From 32b08ac971b2ab66cf83360a9e2d42a99bfe3b70 Mon Sep 17 00:00:00 2001 From: Gael Guennebaud Date: Fri, 17 Jul 2009 16:22:39 +0200 Subject: re-implement stableNorm using a homemade blocky and vectorization friendly algorithm (slow if no vectorization) --- Eigen/Core | 1 + Eigen/src/Core/Block.h | 4 +- Eigen/src/Core/Dot.h | 125 ---------------------------- Eigen/src/Core/Functors.h | 3 - Eigen/src/Core/MatrixBase.h | 1 + Eigen/src/Core/StableNorm.h | 194 ++++++++++++++++++++++++++++++++++++++++++++ bench/bench_norm.cpp | 106 ++++++++++++------------ test/adjoint.cpp | 4 +- 8 files changed, 256 insertions(+), 182 deletions(-) create mode 100644 Eigen/src/Core/StableNorm.h diff --git a/Eigen/Core b/Eigen/Core index a7a2a768a..ad606ec38 100644 --- a/Eigen/Core +++ b/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" diff --git a/Eigen/src/Core/Block.h b/Eigen/src/Core/Block.h index 382518696..ffc0707ee 100644 --- a/Eigen/src/Core/Block.h +++ b/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. - * The default is AsRequested. This parameter is internaly used by Eigen + * \param _PacketAccess allows to enforce aligned loads and stores if set to \b ForceAligned. + * 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 diff --git a/Eigen/src/Core/Dot.h b/Eigen/src/Core/Dot.h index d97f5837f..9e84d72bb 100644 --- a/Eigen/src/Core/Dot.h +++ b/Eigen/src/Core/Dot.h @@ -292,131 +292,6 @@ inline typename NumTraits::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 -inline typename NumTraits::Scalar>::Real -MatrixBase::stableNorm() const -{ - return this->cwise().abs().redux(ei_scalar_hypot_op()); -} - -/** \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 -inline typename NumTraits::Scalar>::Real -MatrixBase::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::max(); // largest integer - ibeta = std::numeric_limits::radix; //NumTraits::Base; // base for floating-point numbers - it = std::numeric_limits::digits; //NumTraits::Mantissa; // number of base-beta digits in mantissa - iemin = std::numeric_limits::min_exponent; // minimum exponent - iemax = std::numeric_limits::max_exponent; // maximum exponent - rbig = std::numeric_limits::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 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 diff --git a/Eigen/src/Core/Functors.h b/Eigen/src/Core/Functors.h index 89badb353..a4c9604df 100644 --- a/Eigen/src/Core/Functors.h +++ b/Eigen/src/Core/Functors.h @@ -124,9 +124,6 @@ template struct ei_scalar_hypot_op EIGEN_EMPTY_STRUCT { // typedef typename NumTraits::Real result_type; EIGEN_STRONG_INLINE const Scalar operator() (const Scalar& _x, const Scalar& _y) const { -// typedef typename NumTraits::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; diff --git a/Eigen/src/Core/MatrixBase.h b/Eigen/src/Core/MatrixBase.h index b8273ca22..5298ae271 100644 --- a/Eigen/src/Core/MatrixBase.h +++ b/Eigen/src/Core/MatrixBase.h @@ -384,6 +384,7 @@ template class MatrixBase RealScalar norm() const; RealScalar stableNorm() const; RealScalar blueNorm() const; + RealScalar hypotNorm() const; const PlainMatrixType normalized() const; void normalize(); diff --git a/Eigen/src/Core/StableNorm.h b/Eigen/src/Core/StableNorm.h new file mode 100644 index 000000000..383c64f01 --- /dev/null +++ b/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 +// +// 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 . + +#ifndef EIGEN_STABLENORM_H +#define EIGEN_STABLENORM_H + +template +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 +inline typename NumTraits::Scalar>::Real +MatrixBase::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(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 +inline typename NumTraits::Scalar>::Real +MatrixBase::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::max(); // largest integer + ibeta = std::numeric_limits::radix; // base for floating-point numbers + it = std::numeric_limits::digits; // number of base-beta digits in mantissa + iemin = std::numeric_limits::min_exponent; // minimum exponent + iemax = std::numeric_limits::max_exponent; // maximum exponent + rbig = std::numeric_limits::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 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 +inline typename NumTraits::Scalar>::Real +MatrixBase::hypotNorm() const +{ + return this->cwise().abs().redux(ei_scalar_hypot_op()); +} + +#endif // EIGEN_STABLENORM_H diff --git a/bench/bench_norm.cpp b/bench/bench_norm.cpp index 8e4fefdd7..7a3dc2e68 100644 --- a/bench/bench_norm.cpp +++ b/bench/bench_norm.cpp @@ -13,7 +13,7 @@ EIGEN_DONT_INLINE typename T::Scalar sqsumNorm(const T& v) template EIGEN_DONT_INLINE typename T::Scalar hypotNorm(const T& v) { - return v.stableNorm(); + return v.hypotNorm(); } template @@ -56,23 +56,7 @@ EIGEN_DONT_INLINE typename T::Scalar twopassNorm(T& v) template EIGEN_DONT_INLINE typename T::Scalar bl2passNorm(T& v) { - const int blockSize = 4096; - typedef typename T::Scalar Scalar; - Scalar s = 0; - Scalar ssq = 0; - for (int bi=0; bi::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); + return v.stableNorm(); } template @@ -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()) << "\t" << blueNorm(vd.cast()) << "\n"; std::cout << "lapackNorm\t" << lapackNorm(vf) << "\t" << lapackNorm(vd) << "\t" << lapackNorm(vf.cast()) << "\t" << lapackNorm(vd.cast()) << "\n"; std::cout << "twopassNorm\t" << twopassNorm(vf) << "\t" << twopassNorm(vd) << "\t" << twopassNorm(vf.cast()) << "\t" << twopassNorm(vd.cast()) << "\n"; - std::cout << "bl2passNorm\t" << bl2passNorm(vf) << "\t" << bl2passNorm(vd) << "\t" << bl2passNorm(vf.cast()) << "\t" << bl2passNorm(vd.cast()) << "\n"; +// std::cout << "bl2passNorm\t" << bl2passNorm(vf) << "\t" << bl2passNorm(vd) << "\t" << bl2passNorm(vf.cast()) << "\t" << bl2passNorm(vd.cast()) << "\n"; } int main(int argc, char** argv) @@ -273,34 +275,7 @@ int main(int argc, char** argv) double y = 1.1345743233455785456788e12 * ei_random(); VectorXf v = VectorXf::Ones(1024) * y; - 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; - 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; +// 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); + } } diff --git a/test/adjoint.cpp b/test/adjoint.cpp index 1f4aa7427..47e1bd740 100644 --- a/test/adjoint.cpp +++ b/test/adjoint.cpp @@ -76,8 +76,8 @@ template void adjoint(const MatrixType& m) { VERIFY_IS_MUCH_SMALLER_THAN(vzero.norm(), static_cast(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 -- cgit v1.2.3