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 using std::max;
21 Scalar maxCoeff = bl.cwiseAbs().maxCoeff();
22
23 if (maxCoeff>scale)
24 {
25 ssq = ssq * numext::abs2(scale/maxCoeff);
26 Scalar tmp = Scalar(1)/maxCoeff;
27 if(tmp > NumTraits<Scalar>::highest())
28 {
29 invScale = NumTraits<Scalar>::highest();
30 scale = Scalar(1)/invScale;
31 }
32 else
33 {
34 scale = maxCoeff;
35 invScale = tmp;
36 }
37 }
38
39 // TODO if the maxCoeff is much much smaller than the current scale,
40 // then we can neglect this sub vector
41 if(scale>Scalar(0)) // if scale==0, then bl is 0
42 ssq += (bl*invScale).squaredNorm();
43 }
44
45 template<typename Derived>
46 inline typename NumTraits<typename traits<Derived>::Scalar>::Real
blueNorm_impl(const EigenBase<Derived> & _vec)47 blueNorm_impl(const EigenBase<Derived>& _vec)
48 {
49 typedef typename Derived::RealScalar RealScalar;
50 typedef typename Derived::Index Index;
51 using std::pow;
52 using std::min;
53 using std::max;
54 using std::sqrt;
55 using std::abs;
56 const Derived& vec(_vec.derived());
57 static bool initialized = false;
58 static RealScalar b1, b2, s1m, s2m, overfl, rbig, relerr;
59 if(!initialized)
60 {
61 int ibeta, it, iemin, iemax, iexp;
62 RealScalar eps;
63 // This program calculates the machine-dependent constants
64 // bl, b2, slm, s2m, relerr overfl
65 // from the "basic" machine-dependent numbers
66 // nbig, ibeta, it, iemin, iemax, rbig.
67 // The following define the basic machine-dependent constants.
68 // For portability, the PORT subprograms "ilmaeh" and "rlmach"
69 // are used. For any specific computer, each of the assignment
70 // statements can be replaced
71 ibeta = std::numeric_limits<RealScalar>::radix; // base for floating-point numbers
72 it = std::numeric_limits<RealScalar>::digits; // number of base-beta digits in mantissa
73 iemin = std::numeric_limits<RealScalar>::min_exponent; // minimum exponent
74 iemax = std::numeric_limits<RealScalar>::max_exponent; // maximum exponent
75 rbig = (std::numeric_limits<RealScalar>::max)(); // largest floating-point number
76
77 iexp = -((1-iemin)/2);
78 b1 = RealScalar(pow(RealScalar(ibeta),RealScalar(iexp))); // lower boundary of midrange
79 iexp = (iemax + 1 - it)/2;
80 b2 = RealScalar(pow(RealScalar(ibeta),RealScalar(iexp))); // upper boundary of midrange
81
82 iexp = (2-iemin)/2;
83 s1m = RealScalar(pow(RealScalar(ibeta),RealScalar(iexp))); // scaling factor for lower range
84 iexp = - ((iemax+it)/2);
85 s2m = RealScalar(pow(RealScalar(ibeta),RealScalar(iexp))); // scaling factor for upper range
86
87 overfl = rbig*s2m; // overflow boundary for abig
88 eps = RealScalar(pow(double(ibeta), 1-it));
89 relerr = sqrt(eps); // tolerance for neglecting asml
90 initialized = true;
91 }
92 Index n = vec.size();
93 RealScalar ab2 = b2 / RealScalar(n);
94 RealScalar asml = RealScalar(0);
95 RealScalar amed = RealScalar(0);
96 RealScalar abig = RealScalar(0);
97 for(typename Derived::InnerIterator it(vec, 0); it; ++it)
98 {
99 RealScalar ax = abs(it.value());
100 if(ax > ab2) abig += numext::abs2(ax*s2m);
101 else if(ax < b1) asml += numext::abs2(ax*s1m);
102 else amed += numext::abs2(ax);
103 }
104 if(abig > RealScalar(0))
105 {
106 abig = sqrt(abig);
107 if(abig > overfl)
108 {
109 return rbig;
110 }
111 if(amed > RealScalar(0))
112 {
113 abig = abig/s2m;
114 amed = sqrt(amed);
115 }
116 else
117 return abig/s2m;
118 }
119 else if(asml > RealScalar(0))
120 {
121 if (amed > RealScalar(0))
122 {
123 abig = sqrt(amed);
124 amed = sqrt(asml) / s1m;
125 }
126 else
127 return sqrt(asml)/s1m;
128 }
129 else
130 return sqrt(amed);
131 asml = (min)(abig, amed);
132 abig = (max)(abig, amed);
133 if(asml <= abig*relerr)
134 return abig;
135 else
136 return abig * sqrt(RealScalar(1) + numext::abs2(asml/abig));
137 }
138
139 } // end namespace internal
140
141 /** \returns the \em l2 norm of \c *this avoiding underflow and overflow.
142 * This version use a blockwise two passes algorithm:
143 * 1 - find the absolute largest coefficient \c s
144 * 2 - compute \f$ s \Vert \frac{*this}{s} \Vert \f$ in a standard way
145 *
146 * For architecture/scalar types supporting vectorization, this version
147 * is faster than blueNorm(). Otherwise the blueNorm() is much faster.
148 *
149 * \sa norm(), blueNorm(), hypotNorm()
150 */
151 template<typename Derived>
152 inline typename NumTraits<typename internal::traits<Derived>::Scalar>::Real
stableNorm()153 MatrixBase<Derived>::stableNorm() const
154 {
155 using std::min;
156 using std::sqrt;
157 const Index blockSize = 4096;
158 RealScalar scale(0);
159 RealScalar invScale(1);
160 RealScalar ssq(0); // sum of square
161 enum {
162 Alignment = (int(Flags)&DirectAccessBit) || (int(Flags)&AlignedBit) ? 1 : 0
163 };
164 Index n = size();
165 Index bi = internal::first_aligned(derived());
166 if (bi>0)
167 internal::stable_norm_kernel(this->head(bi), ssq, scale, invScale);
168 for (; bi<n; bi+=blockSize)
169 internal::stable_norm_kernel(this->segment(bi,(min)(blockSize, n - bi)).template forceAlignedAccessIf<Alignment>(), ssq, scale, invScale);
170 return scale * sqrt(ssq);
171 }
172
173 /** \returns the \em l2 norm of \c *this using the Blue's algorithm.
174 * A Portable Fortran Program to Find the Euclidean Norm of a Vector,
175 * ACM TOMS, Vol 4, Issue 1, 1978.
176 *
177 * For architecture/scalar types without vectorization, this version
178 * is much faster than stableNorm(). Otherwise the stableNorm() is faster.
179 *
180 * \sa norm(), stableNorm(), hypotNorm()
181 */
182 template<typename Derived>
183 inline typename NumTraits<typename internal::traits<Derived>::Scalar>::Real
blueNorm()184 MatrixBase<Derived>::blueNorm() const
185 {
186 return internal::blueNorm_impl(*this);
187 }
188
189 /** \returns the \em l2 norm of \c *this avoiding undeflow and overflow.
190 * This version use a concatenation of hypot() calls, and it is very slow.
191 *
192 * \sa norm(), stableNorm()
193 */
194 template<typename Derived>
195 inline typename NumTraits<typename internal::traits<Derived>::Scalar>::Real
hypotNorm()196 MatrixBase<Derived>::hypotNorm() const
197 {
198 return this->cwiseAbs().redux(internal::scalar_hypot_op<RealScalar>());
199 }
200
201 } // end namespace Eigen
202
203 #endif // EIGEN_STABLENORM_H
204