1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
|
// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2006-2008 Benoit Jacob <jacob.benoit.1@gmail.com>
//
// 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_MATHFUNCTIONS_H
#define EIGEN_MATHFUNCTIONS_H
template<typename T> inline typename NumTraits<T>::Real precision();
template<typename T> inline typename NumTraits<T>::Real machine_epsilon();
template<typename T> inline T ei_random(T a, T b);
template<typename T> inline T ei_random();
template<typename T> inline T ei_random_amplitude()
{
if(NumTraits<T>::HasFloatingPoint) return static_cast<T>(1);
else return static_cast<T>(10);
}
template<typename T> inline typename NumTraits<T>::Real ei_hypot(T x, T y)
{
typedef typename NumTraits<T>::Real RealScalar;
RealScalar _x = ei_abs(x);
RealScalar _y = ei_abs(y);
T p = std::max(_x, _y);
T q = std::min(_x, _y);
T qp = q/p;
return p * ei_sqrt(T(1) + qp*qp);
}
/**************
*** int ***
**************/
template<> inline int precision<int>() { return 0; }
template<> inline int machine_epsilon<int>() { return 0; }
inline int ei_real(int x) { return x; }
inline int& ei_real_ref(int& x) { return x; }
inline int ei_imag(int) { return 0; }
inline int ei_conj(int x) { return x; }
inline int ei_abs(int x) { return abs(x); }
inline int ei_abs2(int x) { return x*x; }
inline int ei_sqrt(int) { ei_assert(false); return 0; }
inline int ei_exp(int) { ei_assert(false); return 0; }
inline int ei_log(int) { ei_assert(false); return 0; }
inline int ei_sin(int) { ei_assert(false); return 0; }
inline int ei_cos(int) { ei_assert(false); return 0; }
inline int ei_pow(int x, int y)
{
int res = 1;
if(y < 0) return 0;
if(y & 1) res *= x;
y >>= 1;
while(y)
{
x *= x;
if(y&1) res *= x;
y >>= 1;
}
return res;
}
template<> inline int ei_random(int a, int b)
{
// We can't just do rand()%n as only the high-order bits are really random
return a + static_cast<int>((b-a+1) * (rand() / (RAND_MAX + 1.0)));
}
template<> inline int ei_random()
{
return ei_random<int>(-ei_random_amplitude<int>(), ei_random_amplitude<int>());
}
inline bool ei_isMuchSmallerThan(int a, int, int = precision<int>())
{
return a == 0;
}
inline bool ei_isApprox(int a, int b, int = precision<int>())
{
return a == b;
}
inline bool ei_isApproxOrLessThan(int a, int b, int = precision<int>())
{
return a <= b;
}
/**************
*** float ***
**************/
template<> inline float precision<float>() { return 1e-5f; }
template<> inline float machine_epsilon<float>() { return 1.192e-07f; }
inline float ei_real(float x) { return x; }
inline float& ei_real_ref(float& x) { return x; }
inline float ei_imag(float) { return 0.f; }
inline float ei_conj(float x) { return x; }
inline float ei_abs(float x) { return std::abs(x); }
inline float ei_abs2(float x) { return x*x; }
inline float ei_sqrt(float x) { return std::sqrt(x); }
inline float ei_exp(float x) { return std::exp(x); }
inline float ei_log(float x) { return std::log(x); }
inline float ei_sin(float x) { return std::sin(x); }
inline float ei_cos(float x) { return std::cos(x); }
inline float ei_pow(float x, float y) { return std::pow(x, y); }
template<> inline float ei_random(float a, float b)
{
#ifdef EIGEN_NICE_RANDOM
int i;
do { i = ei_random<int>(256*int(a),256*int(b));
} while(i==0);
return float(i)/256.f;
#else
return a + (b-a) * float(std::rand()) / float(RAND_MAX);
#endif
}
template<> inline float ei_random()
{
return ei_random<float>(-ei_random_amplitude<float>(), ei_random_amplitude<float>());
}
inline bool ei_isMuchSmallerThan(float a, float b, float prec = precision<float>())
{
return ei_abs(a) <= ei_abs(b) * prec;
}
inline bool ei_isApprox(float a, float b, float prec = precision<float>())
{
return ei_abs(a - b) <= std::min(ei_abs(a), ei_abs(b)) * prec;
}
inline bool ei_isApproxOrLessThan(float a, float b, float prec = precision<float>())
{
return a <= b || ei_isApprox(a, b, prec);
}
/**************
*** double ***
**************/
template<> inline double precision<double>() { return 1e-12; }
template<> inline double machine_epsilon<double>() { return 2.220e-16; }
inline double ei_real(double x) { return x; }
inline double& ei_real_ref(double& x) { return x; }
inline double ei_imag(double) { return 0.; }
inline double ei_conj(double x) { return x; }
inline double ei_abs(double x) { return std::abs(x); }
inline double ei_abs2(double x) { return x*x; }
inline double ei_sqrt(double x) { return std::sqrt(x); }
inline double ei_exp(double x) { return std::exp(x); }
inline double ei_log(double x) { return std::log(x); }
inline double ei_sin(double x) { return std::sin(x); }
inline double ei_cos(double x) { return std::cos(x); }
inline double ei_pow(double x, double y) { return std::pow(x, y); }
template<> inline double ei_random(double a, double b)
{
#ifdef EIGEN_NICE_RANDOM
int i;
do { i= ei_random<int>(256*int(a),256*int(b));
} while(i==0);
return i/256.;
#else
return a + (b-a) * std::rand() / RAND_MAX;
#endif
}
template<> inline double ei_random()
{
return ei_random<double>(-ei_random_amplitude<double>(), ei_random_amplitude<double>());
}
inline bool ei_isMuchSmallerThan(double a, double b, double prec = precision<double>())
{
return ei_abs(a) <= ei_abs(b) * prec;
}
inline bool ei_isApprox(double a, double b, double prec = precision<double>())
{
return ei_abs(a - b) <= std::min(ei_abs(a), ei_abs(b)) * prec;
}
inline bool ei_isApproxOrLessThan(double a, double b, double prec = precision<double>())
{
return a <= b || ei_isApprox(a, b, prec);
}
/*********************
*** complex<float> ***
*********************/
template<> inline float precision<std::complex<float> >() { return precision<float>(); }
template<> inline float machine_epsilon<std::complex<float> >() { return machine_epsilon<float>(); }
inline float ei_real(const std::complex<float>& x) { return std::real(x); }
inline float ei_imag(const std::complex<float>& x) { return std::imag(x); }
inline float& ei_real_ref(std::complex<float>& x) { return reinterpret_cast<float*>(&x)[0]; }
inline float& ei_imag_ref(std::complex<float>& x) { return reinterpret_cast<float*>(&x)[1]; }
inline std::complex<float> ei_conj(const std::complex<float>& x) { return std::conj(x); }
inline float ei_abs(const std::complex<float>& x) { return std::abs(x); }
inline float ei_abs2(const std::complex<float>& x) { return std::norm(x); }
inline std::complex<float> ei_exp(std::complex<float> x) { return std::exp(x); }
inline std::complex<float> ei_sin(std::complex<float> x) { return std::sin(x); }
inline std::complex<float> ei_cos(std::complex<float> x) { return std::cos(x); }
template<> inline std::complex<float> ei_random()
{
return std::complex<float>(ei_random<float>(), ei_random<float>());
}
inline bool ei_isMuchSmallerThan(const std::complex<float>& a, const std::complex<float>& b, float prec = precision<float>())
{
return ei_abs2(a) <= ei_abs2(b) * prec * prec;
}
inline bool ei_isMuchSmallerThan(const std::complex<float>& a, float b, float prec = precision<float>())
{
return ei_abs2(a) <= ei_abs2(b) * prec * prec;
}
inline bool ei_isApprox(const std::complex<float>& a, const std::complex<float>& b, float prec = precision<float>())
{
return ei_isApprox(ei_real(a), ei_real(b), prec)
&& ei_isApprox(ei_imag(a), ei_imag(b), prec);
}
// ei_isApproxOrLessThan wouldn't make sense for complex numbers
/**********************
*** complex<double> ***
**********************/
template<> inline double precision<std::complex<double> >() { return precision<double>(); }
template<> inline double machine_epsilon<std::complex<double> >() { return machine_epsilon<double>(); }
inline double ei_real(const std::complex<double>& x) { return std::real(x); }
inline double ei_imag(const std::complex<double>& x) { return std::imag(x); }
inline double& ei_real_ref(std::complex<double>& x) { return reinterpret_cast<double*>(&x)[0]; }
inline double& ei_imag_ref(std::complex<double>& x) { return reinterpret_cast<double*>(&x)[1]; }
inline std::complex<double> ei_conj(const std::complex<double>& x) { return std::conj(x); }
inline double ei_abs(const std::complex<double>& x) { return std::abs(x); }
inline double ei_abs2(const std::complex<double>& x) { return std::norm(x); }
inline std::complex<double> ei_exp(std::complex<double> x) { return std::exp(x); }
inline std::complex<double> ei_sin(std::complex<double> x) { return std::sin(x); }
inline std::complex<double> ei_cos(std::complex<double> x) { return std::cos(x); }
template<> inline std::complex<double> ei_random()
{
return std::complex<double>(ei_random<double>(), ei_random<double>());
}
inline bool ei_isMuchSmallerThan(const std::complex<double>& a, const std::complex<double>& b, double prec = precision<double>())
{
return ei_abs2(a) <= ei_abs2(b) * prec * prec;
}
inline bool ei_isMuchSmallerThan(const std::complex<double>& a, double b, double prec = precision<double>())
{
return ei_abs2(a) <= ei_abs2(b) * prec * prec;
}
inline bool ei_isApprox(const std::complex<double>& a, const std::complex<double>& b, double prec = precision<double>())
{
return ei_isApprox(ei_real(a), ei_real(b), prec)
&& ei_isApprox(ei_imag(a), ei_imag(b), prec);
}
// ei_isApproxOrLessThan wouldn't make sense for complex numbers
/******************
*** long double ***
******************/
template<> inline long double precision<long double>() { return precision<double>(); }
template<> inline long double machine_epsilon<long double>() { return 1.084e-19l; }
inline long double ei_real(long double x) { return x; }
inline long double& ei_real_ref(long double& x) { return x; }
inline long double ei_imag(long double) { return 0.; }
inline long double ei_conj(long double x) { return x; }
inline long double ei_abs(long double x) { return std::abs(x); }
inline long double ei_abs2(long double x) { return x*x; }
inline long double ei_sqrt(long double x) { return std::sqrt(x); }
inline long double ei_exp(long double x) { return std::exp(x); }
inline long double ei_log(long double x) { return std::log(x); }
inline long double ei_sin(long double x) { return std::sin(x); }
inline long double ei_cos(long double x) { return std::cos(x); }
inline long double ei_pow(long double x, long double y) { return std::pow(x, y); }
template<> inline long double ei_random(long double a, long double b)
{
return ei_random<double>(static_cast<double>(a),static_cast<double>(b));
}
template<> inline long double ei_random()
{
return ei_random<double>(-ei_random_amplitude<double>(), ei_random_amplitude<double>());
}
inline bool ei_isMuchSmallerThan(long double a, long double b, long double prec = precision<long double>())
{
return ei_abs(a) <= ei_abs(b) * prec;
}
inline bool ei_isApprox(long double a, long double b, long double prec = precision<long double>())
{
return ei_abs(a - b) <= std::min(ei_abs(a), ei_abs(b)) * prec;
}
inline bool ei_isApproxOrLessThan(long double a, long double b, long double prec = precision<long double>())
{
return a <= b || ei_isApprox(a, b, prec);
}
#endif // EIGEN_MATHFUNCTIONS_H
|
