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
13namespace Eigen {
14
15namespace internal {
16
17template<typename ExpressionType, typename Scalar>
18inline 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
53template<typename Derived>
54inline typename NumTraits<typename traits<Derived>::Scalar>::Real
55blueNorm_impl(const EigenBase<Derived>& _vec)
56{
57 typedef typename Derived::RealScalar RealScalar;
58 using std::pow;
59 using std::sqrt;
60 using std::abs;
61 const Derived& vec(_vec.derived());
62 static bool initialized = false;
63 static RealScalar b1, b2, s1m, s2m, rbig, relerr;
64 if(!initialized)
65 {
66 int ibeta, it, iemin, iemax, iexp;
67 RealScalar eps;
68 // This program calculates the machine-dependent constants
69 // bl, b2, slm, s2m, relerr overfl
70 // from the "basic" machine-dependent numbers
71 // nbig, ibeta, it, iemin, iemax, rbig.
72 // The following define the basic machine-dependent constants.
73 // For portability, the PORT subprograms "ilmaeh" and "rlmach"
74 // are used. For any specific computer, each of the assignment
75 // statements can be replaced
76 ibeta = std::numeric_limits<RealScalar>::radix; // base for floating-point numbers
77 it = std::numeric_limits<RealScalar>::digits; // number of base-beta digits in mantissa
78 iemin = std::numeric_limits<RealScalar>::min_exponent; // minimum exponent
79 iemax = std::numeric_limits<RealScalar>::max_exponent; // maximum exponent
80 rbig = (std::numeric_limits<RealScalar>::max)(); // largest floating-point number
81
82 iexp = -((1-iemin)/2);
83 b1 = RealScalar(pow(RealScalar(ibeta),RealScalar(iexp))); // lower boundary of midrange
84 iexp = (iemax + 1 - it)/2;
85 b2 = RealScalar(pow(RealScalar(ibeta),RealScalar(iexp))); // upper boundary of midrange
86
87 iexp = (2-iemin)/2;
88 s1m = RealScalar(pow(RealScalar(ibeta),RealScalar(iexp))); // scaling factor for lower range
89 iexp = - ((iemax+it)/2);
90 s2m = RealScalar(pow(RealScalar(ibeta),RealScalar(iexp))); // scaling factor for upper range
91
92 eps = RealScalar(pow(double(ibeta), 1-it));
93 relerr = sqrt(eps); // tolerance for neglecting asml
94 initialized = true;
95 }
96 Index n = vec.size();
97 RealScalar ab2 = b2 / RealScalar(n);
98 RealScalar asml = RealScalar(0);
99 RealScalar amed = RealScalar(0);
100 RealScalar abig = RealScalar(0);
101 for(typename Derived::InnerIterator it(vec, 0); it; ++it)
102 {
103 RealScalar ax = abs(it.value());
104 if(ax > ab2) abig += numext::abs2(ax*s2m);
105 else if(ax < b1) asml += numext::abs2(ax*s1m);
106 else amed += numext::abs2(ax);
107 }
108 if(amed!=amed)
109 return amed; // we got a NaN
110 if(abig > RealScalar(0))
111 {
112 abig = sqrt(abig);
113 if(abig > rbig) // overflow, or *this contains INF values
114 return abig; // return INF
115 if(amed > RealScalar(0))
116 {
117 abig = abig/s2m;
118 amed = sqrt(amed);
119 }
120 else
121 return abig/s2m;
122 }
123 else if(asml > RealScalar(0))
124 {
125 if (amed > RealScalar(0))
126 {
127 abig = sqrt(amed);
128 amed = sqrt(asml) / s1m;
129 }
130 else
131 return sqrt(asml)/s1m;
132 }
133 else
134 return sqrt(amed);
135 asml = numext::mini(abig, amed);
136 abig = numext::maxi(abig, amed);
137 if(asml <= abig*relerr)
138 return abig;
139 else
140 return abig * sqrt(RealScalar(1) + numext::abs2(asml/abig));
141}
142
143} // end namespace internal
144
145/** \returns the \em l2 norm of \c *this avoiding underflow and overflow.
146 * This version use a blockwise two passes algorithm:
147 * 1 - find the absolute largest coefficient \c s
148 * 2 - compute \f$ s \Vert \frac{*this}{s} \Vert \f$ in a standard way
149 *
150 * For architecture/scalar types supporting vectorization, this version
151 * is faster than blueNorm(). Otherwise the blueNorm() is much faster.
152 *
153 * \sa norm(), blueNorm(), hypotNorm()
154 */
155template<typename Derived>
156inline typename NumTraits<typename internal::traits<Derived>::Scalar>::Real
157MatrixBase<Derived>::stableNorm() const
158{
159 using std::sqrt;
160 using std::abs;
161 const Index blockSize = 4096;
162 RealScalar scale(0);
163 RealScalar invScale(1);
164 RealScalar ssq(0); // sum of square
165
166 typedef typename internal::nested_eval<Derived,2>::type DerivedCopy;
167 typedef typename internal::remove_all<DerivedCopy>::type DerivedCopyClean;
168 const DerivedCopy copy(derived());
169
170 enum {
171 CanAlign = ( (int(DerivedCopyClean::Flags)&DirectAccessBit)
172 || (int(internal::evaluator<DerivedCopyClean>::Alignment)>0) // FIXME Alignment)>0 might not be enough
173 ) && (blockSize*sizeof(Scalar)*2<EIGEN_STACK_ALLOCATION_LIMIT)
174 && (EIGEN_MAX_STATIC_ALIGN_BYTES>0) // if we cannot allocate on the stack, then let's not bother about this optimization
175 };
176 typedef typename internal::conditional<CanAlign, Ref<const Matrix<Scalar,Dynamic,1,0,blockSize,1>, internal::evaluator<DerivedCopyClean>::Alignment>,
177 typename DerivedCopyClean::ConstSegmentReturnType>::type SegmentWrapper;
178 Index n = size();
179
180 if(n==1)
181 return abs(this->coeff(0));
182
183 Index bi = internal::first_default_aligned(copy);
184 if (bi>0)
185 internal::stable_norm_kernel(copy.head(bi), ssq, scale, invScale);
186 for (; bi<n; bi+=blockSize)
187 internal::stable_norm_kernel(SegmentWrapper(copy.segment(bi,numext::mini(blockSize, n - bi))), ssq, scale, invScale);
188 return scale * sqrt(ssq);
189}
190
191/** \returns the \em l2 norm of \c *this using the Blue's algorithm.
192 * A Portable Fortran Program to Find the Euclidean Norm of a Vector,
193 * ACM TOMS, Vol 4, Issue 1, 1978.
194 *
195 * For architecture/scalar types without vectorization, this version
196 * is much faster than stableNorm(). Otherwise the stableNorm() is faster.
197 *
198 * \sa norm(), stableNorm(), hypotNorm()
199 */
200template<typename Derived>
201inline typename NumTraits<typename internal::traits<Derived>::Scalar>::Real
202MatrixBase<Derived>::blueNorm() const
203{
204 return internal::blueNorm_impl(*this);
205}
206
207/** \returns the \em l2 norm of \c *this avoiding undeflow and overflow.
208 * This version use a concatenation of hypot() calls, and it is very slow.
209 *
210 * \sa norm(), stableNorm()
211 */
212template<typename Derived>
213inline typename NumTraits<typename internal::traits<Derived>::Scalar>::Real
214MatrixBase<Derived>::hypotNorm() const
215{
216 return this->cwiseAbs().redux(internal::scalar_hypot_op<RealScalar>());
217}
218
219} // end namespace Eigen
220
221#endif // EIGEN_STABLENORM_H
222