (int(Flags)&AlignedBit) ? ForceAligned : AsRequested
+  };
+  int n = size();
+  int bi=0;
+  if ((int(Flags)&DirectAccessBit) && !(int(Flags)&AlignedBit))
+  {
+    bi = ei_alignmentOffset(&const_cast_derived().coeffRef(0), n);
+    if (bi>0)
+      ei_stable_norm_kernel(start(bi), ssq, scale, invScale);
+  }
+  for (; bi<n; bi+=blockSize)
+    ei_stable_norm_kernel(VectorBlock<Derived,Dynamic,Alignment>(derived(),bi,std::min(blockSize, n - bi)), ssq, scale, invScale);
+  return scale * ei_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 ei_traits<Derived>::Scalar>::Real
+MatrixBase<Derived>::blueNorm() const
+{
+  static int 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<int>::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
+
+    // Check the basic machine-dependent constants.
+    if(iemin > 1 - 2*it || 1+it>iemax || (it==2 && ibeta<5)
+      || (it<=4 && ibeta <= 3 ) || it<2)
+    {
+      ei_assert(false && "the algorithm cannot be guaranteed on this computer");
+    }
+    iexp  = -((1-iemin)/2);
+    b1    = std::pow(ibeta, iexp);  // lower boundary of midrange
+    iexp  = (iemax + 1 - it)/2;
+    b2    = std::pow(ibeta,iexp);   // upper boundary of midrange
+
+    iexp  = (2-iemin)/2;
+    s1m   = std::pow(ibeta,iexp);   // scaling factor for lower range
+    iexp  = - ((iemax+it)/2);
+    s2m   = std::pow(ibeta,iexp);   // scaling factor for upper range
+
+    overfl  = rbig*s2m;             // overfow boundary for abig
+    eps     = std::pow(ibeta, 1-it);
+    relerr  = ei_sqrt(eps);         // tolerance for neglecting asml
+    abig    = 1.0/eps - 1.0;
+    if (RealScalar(nbig)>abig)  nmax = abig;  // largest safe n
+    else                        nmax = nbig;
+  }
+  int n = size();
+  RealScalar ab2 = b2 / RealScalar(n);
+  RealScalar asml = RealScalar(0);
+  RealScalar amed = RealScalar(0);
+  RealScalar abig = RealScalar(0);
+  for(int j=0; j<n; ++j)
+  {
+    RealScalar ax = ei_abs(coeff(j));
+    if(ax > ab2)     abig += ei_abs2(ax*s2m);
+    else if(ax < b1) asml += ei_abs2(ax*s1m);
+    else             amed += ei_abs2(ax);
+  }
+  if(abig > RealScalar(0))
+  {
+    abig = ei_sqrt(abig);
+    if(abig > overfl)
+    {
+      ei_assert(false && "overflow");
+      return rbig;
+    }
+    if(amed > RealScalar(0))
+    {
+      abig = abig/s2m;
+      amed = ei_sqrt(amed);
+    }
+    else
+      return abig/s2m;
+  }
+  else if(asml > RealScalar(0))
+  {
+    if (amed > RealScalar(0))
+    {
+      abig = ei_sqrt(amed);
+      amed = ei_sqrt(asml) / s1m;
+    }
+    else
+      return ei_sqrt(asml)/s1m;
+  }
+  else
+    return ei_sqrt(amed);
+  asml = std::min(abig, amed);
+  abig = std::max(abig, amed);
+  if(asml <= abig*relerr)
+    return abig;
+  else
+    return abig * ei_sqrt(RealScalar(1) + ei_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 ei_traits<Derived>::Scalar>::Real
+MatrixBase<Derived>::hypotNorm() const
+{
+  return this->cwise().abs().redux(ei_scalar_hypot_op<RealScalar>());
+}
+
+#endif // EIGEN_STABLENORM_H

bench/bench_norm.cpp

@@ -13,7 +13,7 @@ EIGEN_DONT_INLINE typename T::Scalar sqsumNorm(const T& v)
 template<typename T>
 EIGEN_DONT_INLINE typename T::Scalar hypotNorm(const T& v)
 {
-  return v.stableNorm();
+  return v.hypotNorm();
 }
 
 template<typename T>
@@ -56,23 +56,7 @@ EIGEN_DONT_INLINE typename T::Scalar twopassNorm(T& v)
 template<typename T>
 EIGEN_DONT_INLINE typename T::Scalar bl2passNorm(T& v)
 {
-  const int blockSize = 4096;
+  return v.stableNorm();
-  typedef typename T::Scalar Scalar;
-  Scalar s = 0;
-  Scalar ssq = 0;
-  for (int bi=0; bi<v.size(); bi+=blockSize)
-  {
-    int r = std::min(blockSize, v.size() - bi);
-    Eigen::Block<typename ei_cleantype<T>::type,Eigen::Dynamic,1,Eigen::ForceAligned> sv(v,bi,0,r,1);
-    Scalar m = sv.cwise().abs().maxCoeff();
-    if (m>s)
-    {
-      ssq = ssq * ei_abs2(s/m);
-      s = m;
-    }
-    ssq += (sv/s).squaredNorm();
-  }
-  return s*ei_sqrt(ssq);
 }
 
 template<typename T>
@@ -163,6 +147,19 @@ EIGEN_DONT_INLINE typename T::Scalar pblueNorm(const T& v)
     Packet ax_s1m = ei_pmul(ax,ps1m);
     Packet maskBig = ei_plt(pb2,ax);
     Packet maskSml = ei_plt(ax,pb1);
+
+//     Packet maskMed = ei_pand(maskSml,maskBig);
+//     Packet scale = ei_pset1(Scalar(0));
+//     scale = ei_por(scale, ei_pand(maskBig,ps2m));
+//     scale = ei_por(scale, ei_pand(maskSml,ps1m));
+//     scale = ei_por(scale, ei_pandnot(ei_pset1(Scalar(1)),maskMed));
+//     ax = ei_pmul(ax,scale);
+//     ax = ei_pmul(ax,ax);
+//     pabig = ei_padd(pabig, ei_pand(maskBig, ax));
+//     pasml = ei_padd(pasml, ei_pand(maskSml, ax));
+//     pamed = ei_padd(pamed, ei_pandnot(ax,maskMed));
+
+
     pabig = ei_padd(pabig, ei_pand(maskBig, ei_pmul(ax_s2m,ax_s2m)));
     pasml = ei_padd(pasml, ei_pand(maskSml, ei_pmul(ax_s1m,ax_s1m)));
     pamed = ei_padd(pamed, ei_pandnot(ei_pmul(ax,ax),ei_pand(maskSml,maskBig)));
@@ -215,7 +212,7 @@ EIGEN_DONT_INLINE typename T::Scalar pblueNorm(const T& v)
 }
 
 #define BENCH_PERF(NRM) { \
-  Eigen::BenchTimer tf, td; tf.reset(); td.reset();\
+  Eigen::BenchTimer tf, td, tcf; tf.reset(); td.reset(); tcf.reset();\
   for (int k=0; k<tries; ++k) { \
     tf.start(); \
     for (int i=0; i<iters; ++i) NRM(vf); \
@@ -226,7 +223,12 @@ EIGEN_DONT_INLINE typename T::Scalar pblueNorm(const T& v)
     for (int i=0; i<iters; ++i) NRM(vd); \
     td.stop(); \
   } \
-  std::cout << #NRM << "\t" << tf.value() << "   " << td.value() << "\n"; \
+  for (int k=0; k<std::max(1,tries/3); ++k) { \
+    tcf.start(); \
+    for (int i=0; i<iters; ++i) NRM(vcf); \
+    tcf.stop(); \
+  } \
+  std::cout << #NRM << "\t" << tf.value() << "   " << td.value() <<  "    " << tcf.value() << "\n"; \
 }
 
 void check_accuracy(double basef, double based, int s)
@@ -263,7 +265,7 @@ void check_accuracy_var(int ef0, int ef1, int ed0, int ed1, int s)
   std::cout << "pblueNorm\t"  << pblueNorm(vf)  << "\t" << pblueNorm(vd)  << "\t" << blueNorm(vf.cast<long double>()) << "\t" << blueNorm(vd.cast<long double>()) << "\n";
   std::cout << "lapackNorm\t" << lapackNorm(vf) << "\t" << lapackNorm(vd) << "\t" << lapackNorm(vf.cast<long double>()) << "\t" << lapackNorm(vd.cast<long double>()) << "\n";
   std::cout << "twopassNorm\t" << twopassNorm(vf) << "\t" << twopassNorm(vd) << "\t" << twopassNorm(vf.cast<long double>()) << "\t" << twopassNorm(vd.cast<long double>()) << "\n";
-  std::cout << "bl2passNorm\t" << bl2passNorm(vf) << "\t" << bl2passNorm(vd) << "\t" << bl2passNorm(vf.cast<long double>()) << "\t" << bl2passNorm(vd.cast<long double>()) << "\n";
+//   std::cout << "bl2passNorm\t" << bl2passNorm(vf) << "\t" << bl2passNorm(vd) << "\t" << bl2passNorm(vf.cast<long double>()) << "\t" << bl2passNorm(vd.cast<long double>()) << "\n";
 }
 
 int main(int argc, char** argv)
@@ -273,34 +275,7 @@ int main(int argc, char** argv)
   double y = 1.1345743233455785456788e12 * ei_random<double>();
   VectorXf v = VectorXf::Ones(1024) * y;
 
-  std::cerr << "Performance (out of cache):\n";
+// return 0;
-  {
-    int iters = 1;
-    VectorXf vf = VectorXf::Random(1024*1024*32) * y;
-    VectorXd vd = VectorXd::Random(1024*1024*32) * y;
-    BENCH_PERF(sqsumNorm);
-    BENCH_PERF(blueNorm);
-    BENCH_PERF(pblueNorm);
-    BENCH_PERF(lapackNorm);
-    BENCH_PERF(hypotNorm);
-    BENCH_PERF(twopassNorm);
-    BENCH_PERF(bl2passNorm);
-  }
-
-  std::cerr << "\nPerformance (in cache):\n";
-  {
-    int iters = 100000;
-    VectorXf vf = VectorXf::Random(512) * y;
-    VectorXd vd = VectorXd::Random(512) * y;
-    BENCH_PERF(sqsumNorm);
-    BENCH_PERF(blueNorm);
-    BENCH_PERF(pblueNorm);
-    BENCH_PERF(lapackNorm);
-    BENCH_PERF(hypotNorm);
-    BENCH_PERF(twopassNorm);
-    BENCH_PERF(bl2passNorm);
-  }
-return 0;
   int s = 10000;
   double basef_ok = 1.1345743233455785456788e15;
   double based_ok = 1.1345743233455785456788e95;
@@ -317,7 +292,7 @@ return 0;
   check_accuracy(basef_ok, based_ok, s);
 
   std::cerr << "\nUnderflow:\n";
-  check_accuracy(basef_under, based_under, 1);
+  check_accuracy(basef_under, based_under, s);
 
   std::cerr << "\nOverflow:\n";
   check_accuracy(basef_over, based_over, s);
@@ -335,4 +310,35 @@ return 0;
     check_accuracy_var(-27,20,-302,-190,s);
     std::cout << "\n";
   }
+
+  std::cout.precision(4);
+  std::cerr << "Performance (out of cache):\n";
+  {
+    int iters = 1;
+    VectorXf vf = VectorXf::Random(1024*1024*32) * y;
+    VectorXd vd = VectorXd::Random(1024*1024*32) * y;
+    VectorXcf vcf = VectorXcf::Random(1024*1024*32) * y;
+    BENCH_PERF(sqsumNorm);
+    BENCH_PERF(blueNorm);
+//     BENCH_PERF(pblueNorm);
+//     BENCH_PERF(lapackNorm);
+//     BENCH_PERF(hypotNorm);
+//     BENCH_PERF(twopassNorm);
+    BENCH_PERF(bl2passNorm);
+  }
+
+  std::cerr << "\nPerformance (in cache):\n";
+  {
+    int iters = 100000;
+    VectorXf vf = VectorXf::Random(512) * y;
+    VectorXd vd = VectorXd::Random(512) * y;
+    VectorXcf vcf = VectorXcf::Random(512) * y;
+    BENCH_PERF(sqsumNorm);
+    BENCH_PERF(blueNorm);
+//     BENCH_PERF(pblueNorm);
+//     BENCH_PERF(lapackNorm);
+//     BENCH_PERF(hypotNorm);
+//     BENCH_PERF(twopassNorm);
+    BENCH_PERF(bl2passNorm);
+  }
 }

test/adjoint.cpp

@@ -76,8 +76,8 @@ template<typename MatrixType> void adjoint(const MatrixType& m)
   {
     VERIFY_IS_MUCH_SMALLER_THAN(vzero.norm(),         static_cast<RealScalar>(1));
     VERIFY_IS_APPROX(v1.norm(),                       v1.stableNorm());
-    // NOTE disabled because it currently compiles for float and double only
+    VERIFY_IS_APPROX(v1.blueNorm(),                   v1.stableNorm());
-    // VERIFY_IS_APPROX(v1.blueNorm(),                       v1.stableNorm());
+    VERIFY_IS_APPROX(v1.hypotNorm(),                  v1.stableNorm());
   }
 
   // check compatibility of dot and adjoint