1 // This file is part of Eigen, a lightweight C++ template library
2 // for linear algebra.
3 //
4 // Copyright (C) 2009 Gael Guennebaud <gael.guennebaud@inria.fr>
5 //
6 // This Source Code Form is subject to the terms of the Mozilla
7 // Public License v. 2.0. If a copy of the MPL was not distributed
8 // with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
9
10 #ifndef EIGEN_STABLENORM_H
11 #define EIGEN_STABLENORM_H
12
13 namespace Eigen {
14
15 namespace internal {
16
17 template<typename ExpressionType, typename Scalar>
stable_norm_kernel(const ExpressionType & bl,Scalar & ssq,Scalar & scale,Scalar & invScale)18 inline void stable_norm_kernel(const ExpressionType& bl, Scalar& ssq, Scalar& scale, Scalar& invScale)
19 {
20 Scalar maxCoeff = bl.cwiseAbs().maxCoeff();
21
22 if(maxCoeff>scale)
23 {
24 ssq = ssq * numext::abs2(scale/maxCoeff);
25 Scalar tmp = Scalar(1)/maxCoeff;
26 if(tmp > NumTraits<Scalar>::highest())
27 {
28 invScale = NumTraits<Scalar>::highest();
29 scale = Scalar(1)/invScale;
30 }
31 else if(maxCoeff>NumTraits<Scalar>::highest()) // we got a INF
32 {
33 invScale = Scalar(1);
34 scale = maxCoeff;
35 }
36 else
37 {
38 scale = maxCoeff;
39 invScale = tmp;
40 }
41 }
42 else if(maxCoeff!=maxCoeff) // we got a NaN
43 {
44 scale = maxCoeff;
45 }
46
47 // TODO if the maxCoeff is much much smaller than the current scale,
48 // then we can neglect this sub vector
49 if(scale>Scalar(0)) // if scale==0, then bl is 0
50 ssq += (bl*invScale).squaredNorm();
51 }
52
53 template<typename VectorType, typename RealScalar>
stable_norm_impl_inner_step(const VectorType & vec,RealScalar & ssq,RealScalar & scale,RealScalar & invScale)54 void stable_norm_impl_inner_step(const VectorType &vec, RealScalar& ssq, RealScalar& scale, RealScalar& invScale)
55 {
56 typedef typename VectorType::Scalar Scalar;
57 const Index blockSize = 4096;
58
59 typedef typename internal::nested_eval<VectorType,2>::type VectorTypeCopy;
60 typedef typename internal::remove_all<VectorTypeCopy>::type VectorTypeCopyClean;
61 const VectorTypeCopy copy(vec);
62
63 enum {
64 CanAlign = ( (int(VectorTypeCopyClean::Flags)&DirectAccessBit)
65 || (int(internal::evaluator<VectorTypeCopyClean>::Alignment)>0) // FIXME Alignment)>0 might not be enough
66 ) && (blockSize*sizeof(Scalar)*2<EIGEN_STACK_ALLOCATION_LIMIT)
67 && (EIGEN_MAX_STATIC_ALIGN_BYTES>0) // if we cannot allocate on the stack, then let's not bother about this optimization
68 };
69 typedef typename internal::conditional<CanAlign, Ref<const Matrix<Scalar,Dynamic,1,0,blockSize,1>, internal::evaluator<VectorTypeCopyClean>::Alignment>,
70 typename VectorTypeCopyClean::ConstSegmentReturnType>::type SegmentWrapper;
71 Index n = vec.size();
72
73 Index bi = internal::first_default_aligned(copy);
74 if (bi>0)
75 internal::stable_norm_kernel(copy.head(bi), ssq, scale, invScale);
76 for (; bi<n; bi+=blockSize)
77 internal::stable_norm_kernel(SegmentWrapper(copy.segment(bi,numext::mini(blockSize, n - bi))), ssq, scale, invScale);
78 }
79
80 template<typename VectorType>
81 typename VectorType::RealScalar
82 stable_norm_impl(const VectorType &vec, typename enable_if<VectorType::IsVectorAtCompileTime>::type* = 0 )
83 {
84 using std::sqrt;
85 using std::abs;
86
87 Index n = vec.size();
88
89 if(n==1)
90 return abs(vec.coeff(0));
91
92 typedef typename VectorType::RealScalar RealScalar;
93 RealScalar scale(0);
94 RealScalar invScale(1);
95 RealScalar ssq(0); // sum of squares
96
97 stable_norm_impl_inner_step(vec, ssq, scale, invScale);
98
99 return scale * sqrt(ssq);
100 }
101
102 template<typename MatrixType>
103 typename MatrixType::RealScalar
104 stable_norm_impl(const MatrixType &mat, typename enable_if<!MatrixType::IsVectorAtCompileTime>::type* = 0 )
105 {
106 using std::sqrt;
107
108 typedef typename MatrixType::RealScalar RealScalar;
109 RealScalar scale(0);
110 RealScalar invScale(1);
111 RealScalar ssq(0); // sum of squares
112
113 for(Index j=0; j<mat.outerSize(); ++j)
114 stable_norm_impl_inner_step(mat.innerVector(j), ssq, scale, invScale);
115 return scale * sqrt(ssq);
116 }
117
118 template<typename Derived>
119 inline typename NumTraits<typename traits<Derived>::Scalar>::Real
blueNorm_impl(const EigenBase<Derived> & _vec)120 blueNorm_impl(const EigenBase<Derived>& _vec)
121 {
122 typedef typename Derived::RealScalar RealScalar;
123 using std::pow;
124 using std::sqrt;
125 using std::abs;
126
127 // This program calculates the machine-dependent constants
128 // bl, b2, slm, s2m, relerr overfl
129 // from the "basic" machine-dependent numbers
130 // nbig, ibeta, it, iemin, iemax, rbig.
131 // The following define the basic machine-dependent constants.
132 // For portability, the PORT subprograms "ilmaeh" and "rlmach"
133 // are used. For any specific computer, each of the assignment
134 // statements can be replaced
135 static const int ibeta = std::numeric_limits<RealScalar>::radix; // base for floating-point numbers
136 static const int it = NumTraits<RealScalar>::digits(); // number of base-beta digits in mantissa
137 static const int iemin = NumTraits<RealScalar>::min_exponent(); // minimum exponent
138 static const int iemax = NumTraits<RealScalar>::max_exponent(); // maximum exponent
139 static const RealScalar rbig = NumTraits<RealScalar>::highest(); // largest floating-point number
140 static const RealScalar b1 = RealScalar(pow(RealScalar(ibeta),RealScalar(-((1-iemin)/2)))); // lower boundary of midrange
141 static const RealScalar b2 = RealScalar(pow(RealScalar(ibeta),RealScalar((iemax + 1 - it)/2))); // upper boundary of midrange
142 static const RealScalar s1m = RealScalar(pow(RealScalar(ibeta),RealScalar((2-iemin)/2))); // scaling factor for lower range
143 static const RealScalar s2m = RealScalar(pow(RealScalar(ibeta),RealScalar(- ((iemax+it)/2)))); // scaling factor for upper range
144 static const RealScalar eps = RealScalar(pow(double(ibeta), 1-it));
145 static const RealScalar relerr = sqrt(eps); // tolerance for neglecting asml
146
147 const Derived& vec(_vec.derived());
148 Index n = vec.size();
149 RealScalar ab2 = b2 / RealScalar(n);
150 RealScalar asml = RealScalar(0);
151 RealScalar amed = RealScalar(0);
152 RealScalar abig = RealScalar(0);
153
154 for(Index j=0; j<vec.outerSize(); ++j)
155 {
156 for(typename Derived::InnerIterator iter(vec, j); iter; ++iter)
157 {
158 RealScalar ax = abs(iter.value());
159 if(ax > ab2) abig += numext::abs2(ax*s2m);
160 else if(ax < b1) asml += numext::abs2(ax*s1m);
161 else amed += numext::abs2(ax);
162 }
163 }
164 if(amed!=amed)
165 return amed; // we got a NaN
166 if(abig > RealScalar(0))
167 {
168 abig = sqrt(abig);
169 if(abig > rbig) // overflow, or *this contains INF values
170 return abig; // return INF
171 if(amed > RealScalar(0))
172 {
173 abig = abig/s2m;
174 amed = sqrt(amed);
175 }
176 else
177 return abig/s2m;
178 }
179 else if(asml > RealScalar(0))
180 {
181 if (amed > RealScalar(0))
182 {
183 abig = sqrt(amed);
184 amed = sqrt(asml) / s1m;
185 }
186 else
187 return sqrt(asml)/s1m;
188 }
189 else
190 return sqrt(amed);
191 asml = numext::mini(abig, amed);
192 abig = numext::maxi(abig, amed);
193 if(asml <= abig*relerr)
194 return abig;
195 else
196 return abig * sqrt(RealScalar(1) + numext::abs2(asml/abig));
197 }
198
199 } // end namespace internal
200
201 /** \returns the \em l2 norm of \c *this avoiding underflow and overflow.
202 * This version use a blockwise two passes algorithm:
203 * 1 - find the absolute largest coefficient \c s
204 * 2 - compute \f$ s \Vert \frac{*this}{s} \Vert \f$ in a standard way
205 *
206 * For architecture/scalar types supporting vectorization, this version
207 * is faster than blueNorm(). Otherwise the blueNorm() is much faster.
208 *
209 * \sa norm(), blueNorm(), hypotNorm()
210 */
211 template<typename Derived>
212 inline typename NumTraits<typename internal::traits<Derived>::Scalar>::Real
stableNorm()213 MatrixBase<Derived>::stableNorm() const
214 {
215 return internal::stable_norm_impl(derived());
216 }
217
218 /** \returns the \em l2 norm of \c *this using the Blue's algorithm.
219 * A Portable Fortran Program to Find the Euclidean Norm of a Vector,
220 * ACM TOMS, Vol 4, Issue 1, 1978.
221 *
222 * For architecture/scalar types without vectorization, this version
223 * is much faster than stableNorm(). Otherwise the stableNorm() is faster.
224 *
225 * \sa norm(), stableNorm(), hypotNorm()
226 */
227 template<typename Derived>
228 inline typename NumTraits<typename internal::traits<Derived>::Scalar>::Real
blueNorm()229 MatrixBase<Derived>::blueNorm() const
230 {
231 return internal::blueNorm_impl(*this);
232 }
233
234 /** \returns the \em l2 norm of \c *this avoiding undeflow and overflow.
235 * This version use a concatenation of hypot() calls, and it is very slow.
236 *
237 * \sa norm(), stableNorm()
238 */
239 template<typename Derived>
240 inline typename NumTraits<typename internal::traits<Derived>::Scalar>::Real
hypotNorm()241 MatrixBase<Derived>::hypotNorm() const
242 {
243 if(size()==1)
244 return numext::abs(coeff(0,0));
245 else
246 return this->cwiseAbs().redux(internal::scalar_hypot_op<RealScalar>());
247 }
248
249 } // end namespace Eigen
250
251 #endif // EIGEN_STABLENORM_H
252