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3 /* Copyright 1998-2011 by Ullrich Koethe */
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35
36 #ifndef VIGRA_MATHUTIL_HXX
37 #define VIGRA_MATHUTIL_HXX
38
39 #ifdef _MSC_VER
40 # pragma warning (disable: 4996) // hypot/_hypot confusion
41 #endif
42
43 #include <cmath>
44 #include <cstdlib>
45 #include <complex>
46 #include "config.hxx"
47 #include "error.hxx"
48 #include "tuple.hxx"
49 #include "sized_int.hxx"
50 #include "numerictraits.hxx"
51 #include "algorithm.hxx"
52
53 /** \page MathConstants Mathematical Constants
54
55 <TT>M_PI, M_SQRT2 etc.</TT>
56
57 <b>\#include</b> <vigra/mathutil.hxx>
58
59 Since mathematical constants such as <TT>M_PI</TT> and <TT>M_SQRT2</TT>
60 are not officially standardized, we provide definitions here for those
61 compilers that don't support them.
62
63 \code
64 #ifndef M_PI
65 # define M_PI 3.14159265358979323846
66 #endif
67
68 #ifndef M_SQRT2
69 # define M_2_PI 0.63661977236758134308
70 #endif
71
72 #ifndef M_PI_2
73 # define M_PI_2 1.57079632679489661923
74 #endif
75
76 #ifndef M_PI_4
77 # define M_PI_4 0.78539816339744830962
78 #endif
79
80 #ifndef M_SQRT2
81 # define M_SQRT2 1.41421356237309504880
82 #endif
83
84 #ifndef M_EULER_GAMMA
85 # define M_EULER_GAMMA 0.5772156649015329
86 #endif
87 \endcode
88 */
89 #ifndef M_PI
90 # define M_PI 3.14159265358979323846
91 #endif
92
93 #ifndef M_2_PI
94 # define M_2_PI 0.63661977236758134308
95 #endif
96
97 #ifndef M_PI_2
98 # define M_PI_2 1.57079632679489661923
99 #endif
100
101 #ifndef M_PI_4
102 # define M_PI_4 0.78539816339744830962
103 #endif
104
105 #ifndef M_SQRT2
106 # define M_SQRT2 1.41421356237309504880
107 #endif
108
109 #ifndef M_E
110 # define M_E 2.71828182845904523536
111 #endif
112
113 #ifndef M_EULER_GAMMA
114 # define M_EULER_GAMMA 0.5772156649015329
115 #endif
116
117 namespace vigra {
118
119 /** \addtogroup MathFunctions Mathematical Functions
120
121 Useful mathematical functions and functors.
122 */
123 //@{
124 // import functions into namespace vigra which VIGRA is going to overload
125
126 using VIGRA_CSTD::pow;
127 using VIGRA_CSTD::floor;
128 using VIGRA_CSTD::ceil;
129 using VIGRA_CSTD::exp;
130
131 // import abs(float), abs(double), abs(long double) from <cmath>
132 // abs(int), abs(long), abs(long long) from <cstdlib>
133 // abs(std::complex<T>) from <complex>
134 using std::abs;
135
136 // define the missing variants of abs() to avoid 'ambiguous overload'
137 // errors in template functions
138 #define VIGRA_DEFINE_UNSIGNED_ABS(T) \
139 inline T abs(T t) { return t; }
140
141 VIGRA_DEFINE_UNSIGNED_ABS(bool)
142 VIGRA_DEFINE_UNSIGNED_ABS(unsigned char)
143 VIGRA_DEFINE_UNSIGNED_ABS(unsigned short)
144 VIGRA_DEFINE_UNSIGNED_ABS(unsigned int)
145 VIGRA_DEFINE_UNSIGNED_ABS(unsigned long)
146 VIGRA_DEFINE_UNSIGNED_ABS(unsigned long long)
147
148 #undef VIGRA_DEFINE_UNSIGNED_ABS
149
150 #define VIGRA_DEFINE_MISSING_ABS(T) \
151 inline T abs(T t) { return t < 0 ? static_cast<T>(-t) : t; }
152
153 VIGRA_DEFINE_MISSING_ABS(signed char)
154 VIGRA_DEFINE_MISSING_ABS(signed short)
155
156 #if defined(_MSC_VER) && _MSC_VER < 1600
157 VIGRA_DEFINE_MISSING_ABS(signed long long)
158 #endif
159
160 #undef VIGRA_DEFINE_MISSING_ABS
161
162 // scalar dot is needed for generic functions that should work with
163 // scalars and vectors alike
164
165 #define VIGRA_DEFINE_SCALAR_DOT(T) \
166 inline NumericTraits<T>::Promote dot(T l, T r) { return l*r; }
167
168 VIGRA_DEFINE_SCALAR_DOT(unsigned char)
169 VIGRA_DEFINE_SCALAR_DOT(unsigned short)
170 VIGRA_DEFINE_SCALAR_DOT(unsigned int)
171 VIGRA_DEFINE_SCALAR_DOT(unsigned long)
172 VIGRA_DEFINE_SCALAR_DOT(unsigned long long)
173 VIGRA_DEFINE_SCALAR_DOT(signed char)
174 VIGRA_DEFINE_SCALAR_DOT(signed short)
175 VIGRA_DEFINE_SCALAR_DOT(signed int)
176 VIGRA_DEFINE_SCALAR_DOT(signed long)
177 VIGRA_DEFINE_SCALAR_DOT(signed long long)
178 VIGRA_DEFINE_SCALAR_DOT(float)
179 VIGRA_DEFINE_SCALAR_DOT(double)
180 VIGRA_DEFINE_SCALAR_DOT(long double)
181
182 #undef VIGRA_DEFINE_SCALAR_DOT
183
184 using std::pow;
185
186 // support 'double' exponents for all floating point versions of pow()
187
188 inline float pow(float v, double e)
189 {
190 return std::pow(v, (float)e);
191 }
192
193 inline long double pow(long double v, double e)
194 {
195 return std::pow(v, (long double)e);
196 }
197
198 /** \brief The rounding function.
199
200 Defined for all floating point types. Rounds towards the nearest integer
201 such that <tt>abs(round(t)) == round(abs(t))</tt> for all <tt>t</tt>.
202
203 <b>\#include</b> <vigra/mathutil.hxx><br>
204 Namespace: vigra
205 */
206 #ifdef DOXYGEN // only for documentation
207 REAL round(REAL v);
208 #endif
209
210 inline float round(float t)
211 {
212 return t >= 0.0
213 ? floor(t + 0.5f)
214 : ceil(t - 0.5f);
215 }
216
217 inline double round(double t)
218 {
219 return t >= 0.0
220 ? floor(t + 0.5)
221 : ceil(t - 0.5);
222 }
223
224 inline long double round(long double t)
225 {
226 return t >= 0.0
227 ? floor(t + 0.5)
228 : ceil(t - 0.5);
229 }
230
231
232 /** \brief Round and cast to integer.
233
234 Rounds to the nearest integer like round(), but casts the result to
235 <tt>int</tt> (this will be faster and is usually needed anyway).
236
237 <b>\#include</b> <vigra/mathutil.hxx><br>
238 Namespace: vigra
239 */
240 inline int roundi(double t)
241 {
242 return t >= 0.0
243 ? int(t + 0.5)
244 : int(t - 0.5);
245 }
246
247 /** \brief Round up to the nearest power of 2.
248
249 Efficient algorithm for finding the smallest power of 2 which is not smaller than \a x
250 (function clp2() from Henry Warren: "Hacker's Delight", Addison-Wesley, 2003,
251 see http://www.hackersdelight.org/).
252 If \a x > 2^31, the function will return 0 because integer arithmetic is defined modulo 2^32.
253
254 <b>\#include</b> <vigra/mathutil.hxx><br>
255 Namespace: vigra
256 */
257 inline UInt32 ceilPower2(UInt32 x)
258 {
259 if(x == 0) return 0;
260
261 x = x - 1;
262 x = x | (x >> 1);
263 x = x | (x >> 2);
264 x = x | (x >> 4);
265 x = x | (x >> 8);
266 x = x | (x >>16);
267 return x + 1;
268 }
269
270 /** \brief Round down to the nearest power of 2.
271
272 Efficient algorithm for finding the largest power of 2 which is not greater than \a x
273 (function flp2() from Henry Warren: "Hacker's Delight", Addison-Wesley, 2003,
274 see http://www.hackersdelight.org/).
275
276 <b>\#include</b> <vigra/mathutil.hxx><br>
277 Namespace: vigra
278 */
279 inline UInt32 floorPower2(UInt32 x)
280 {
281 x = x | (x >> 1);
282 x = x | (x >> 2);
283 x = x | (x >> 4);
284 x = x | (x >> 8);
285 x = x | (x >>16);
286 return x - (x >> 1);
287 }
288
289 namespace detail {
290
291 template <class T>
292 struct IntLog2
293 {
294 static Int32 table[64];
295 };
296
297 template <class T>
298 Int32 IntLog2<T>::table[64] = {
299 -1, 0, -1, 15, -1, 1, 28, -1, 16, -1, -1, -1, 2, 21,
300 29, -1, -1, -1, 19, 17, 10, -1, 12, -1, -1, 3, -1, 6,
301 -1, 22, 30, -1, 14, -1, 27, -1, -1, -1, 20, -1, 18, 9,
302 11, -1, 5, -1, -1, 13, 26, -1, -1, 8, -1, 4, -1, 25,
303 -1, 7, 24, -1, 23, -1, 31, -1};
304
305 } // namespace detail
306
307 /** \brief Compute the base-2 logarithm of an integer.
308
309 Returns the position of the left-most 1-bit in the given number \a x, or
310 -1 if \a x == 0. That is,
311
312 \code
313 assert(k >= 0 && k < 32 && log2i(1 << k) == k);
314 \endcode
315
316 The function uses Robert Harley's algorithm to determine the number of leading zeros
317 in \a x (algorithm nlz10() at http://www.hackersdelight.org/). But note that the functions
318 \ref floorPower2() or \ref ceilPower2() are more efficient and should be preferred when possible.
319
320 <b>\#include</b> <vigra/mathutil.hxx><br>
321 Namespace: vigra
322 */
323 inline Int32 log2i(UInt32 x)
324 {
325 // Propagate leftmost 1-bit to the right.
326 x = x | (x >> 1);
327 x = x | (x >> 2);
328 x = x | (x >> 4);
329 x = x | (x >> 8);
330 x = x | (x >>16);
331 x = x*0x06EB14F9; // Multiplier is 7*255**3.
332 return detail::IntLog2<Int32>::table[x >> 26];
333 }
334
335 /** \brief The square function.
336
337 <tt>sq(x) = x*x</tt> is needed so often that it makes sense to define it as a function.
338
339 <b>\#include</b> <vigra/mathutil.hxx><br>
340 Namespace: vigra
341 */
342 template <class T>
343 inline
344 typename NumericTraits<T>::Promote sq(T t)
345 {
346 return t*t;
347 }
348
349 namespace detail {
350
351 template <class V, unsigned>
352 struct cond_mult
353 {
354 static V call(const V & x, const V & y) { return x * y; }
355 };
356 template <class V>
357 struct cond_mult<V, 0>
358 {
359 static V call(const V &, const V & y) { return y; }
360 };
361
362 template <class V, unsigned n>
363 struct power_static
364 {
365 static V call(const V & x)
366 {
367 return n / 2
368 ? cond_mult<V, n & 1>::call(x, power_static<V, n / 2>::call(x * x))
369 : n & 1 ? x : V();
370 }
371 };
372 template <class V>
373 struct power_static<V, 0>
374 {
375 static V call(const V & x)
376 {
377 return V(1);
378 }
379 };
380
381 } // namespace detail
382
383 /** \brief Exponentiation to a positive integer power by squaring.
384
385 <b>\#include</b> <vigra/mathutil.hxx><br>
386 Namespace: vigra
387 */
388 template <unsigned n, class V>
389 inline V power(const V & x)
390 {
391 return detail::power_static<V, n>::call(x);
392 }
393 //doxygen_overloaded_function(template <unsigned n, class V> power(const V & x))
394
395 namespace detail {
396
397 template <class T>
398 struct IntSquareRoot
399 {
400 static UInt32 sqq_table[];
401 static UInt32 exec(UInt32 v);
402 };
403
404 template <class T>
405 UInt32 IntSquareRoot<T>::sqq_table[] = {
406 0, 16, 22, 27, 32, 35, 39, 42, 45, 48, 50, 53, 55, 57,
407 59, 61, 64, 65, 67, 69, 71, 73, 75, 76, 78, 80, 81, 83,
408 84, 86, 87, 89, 90, 91, 93, 94, 96, 97, 98, 99, 101, 102,
409 103, 104, 106, 107, 108, 109, 110, 112, 113, 114, 115, 116, 117, 118,
410 119, 120, 121, 122, 123, 124, 125, 126, 128, 128, 129, 130, 131, 132,
411 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 144, 145,
412 146, 147, 148, 149, 150, 150, 151, 152, 153, 154, 155, 155, 156, 157,
413 158, 159, 160, 160, 161, 162, 163, 163, 164, 165, 166, 167, 167, 168,
414 169, 170, 170, 171, 172, 173, 173, 174, 175, 176, 176, 177, 178, 178,
415 179, 180, 181, 181, 182, 183, 183, 184, 185, 185, 186, 187, 187, 188,
416 189, 189, 190, 191, 192, 192, 193, 193, 194, 195, 195, 196, 197, 197,
417 198, 199, 199, 200, 201, 201, 202, 203, 203, 204, 204, 205, 206, 206,
418 207, 208, 208, 209, 209, 210, 211, 211, 212, 212, 213, 214, 214, 215,
419 215, 216, 217, 217, 218, 218, 219, 219, 220, 221, 221, 222, 222, 223,
420 224, 224, 225, 225, 226, 226, 227, 227, 228, 229, 229, 230, 230, 231,
421 231, 232, 232, 233, 234, 234, 235, 235, 236, 236, 237, 237, 238, 238,
422 239, 240, 240, 241, 241, 242, 242, 243, 243, 244, 244, 245, 245, 246,
423 246, 247, 247, 248, 248, 249, 249, 250, 250, 251, 251, 252, 252, 253,
424 253, 254, 254, 255
425 };
426
427 template <class T>
428 UInt32 IntSquareRoot<T>::exec(UInt32 x)
429 {
430 UInt32 xn;
431 if (x >= 0x10000)
432 if (x >= 0x1000000)
433 if (x >= 0x10000000)
434 if (x >= 0x40000000) {
435 if (x >= (UInt32)65535*(UInt32)65535)
436 return 65535;
437 xn = sqq_table[x>>24] << 8;
438 } else
439 xn = sqq_table[x>>22] << 7;
440 else
441 if (x >= 0x4000000)
442 xn = sqq_table[x>>20] << 6;
443 else
444 xn = sqq_table[x>>18] << 5;
445 else {
446 if (x >= 0x100000)
447 if (x >= 0x400000)
448 xn = sqq_table[x>>16] << 4;
449 else
450 xn = sqq_table[x>>14] << 3;
451 else
452 if (x >= 0x40000)
453 xn = sqq_table[x>>12] << 2;
454 else
455 xn = sqq_table[x>>10] << 1;
456
457 goto nr1;
458 }
459 else
460 if (x >= 0x100) {
461 if (x >= 0x1000)
462 if (x >= 0x4000)
463 xn = (sqq_table[x>>8] >> 0) + 1;
464 else
465 xn = (sqq_table[x>>6] >> 1) + 1;
466 else
467 if (x >= 0x400)
468 xn = (sqq_table[x>>4] >> 2) + 1;
469 else
470 xn = (sqq_table[x>>2] >> 3) + 1;
471
472 goto adj;
473 } else
474 return sqq_table[x] >> 4;
475
476 /* Run two iterations of the standard convergence formula */
477
478 xn = (xn + 1 + x / xn) / 2;
479 nr1:
480 xn = (xn + 1 + x / xn) / 2;
481 adj:
482
483 if (xn * xn > x) /* Correct rounding if necessary */
484 xn--;
485
486 return xn;
487 }
488
489 } // namespace detail
490
491 using VIGRA_CSTD::sqrt;
492
493 /** \brief Signed integer square root.
494
495 Useful for fast fixed-point computations.
496
497 <b>\#include</b> <vigra/mathutil.hxx><br>
498 Namespace: vigra
499 */
500 inline Int32 sqrti(Int32 v)
501 {
502 if(v < 0)
503 throw std::domain_error("sqrti(Int32): negative argument.");
504 return (Int32)detail::IntSquareRoot<UInt32>::exec((UInt32)v);
505 }
506
507 /** \brief Unsigned integer square root.
508
509 Useful for fast fixed-point computations.
510
511 <b>\#include</b> <vigra/mathutil.hxx><br>
512 Namespace: vigra
513 */
514 inline UInt32 sqrti(UInt32 v)
515 {
516 return detail::IntSquareRoot<UInt32>::exec(v);
517 }
518
519 #ifdef VIGRA_NO_HYPOT
520 /** \brief Compute the Euclidean distance (length of the hypotenuse of a right-angled triangle).
521
522 The hypot() function returns the sqrt(a*a + b*b).
523 It is implemented in a way that minimizes round-off error.
524
525 <b>\#include</b> <vigra/mathutil.hxx><br>
526 Namespace: vigra
527 */
528 inline double hypot(double a, double b)
529 {
530 double absa = VIGRA_CSTD::fabs(a), absb = VIGRA_CSTD::fabs(b);
531 if (absa > absb)
532 return absa * VIGRA_CSTD::sqrt(1.0 + sq(absb/absa));
533 else
534 return absb == 0.0
535 ? 0.0
536 : absb * VIGRA_CSTD::sqrt(1.0 + sq(absa/absb));
537 }
538
539 #else
540
541 using ::hypot;
542
543 #endif
544
545 /** \brief The sign function.
546
547 Returns 1, 0, or -1 depending on the sign of \a t, but with the same type as \a t.
548
549 <b>\#include</b> <vigra/mathutil.hxx><br>
550 Namespace: vigra
551 */
552 template <class T>
553 inline T sign(T t)
554 {
555 return t > NumericTraits<T>::zero()
556 ? NumericTraits<T>::one()
557 : t < NumericTraits<T>::zero()
558 ? -NumericTraits<T>::one()
559 : NumericTraits<T>::zero();
560 }
561
562 /** \brief The integer sign function.
563
564 Returns 1, 0, or -1 depending on the sign of \a t, converted to int.
565
566 <b>\#include</b> <vigra/mathutil.hxx><br>
567 Namespace: vigra
568 */
569 template <class T>
570 inline int signi(T t)
571 {
572 return t > NumericTraits<T>::zero()
573 ? 1
574 : t < NumericTraits<T>::zero()
575 ? -1
576 : 0;
577 }
578
579 /** \brief The binary sign function.
580
581 Transfers the sign of \a t2 to \a t1.
582
583 <b>\#include</b> <vigra/mathutil.hxx><br>
584 Namespace: vigra
585 */
586 template <class T1, class T2>
587 inline T1 sign(T1 t1, T2 t2)
588 {
589 return t2 >= NumericTraits<T2>::zero()
590 ? abs(t1)
591 : -abs(t1);
592 }
593
594
595 #ifdef DOXYGEN // only for documentation
596 /** \brief Check if an integer is even.
597
598 Defined for all integral types.
599 */
600 bool even(int t);
601
602 /** \brief Check if an integer is odd.
603
604 Defined for all integral types.
605 */
606 bool odd(int t);
607
608 #endif
609
610 #define VIGRA_DEFINE_ODD_EVEN(T) \
611 inline bool even(T t) { return (t&1) == 0; } \
612 inline bool odd(T t) { return (t&1) == 1; }
613
614 VIGRA_DEFINE_ODD_EVEN(char)
615 VIGRA_DEFINE_ODD_EVEN(short)
616 VIGRA_DEFINE_ODD_EVEN(int)
617 VIGRA_DEFINE_ODD_EVEN(long)
618 VIGRA_DEFINE_ODD_EVEN(long long)
619 VIGRA_DEFINE_ODD_EVEN(unsigned char)
620 VIGRA_DEFINE_ODD_EVEN(unsigned short)
621 VIGRA_DEFINE_ODD_EVEN(unsigned int)
622 VIGRA_DEFINE_ODD_EVEN(unsigned long)
623 VIGRA_DEFINE_ODD_EVEN(unsigned long long)
624
625 #undef VIGRA_DEFINE_ODD_EVEN
626
627 #define VIGRA_DEFINE_NORM(T) \
628 inline NormTraits<T>::SquaredNormType squaredNorm(T t) { return sq(t); } \
629 inline NormTraits<T>::NormType norm(T t) { return abs(t); }
630
631 VIGRA_DEFINE_NORM(bool)
632 VIGRA_DEFINE_NORM(signed char)
633 VIGRA_DEFINE_NORM(unsigned char)
634 VIGRA_DEFINE_NORM(short)
635 VIGRA_DEFINE_NORM(unsigned short)
636 VIGRA_DEFINE_NORM(int)
637 VIGRA_DEFINE_NORM(unsigned int)
638 VIGRA_DEFINE_NORM(long)
639 VIGRA_DEFINE_NORM(unsigned long)
640 VIGRA_DEFINE_NORM(long long)
641 VIGRA_DEFINE_NORM(unsigned long long)
642 VIGRA_DEFINE_NORM(float)
643 VIGRA_DEFINE_NORM(double)
644 VIGRA_DEFINE_NORM(long double)
645
646 #undef VIGRA_DEFINE_NORM
647
648 template <class T>
649 inline typename NormTraits<std::complex<T> >::SquaredNormType
650 squaredNorm(std::complex<T> const & t)
651 {
652 return sq(t.real()) + sq(t.imag());
653 }
654
655 #ifdef DOXYGEN // only for documentation
656 /** \brief The squared norm of a numerical object.
657
658 <ul>
659 <li>For scalar types: equals <tt>vigra::sq(t)</tt>.
660 <li>For vectorial types (including TinyVector): equals <tt>vigra::dot(t, t)</tt>.
661 <li>For complex number types: equals <tt>vigra::sq(t.real()) + vigra::sq(t.imag())</tt>.
662 <li>For array and matrix types: results in the squared Frobenius norm (sum of squares of the matrix elements).
663 </ul>
664 */
665 NormTraits<T>::SquaredNormType squaredNorm(T const & t);
666
667 #endif
668
669 /** \brief The norm of a numerical object.
670
671 For scalar types: implemented as <tt>abs(t)</tt><br>
672 otherwise: implemented as <tt>sqrt(squaredNorm(t))</tt>.
673
674 <b>\#include</b> <vigra/mathutil.hxx><br>
675 Namespace: vigra
676 */
677 template <class T>
678 inline typename NormTraits<T>::NormType
679 norm(T const & t)
680 {
681 typedef typename NormTraits<T>::SquaredNormType SNT;
682 return sqrt(static_cast<typename SquareRootTraits<SNT>::SquareRootArgument>(squaredNorm(t)));
683 }
684
685 /** \brief Compute the eigenvalues of a 2x2 real symmetric matrix.
686
687 This uses the analytical eigenvalue formula
688 \f[
689 \lambda_{1,2} = \frac{1}{2}\left(a_{00} + a_{11} \pm \sqrt{(a_{00} - a_{11})^2 + 4 a_{01}^2}\right)
690 \f]
691
692 <b>\#include</b> <vigra/mathutil.hxx><br>
693 Namespace: vigra
694 */
695 template <class T>
696 void symmetric2x2Eigenvalues(T a00, T a01, T a11, T * r0, T * r1)
697 {
698 double d = hypot(a00 - a11, 2.0*a01);
699 *r0 = static_cast<T>(0.5*(a00 + a11 + d));
700 *r1 = static_cast<T>(0.5*(a00 + a11 - d));
701 if(*r0 < *r1)
702 std::swap(*r0, *r1);
703 }
704
705 /** \brief Compute the eigenvalues of a 3x3 real symmetric matrix.
706
707 This uses a numerically stable version of the analytical eigenvalue formula according to
708 <p>
709 David Eberly: <a href="http://www.geometrictools.com/Documentation/EigenSymmetric3x3.pdf">
710 <em>"Eigensystems for 3 × 3 Symmetric Matrices (Revisited)"</em></a>, Geometric Tools Documentation, 2006
711
712 <b>\#include</b> <vigra/mathutil.hxx><br>
713 Namespace: vigra
714 */
715 template <class T>
716 void symmetric3x3Eigenvalues(T a00, T a01, T a02, T a11, T a12, T a22,
717 T * r0, T * r1, T * r2)
718 {
719 double inv3 = 1.0 / 3.0, root3 = std::sqrt(3.0);
720
721 double c0 = a00*a11*a22 + 2.0*a01*a02*a12 - a00*a12*a12 - a11*a02*a02 - a22*a01*a01;
722 double c1 = a00*a11 - a01*a01 + a00*a22 - a02*a02 + a11*a22 - a12*a12;
723 double c2 = a00 + a11 + a22;
724 double c2Div3 = c2*inv3;
725 double aDiv3 = (c1 - c2*c2Div3)*inv3;
726 if (aDiv3 > 0.0)
727 aDiv3 = 0.0;
728 double mbDiv2 = 0.5*(c0 + c2Div3*(2.0*c2Div3*c2Div3 - c1));
729 double q = mbDiv2*mbDiv2 + aDiv3*aDiv3*aDiv3;
730 if (q > 0.0)
731 q = 0.0;
732 double magnitude = std::sqrt(-aDiv3);
733 double angle = std::atan2(std::sqrt(-q),mbDiv2)*inv3;
734 double cs = std::cos(angle);
735 double sn = std::sin(angle);
736 *r0 = static_cast<T>(c2Div3 + 2.0*magnitude*cs);
737 *r1 = static_cast<T>(c2Div3 - magnitude*(cs + root3*sn));
738 *r2 = static_cast<T>(c2Div3 - magnitude*(cs - root3*sn));
739 if(*r0 < *r1)
740 std::swap(*r0, *r1);
741 if(*r0 < *r2)
742 std::swap(*r0, *r2);
743 if(*r1 < *r2)
744 std::swap(*r1, *r2);
745 }
746
747 namespace detail {
748
749 template <class T>
750 T ellipticRD(T x, T y, T z)
751 {
752 double f = 1.0, s = 0.0, X, Y, Z, m;
753 for(;;)
754 {
755 m = (x + y + 3.0*z) / 5.0;
756 X = 1.0 - x/m;
757 Y = 1.0 - y/m;
758 Z = 1.0 - z/m;
759 if(std::max(std::max(VIGRA_CSTD::fabs(X), VIGRA_CSTD::fabs(Y)), VIGRA_CSTD::fabs(Z)) < 0.01)
760 break;
761 double l = VIGRA_CSTD::sqrt(x*y) + VIGRA_CSTD::sqrt(x*z) + VIGRA_CSTD::sqrt(y*z);
762 s += f / (VIGRA_CSTD::sqrt(z)*(z + l));
763 f /= 4.0;
764 x = (x + l)/4.0;
765 y = (y + l)/4.0;
766 z = (z + l)/4.0;
767 }
768 double a = X*Y;
769 double b = sq(Z);
770 double c = a - b;
771 double d = a - 6.0*b;
772 double e = d + 2.0*c;
773 return 3.0*s + f*(1.0+d*(-3.0/14.0+d*9.0/88.0-Z*e*4.5/26.0)
774 +Z*(e/6.0+Z*(-c*9.0/22.0+a*Z*3.0/26.0))) / VIGRA_CSTD::pow(m,1.5);
775 }
776
777 template <class T>
778 T ellipticRF(T x, T y, T z)
779 {
780 double X, Y, Z, m;
781 for(;;)
782 {
783 m = (x + y + z) / 3.0;
784 X = 1.0 - x/m;
785 Y = 1.0 - y/m;
786 Z = 1.0 - z/m;
787 if(std::max(std::max(VIGRA_CSTD::fabs(X), VIGRA_CSTD::fabs(Y)), VIGRA_CSTD::fabs(Z)) < 0.01)
788 break;
789 double l = VIGRA_CSTD::sqrt(x*y) + VIGRA_CSTD::sqrt(x*z) + VIGRA_CSTD::sqrt(y*z);
790 x = (x + l)/4.0;
791 y = (y + l)/4.0;
792 z = (z + l)/4.0;
793 }
794 double d = X*Y - sq(Z);
795 double p = X*Y*Z;
796 return (1.0 - d/10.0 + p/14.0 + sq(d)/24.0 - d*p*3.0/44.0) / VIGRA_CSTD::sqrt(m);
797 }
798
799 } // namespace detail
800
801 /** \brief The incomplete elliptic integral of the first kind.
802
803 This function computes
804
805 \f[
806 \mbox{F}(x, k) = \int_0^x \frac{1}{\sqrt{1 - k^2 \sin(t)^2}} dt
807 \f]
808
809 according to the algorithm given in Press et al. "Numerical Recipes".
810
811 Note: In some libraries (e.g. Mathematica), the second parameter of the elliptic integral
812 functions must be k^2 rather than k. Check the documentation when results disagree!
813
814 <b>\#include</b> <vigra/mathutil.hxx><br>
815 Namespace: vigra
816 */
817 inline double ellipticIntegralF(double x, double k)
818 {
819 double c2 = sq(VIGRA_CSTD::cos(x));
820 double s = VIGRA_CSTD::sin(x);
821 return s*detail::ellipticRF(c2, 1.0 - sq(k*s), 1.0);
822 }
823
824 /** \brief The incomplete elliptic integral of the second kind.
825
826 This function computes
827
828 \f[
829 \mbox{E}(x, k) = \int_0^x \sqrt{1 - k^2 \sin(t)^2} dt
830 \f]
831
832 according to the algorithm given in Press et al. "Numerical Recipes". The
833 complete elliptic integral of the second kind is simply <tt>ellipticIntegralE(M_PI/2, k)</TT>.
834
835 Note: In some libraries (e.g. Mathematica), the second parameter of the elliptic integral
836 functions must be k^2 rather than k. Check the documentation when results disagree!
837
838 <b>\#include</b> <vigra/mathutil.hxx><br>
839 Namespace: vigra
840 */
841 inline double ellipticIntegralE(double x, double k)
842 {
843 double c2 = sq(VIGRA_CSTD::cos(x));
844 double s = VIGRA_CSTD::sin(x);
845 k = sq(k*s);
846 return s*(detail::ellipticRF(c2, 1.0-k, 1.0) - k/3.0*detail::ellipticRD(c2, 1.0-k, 1.0));
847 }
848
849 #ifdef _MSC_VER
850
851 namespace detail {
852
853 template <class T>
854 double erfImpl(T x)
855 {
856 double t = 1.0/(1.0+0.5*VIGRA_CSTD::fabs(x));
857 double ans = t*VIGRA_CSTD::exp(-x*x-1.26551223+t*(1.00002368+t*(0.37409196+
858 t*(0.09678418+t*(-0.18628806+t*(0.27886807+
859 t*(-1.13520398+t*(1.48851587+t*(-0.82215223+
860 t*0.17087277)))))))));
861 if (x >= 0.0)
862 return 1.0 - ans;
863 else
864 return ans - 1.0;
865 }
866
867 } // namespace detail
868
869 /** \brief The error function.
870
871 If <tt>erf()</tt> is not provided in the C standard math library (as it should according to the
872 new C99 standard ?), VIGRA implements <tt>erf()</tt> as an approximation of the error
873 function
874
875 \f[
876 \mbox{erf}(x) = \int_0^x e^{-t^2} dt
877 \f]
878
879 according to the formula given in Press et al. "Numerical Recipes".
880
881 <b>\#include</b> <vigra/mathutil.hxx><br>
882 Namespace: vigra
883 */
884 inline double erf(double x)
885 {
886 return detail::erfImpl(x);
887 }
888
889 #else
890
891 using ::erf;
892
893 #endif
894
895 namespace detail {
896
897 template <class T>
898 double noncentralChi2CDFApprox(unsigned int degreesOfFreedom, T noncentrality, T arg)
899 {
900 double a = degreesOfFreedom + noncentrality;
901 double b = (a + noncentrality) / sq(a);
902 double t = (VIGRA_CSTD::pow((double)arg / a, 1.0/3.0) - (1.0 - 2.0 / 9.0 * b)) / VIGRA_CSTD::sqrt(2.0 / 9.0 * b);
903 return 0.5*(1.0 + erf(t/VIGRA_CSTD::sqrt(2.0)));
904 }
905
906 template <class T>
907 void noncentralChi2OneIteration(T arg, T & lans, T & dans, T & pans, unsigned int & j)
908 {
909 double tol = -50.0;
910 if(lans < tol)
911 {
912 lans = lans + VIGRA_CSTD::log(arg / j);
913 dans = VIGRA_CSTD::exp(lans);
914 }
915 else
916 {
917 dans = dans * arg / j;
918 }
919 pans = pans - dans;
920 j += 2;
921 }
922
923 template <class T>
924 std::pair<double, double> noncentralChi2CDF(unsigned int degreesOfFreedom, T noncentrality, T arg, T eps)
925 {
926 vigra_precondition(noncentrality >= 0.0 && arg >= 0.0 && eps > 0.0,
927 "noncentralChi2P(): parameters must be positive.");
928 if (arg == 0.0 && degreesOfFreedom > 0)
929 return std::make_pair(0.0, 0.0);
930
931 // Determine initial values
932 double b1 = 0.5 * noncentrality,
933 ao = VIGRA_CSTD::exp(-b1),
934 eps2 = eps / ao,
935 lnrtpi2 = 0.22579135264473,
936 probability, density, lans, dans, pans, sum, am, hold;
937 unsigned int maxit = 500,
938 i, m;
939 if(degreesOfFreedom % 2)
940 {
941 i = 1;
942 lans = -0.5 * (arg + VIGRA_CSTD::log(arg)) - lnrtpi2;
943 dans = VIGRA_CSTD::exp(lans);
944 pans = erf(VIGRA_CSTD::sqrt(arg/2.0));
945 }
946 else
947 {
948 i = 2;
949 lans = -0.5 * arg;
950 dans = VIGRA_CSTD::exp(lans);
951 pans = 1.0 - dans;
952 }
953
954 // Evaluate first term
955 if(degreesOfFreedom == 0)
956 {
957 m = 1;
958 degreesOfFreedom = 2;
959 am = b1;
960 sum = 1.0 / ao - 1.0 - am;
961 density = am * dans;
962 probability = 1.0 + am * pans;
963 }
964 else
965 {
966 m = 0;
967 degreesOfFreedom = degreesOfFreedom - 1;
968 am = 1.0;
969 sum = 1.0 / ao - 1.0;
970 while(i < degreesOfFreedom)
971 detail::noncentralChi2OneIteration(arg, lans, dans, pans, i);
972 degreesOfFreedom = degreesOfFreedom + 1;
973 density = dans;
974 probability = pans;
975 }
976 // Evaluate successive terms of the expansion
977 for(++m; m<maxit; ++m)
978 {
979 am = b1 * am / m;
980 detail::noncentralChi2OneIteration(arg, lans, dans, pans, degreesOfFreedom);
981 sum = sum - am;
982 density = density + am * dans;
983 hold = am * pans;
984 probability = probability + hold;
985 if((pans * sum < eps2) && (hold < eps2))
986 break; // converged
987 }
988 if(m == maxit)
989 vigra_fail("noncentralChi2P(): no convergence.");
990 return std::make_pair(0.5 * ao * density, std::min(1.0, std::max(0.0, ao * probability)));
991 }
992
993 } // namespace detail
994
995 /** \brief Chi square distribution.
996
997 Computes the density of a chi square distribution with \a degreesOfFreedom
998 and tolerance \a accuracy at the given argument \a arg
999 by calling <tt>noncentralChi2(degreesOfFreedom, 0.0, arg, accuracy)</tt>.
1000
1001 <b>\#include</b> <vigra/mathutil.hxx><br>
1002 Namespace: vigra
1003 */
1004 inline double chi2(unsigned int degreesOfFreedom, double arg, double accuracy = 1e-7)
1005 {
1006 return detail::noncentralChi2CDF(degreesOfFreedom, 0.0, arg, accuracy).first;
1007 }
1008
1009 /** \brief Cumulative chi square distribution.
1010
1011 Computes the cumulative density of a chi square distribution with \a degreesOfFreedom
1012 and tolerance \a accuracy at the given argument \a arg, i.e. the probability that
1013 a random number drawn from the distribution is below \a arg
1014 by calling <tt>noncentralChi2CDF(degreesOfFreedom, 0.0, arg, accuracy)</tt>.
1015
1016 <b>\#include</b> <vigra/mathutil.hxx><br>
1017 Namespace: vigra
1018 */
1019 inline double chi2CDF(unsigned int degreesOfFreedom, double arg, double accuracy = 1e-7)
1020 {
1021 return detail::noncentralChi2CDF(degreesOfFreedom, 0.0, arg, accuracy).second;
1022 }
1023
1024 /** \brief Non-central chi square distribution.
1025
1026 Computes the density of a non-central chi square distribution with \a degreesOfFreedom,
1027 noncentrality parameter \a noncentrality and tolerance \a accuracy at the given argument
1028 \a arg. It uses Algorithm AS 231 from Appl. Statist. (1987) Vol.36, No.3 (code ported from
1029 http://lib.stat.cmu.edu/apstat/231). The algorithm has linear complexity in the number of
1030 degrees of freedom.
1031
1032 <b>\#include</b> <vigra/mathutil.hxx><br>
1033 Namespace: vigra
1034 */
1035 inline double noncentralChi2(unsigned int degreesOfFreedom,
1036 double noncentrality, double arg, double accuracy = 1e-7)
1037 {
1038 return detail::noncentralChi2CDF(degreesOfFreedom, noncentrality, arg, accuracy).first;
1039 }
1040
1041 /** \brief Cumulative non-central chi square distribution.
1042
1043 Computes the cumulative density of a chi square distribution with \a degreesOfFreedom,
1044 noncentrality parameter \a noncentrality and tolerance \a accuracy at the given argument
1045 \a arg, i.e. the probability that a random number drawn from the distribution is below \a arg
1046 It uses Algorithm AS 231 from Appl. Statist. (1987) Vol.36, No.3 (code ported from
1047 http://lib.stat.cmu.edu/apstat/231). The algorithm has linear complexity in the number of
1048 degrees of freedom (see noncentralChi2CDFApprox() for a constant-time algorithm).
1049
1050 <b>\#include</b> <vigra/mathutil.hxx><br>
1051 Namespace: vigra
1052 */
1053 inline double noncentralChi2CDF(unsigned int degreesOfFreedom,
1054 double noncentrality, double arg, double accuracy = 1e-7)
1055 {
1056 return detail::noncentralChi2CDF(degreesOfFreedom, noncentrality, arg, accuracy).second;
1057 }
1058
1059 /** \brief Cumulative non-central chi square distribution (approximate).
1060
1061 Computes approximate values of the cumulative density of a chi square distribution with \a degreesOfFreedom,
1062 and noncentrality parameter \a noncentrality at the given argument
1063 \a arg, i.e. the probability that a random number drawn from the distribution is below \a arg
1064 It uses the approximate transform into a normal distribution due to Wilson and Hilferty
1065 (see Abramovitz, Stegun: "Handbook of Mathematical Functions", formula 26.3.32).
1066 The algorithm's running time is independent of the inputs, i.e. is should be used
1067 when noncentralChi2CDF() is too slow, and approximate values are sufficient. The accuracy is only
1068 about 0.1 for few degrees of freedom, but reaches about 0.001 above dof = 5.
1069
1070 <b>\#include</b> <vigra/mathutil.hxx><br>
1071 Namespace: vigra
1072 */
1073 inline double noncentralChi2CDFApprox(unsigned int degreesOfFreedom, double noncentrality, double arg)
1074 {
1075 return detail::noncentralChi2CDFApprox(degreesOfFreedom, noncentrality, arg);
1076 }
1077
1078 namespace detail {
1079
1080 // computes (l+m)! / (l-m)!
1081 // l and m must be positive
1082 template <class T>
1083 T facLM(T l, T m)
1084 {
1085 T tmp = NumericTraits<T>::one();
1086 for(T f = l-m+1; f <= l+m; ++f)
1087 tmp *= f;
1088 return tmp;
1089 }
1090
1091 } // namespace detail
1092
1093 /** \brief Associated Legendre polynomial.
1094
1095 Computes the value of the associated Legendre polynomial of order <tt>l, m</tt>
1096 for argument <tt>x</tt>. <tt>x</tt> must be in the range <tt>[-1.0, 1.0]</tt>,
1097 otherwise an exception is thrown. The standard Legendre polynomials are the
1098 special case <tt>m == 0</tt>.
1099
1100 <b>\#include</b> <vigra/mathutil.hxx><br>
1101 Namespace: vigra
1102 */
1103 template <class REAL>
1104 REAL legendre(unsigned int l, int m, REAL x)
1105 {
1106 vigra_precondition(abs(x) <= 1.0, "legendre(): x must be in [-1.0, 1.0].");
1107 if (m < 0)
1108 {
1109 m = -m;
1110 REAL s = odd(m)
1111 ? -1.0
1112 : 1.0;
1113 return legendre(l,m,x) * s / detail::facLM<REAL>(l,m);
1114 }
1115 REAL result = 1.0;
1116 if (m > 0)
1117 {
1118 REAL r = std::sqrt( (1.0-x) * (1.0+x) );
1119 REAL f = 1.0;
1120 for (int i=1; i<=m; i++)
1121 {
1122 result *= (-f) * r;
1123 f += 2.0;
1124 }
1125 }
1126 if((int)l == m)
1127 return result;
1128
1129 REAL result_1 = x * (2.0 * m + 1.0) * result;
1130 if((int)l == m+1)
1131 return result_1;
1132 REAL other = 0.0;
1133 for(unsigned int i = m+2; i <= l; ++i)
1134 {
1135 other = ( (2.0*i-1.0) * x * result_1 - (i+m-1.0)*result) / (i-m);
1136 result = result_1;
1137 result_1 = other;
1138 }
1139 return other;
1140 }
1141
1142 /** \brief \brief Legendre polynomial.
1143
1144 Computes the value of the Legendre polynomial of order <tt>l</tt> for argument <tt>x</tt>.
1145 <tt>x</tt> must be in the range <tt>[-1.0, 1.0]</tt>, otherwise an exception is thrown.
1146
1147 <b>\#include</b> <vigra/mathutil.hxx><br>
1148 Namespace: vigra
1149 */
1150 template <class REAL>
1151 REAL legendre(unsigned int l, REAL x)
1152 {
1153 return legendre(l, 0, x);
1154 }
1155
1156 /** \brief sin(pi*x).
1157
1158 Essentially calls <tt>std::sin(M_PI*x)</tt> but uses a more accurate implementation
1159 to make sure that <tt>sin_pi(1.0) == 0.0</tt> (which does not hold for
1160 <tt>std::sin(M_PI)</tt> due to round-off error), and <tt>sin_pi(0.5) == 1.0</tt>.
1161
1162 <b>\#include</b> <vigra/mathutil.hxx><br>
1163 Namespace: vigra
1164 */
1165 template <class REAL>
1166 REAL sin_pi(REAL x)
1167 {
1168 if(x < 0.0)
1169 return -sin_pi(-x);
1170 if(x < 0.5)
1171 return std::sin(M_PI * x);
1172
1173 bool invert = false;
1174 if(x < 1.0)
1175 {
1176 invert = true;
1177 x = -x;
1178 }
1179
1180 REAL rem = std::floor(x);
1181 if(odd((int)rem))
1182 invert = !invert;
1183 rem = x - rem;
1184 if(rem > 0.5)
1185 rem = 1.0 - rem;
1186 if(rem == 0.5)
1187 rem = NumericTraits<REAL>::one();
1188 else
1189 rem = std::sin(M_PI * rem);
1190 return invert
1191 ? -rem
1192 : rem;
1193 }
1194
1195 /** \brief cos(pi*x).
1196
1197 Essentially calls <tt>std::cos(M_PI*x)</tt> but uses a more accurate implementation
1198 to make sure that <tt>cos_pi(1.0) == -1.0</tt> and <tt>cos_pi(0.5) == 0.0</tt>.
1199
1200 <b>\#include</b> <vigra/mathutil.hxx><br>
1201 Namespace: vigra
1202 */
1203 template <class REAL>
1204 REAL cos_pi(REAL x)
1205 {
1206 return sin_pi(x+0.5);
1207 }
1208
1209 namespace detail {
1210
1211 template <class REAL>
1212 struct GammaImpl
1213 {
1214 static REAL gamma(REAL x);
1215 static REAL loggamma(REAL x);
1216
1217 static double g[];
1218 static double a[];
1219 static double t[];
1220 static double u[];
1221 static double v[];
1222 static double s[];
1223 static double r[];
1224 static double w[];
1225 };
1226
1227 template <class REAL>
1228 double GammaImpl<REAL>::g[] = {
1229 1.0,
1230 0.5772156649015329,
1231 -0.6558780715202538,
1232 -0.420026350340952e-1,
1233 0.1665386113822915,
1234 -0.421977345555443e-1,
1235 -0.9621971527877e-2,
1236 0.7218943246663e-2,
1237 -0.11651675918591e-2,
1238 -0.2152416741149e-3,
1239 0.1280502823882e-3,
1240 -0.201348547807e-4,
1241 -0.12504934821e-5,
1242 0.1133027232e-5,
1243 -0.2056338417e-6,
1244 0.6116095e-8,
1245 0.50020075e-8,
1246 -0.11812746e-8,
1247 0.1043427e-9,
1248 0.77823e-11,
1249 -0.36968e-11,
1250 0.51e-12,
1251 -0.206e-13,
1252 -0.54e-14,
1253 0.14e-14
1254 };
1255
1256 template <class REAL>
1257 double GammaImpl<REAL>::a[] = {
1258 7.72156649015328655494e-02,
1259 3.22467033424113591611e-01,
1260 6.73523010531292681824e-02,
1261 2.05808084325167332806e-02,
1262 7.38555086081402883957e-03,
1263 2.89051383673415629091e-03,
1264 1.19270763183362067845e-03,
1265 5.10069792153511336608e-04,
1266 2.20862790713908385557e-04,
1267 1.08011567247583939954e-04,
1268 2.52144565451257326939e-05,
1269 4.48640949618915160150e-05
1270 };
1271
1272 template <class REAL>
1273 double GammaImpl<REAL>::t[] = {
1274 4.83836122723810047042e-01,
1275 -1.47587722994593911752e-01,
1276 6.46249402391333854778e-02,
1277 -3.27885410759859649565e-02,
1278 1.79706750811820387126e-02,
1279 -1.03142241298341437450e-02,
1280 6.10053870246291332635e-03,
1281 -3.68452016781138256760e-03,
1282 2.25964780900612472250e-03,
1283 -1.40346469989232843813e-03,
1284 8.81081882437654011382e-04,
1285 -5.38595305356740546715e-04,
1286 3.15632070903625950361e-04,
1287 -3.12754168375120860518e-04,
1288 3.35529192635519073543e-04
1289 };
1290
1291 template <class REAL>
1292 double GammaImpl<REAL>::u[] = {
1293 -7.72156649015328655494e-02,
1294 6.32827064025093366517e-01,
1295 1.45492250137234768737e+00,
1296 9.77717527963372745603e-01,
1297 2.28963728064692451092e-01,
1298 1.33810918536787660377e-02
1299 };
1300
1301 template <class REAL>
1302 double GammaImpl<REAL>::v[] = {
1303 0.0,
1304 2.45597793713041134822e+00,
1305 2.12848976379893395361e+00,
1306 7.69285150456672783825e-01,
1307 1.04222645593369134254e-01,
1308 3.21709242282423911810e-03
1309 };
1310
1311 template <class REAL>
1312 double GammaImpl<REAL>::s[] = {
1313 -7.72156649015328655494e-02,
1314 2.14982415960608852501e-01,
1315 3.25778796408930981787e-01,
1316 1.46350472652464452805e-01,
1317 2.66422703033638609560e-02,
1318 1.84028451407337715652e-03,
1319 3.19475326584100867617e-05
1320 };
1321
1322 template <class REAL>
1323 double GammaImpl<REAL>::r[] = {
1324 0.0,
1325 1.39200533467621045958e+00,
1326 7.21935547567138069525e-01,
1327 1.71933865632803078993e-01,
1328 1.86459191715652901344e-02,
1329 7.77942496381893596434e-04,
1330 7.32668430744625636189e-06
1331 };
1332
1333 template <class REAL>
1334 double GammaImpl<REAL>::w[] = {
1335 4.18938533204672725052e-01,
1336 8.33333333333329678849e-02,
1337 -2.77777777728775536470e-03,
1338 7.93650558643019558500e-04,
1339 -5.95187557450339963135e-04,
1340 8.36339918996282139126e-04,
1341 -1.63092934096575273989e-03
1342 };
1343
1344 template <class REAL>
1345 REAL GammaImpl<REAL>::gamma(REAL x)
1346 {
1347 int i, k, m, ix = (int)x;
1348 double ga = 0.0, gr = 0.0, r = 0.0, z = 0.0;
1349
1350 vigra_precondition(x <= 171.0,
1351 "gamma(): argument cannot exceed 171.0.");
1352
1353 if (x == ix)
1354 {
1355 if (ix > 0)
1356 {
1357 ga = 1.0; // use factorial
1358 for (i=2; i<ix; ++i)
1359 {
1360 ga *= i;
1361 }
1362 }
1363 else
1364 {
1365 vigra_precondition(false,
1366 "gamma(): gamma function is undefined for 0 and negative integers.");
1367 }
1368 }
1369 else
1370 {
1371 if (abs(x) > 1.0)
1372 {
1373 z = abs(x);
1374 m = (int)z;
1375 r = 1.0;
1376 for (k=1; k<=m; ++k)
1377 {
1378 r *= (z-k);
1379 }
1380 z -= m;
1381 }
1382 else
1383 {
1384 z = x;
1385 }
1386 gr = g[24];
1387 for (k=23; k>=0; --k)
1388 {
1389 gr = gr*z+g[k];
1390 }
1391 ga = 1.0/(gr*z);
1392 if (abs(x) > 1.0)
1393 {
1394 ga *= r;
1395 if (x < 0.0)
1396 {
1397 ga = -M_PI/(x*ga*sin_pi(x));
1398 }
1399 }
1400 }
1401 return ga;
1402 }
1403
1404 /*
1405 * the following code is derived from lgamma_r() by Sun
1406 *
1407 * ====================================================
1408 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
1409 *
1410 * Developed at SunPro, a Sun Microsystems, Inc. business.
1411 * Permission to use, copy, modify, and distribute this
1412 * software is freely granted, provided that this notice
1413 * is preserved.
1414 * ====================================================
1415 *
1416 */
1417 template <class REAL>
1418 REAL GammaImpl<REAL>::loggamma(REAL x)
1419 {
1420 vigra_precondition(x > 0.0,
1421 "loggamma(): argument must be positive.");
1422
1423 vigra_precondition(x <= 1.0e307,
1424 "loggamma(): argument must not exceed 1e307.");
1425
1426 double res;
1427
1428 if (x < 4.2351647362715017e-22)
1429 {
1430 res = -std::log(x);
1431 }
1432 else if ((x == 2.0) || (x == 1.0))
1433 {
1434 res = 0.0;
1435 }
1436 else if (x < 2.0)
1437 {
1438 const double tc = 1.46163214496836224576e+00;
1439 const double tf = -1.21486290535849611461e-01;
1440 const double tt = -3.63867699703950536541e-18;
1441 if (x <= 0.9)
1442 {
1443 res = -std::log(x);
1444 if (x >= 0.7316)
1445 {
1446 double y = 1.0-x;
1447 double z = y*y;
1448 double p1 = a[0]+z*(a[2]+z*(a[4]+z*(a[6]+z*(a[8]+z*a[10]))));
1449 double p2 = z*(a[1]+z*(a[3]+z*(a[5]+z*(a[7]+z*(a[9]+z*a[11])))));
1450 double p = y*p1+p2;
1451 res += (p-0.5*y);
1452 }
1453 else if (x >= 0.23164)
1454 {
1455 double y = x-(tc-1.0);
1456 double z = y*y;
1457 double w = z*y;
1458 double p1 = t[0]+w*(t[3]+w*(t[6]+w*(t[9] +w*t[12])));
1459 double p2 = t[1]+w*(t[4]+w*(t[7]+w*(t[10]+w*t[13])));
1460 double p3 = t[2]+w*(t[5]+w*(t[8]+w*(t[11]+w*t[14])));
1461 double p = z*p1-(tt-w*(p2+y*p3));
1462 res += (tf + p);
1463 }
1464 else
1465 {
1466 double y = x;
1467 double p1 = y*(u[0]+y*(u[1]+y*(u[2]+y*(u[3]+y*(u[4]+y*u[5])))));
1468 double p2 = 1.0+y*(v[1]+y*(v[2]+y*(v[3]+y*(v[4]+y*v[5]))));
1469 res += (-0.5*y + p1/p2);
1470 }
1471 }
1472 else
1473 {
1474 res = 0.0;
1475 if (x >= 1.7316)
1476 {
1477 double y = 2.0-x;
1478 double z = y*y;
1479 double p1 = a[0]+z*(a[2]+z*(a[4]+z*(a[6]+z*(a[8]+z*a[10]))));
1480 double p2 = z*(a[1]+z*(a[3]+z*(a[5]+z*(a[7]+z*(a[9]+z*a[11])))));
1481 double p = y*p1+p2;
1482 res += (p-0.5*y);
1483 }
1484 else if(x >= 1.23164)
1485 {
1486 double y = x-tc;
1487 double z = y*y;
1488 double w = z*y;
1489 double p1 = t[0]+w*(t[3]+w*(t[6]+w*(t[9] +w*t[12])));
1490 double p2 = t[1]+w*(t[4]+w*(t[7]+w*(t[10]+w*t[13])));
1491 double p3 = t[2]+w*(t[5]+w*(t[8]+w*(t[11]+w*t[14])));
1492 double p = z*p1-(tt-w*(p2+y*p3));
1493 res += (tf + p);
1494 }
1495 else
1496 {
1497 double y = x-1.0;
1498 double p1 = y*(u[0]+y*(u[1]+y*(u[2]+y*(u[3]+y*(u[4]+y*u[5])))));
1499 double p2 = 1.0+y*(v[1]+y*(v[2]+y*(v[3]+y*(v[4]+y*v[5]))));
1500 res += (-0.5*y + p1/p2);
1501 }
1502 }
1503 }
1504 else if(x < 8.0)
1505 {
1506 double i = std::floor(x);
1507 double y = x-i;
1508 double p = y*(s[0]+y*(s[1]+y*(s[2]+y*(s[3]+y*(s[4]+y*(s[5]+y*s[6]))))));
1509 double q = 1.0+y*(r[1]+y*(r[2]+y*(r[3]+y*(r[4]+y*(r[5]+y*r[6])))));
1510 res = 0.5*y+p/q;
1511 double z = 1.0;
1512 while (i > 2.0)
1513 {
1514 --i;
1515 z *= (y+i);
1516 }
1517 res += std::log(z);
1518 }
1519 else if (x < 2.8823037615171174e+17)
1520 {
1521 double t = std::log(x);
1522 double z = 1.0/x;
1523 double y = z*z;
1524 double yy = w[0]+z*(w[1]+y*(w[2]+y*(w[3]+y*(w[4]+y*(w[5]+y*w[6])))));
1525 res = (x-0.5)*(t-1.0)+yy;
1526 }
1527 else
1528 {
1529 res = x*(std::log(x) - 1.0);
1530 }
1531
1532 return res;
1533 }
1534
1535
1536 } // namespace detail
1537
1538 /** \brief The gamma function.
1539
1540 This function implements the algorithm from<br>
1541 Zhang and Jin: "Computation of Special Functions", John Wiley and Sons, 1996.
1542
1543 The argument must be <= 171.0 and cannot be zero or a negative integer. An
1544 exception is thrown when these conditions are violated.
1545
1546 <b>\#include</b> <vigra/mathutil.hxx><br>
1547 Namespace: vigra
1548 */
1549 inline double gamma(double x)
1550 {
1551 return detail::GammaImpl<double>::gamma(x);
1552 }
1553
1554 /** \brief The natural logarithm of the gamma function.
1555
1556 This function is based on a free implementation by Sun Microsystems, Inc., see
1557 <a href="http://www.sourceware.org/cgi-bin/cvsweb.cgi/~checkout~/src/newlib/libm/mathfp/er_lgamma.c?rev=1.6&content-type=text/plain&cvsroot=src">sourceware.org</a> archive. It can be removed once all compilers support the new C99
1558 math functions.
1559
1560 The argument must be positive and < 1e30. An exception is thrown when these conditions are violated.
1561
1562 <b>\#include</b> <vigra/mathutil.hxx><br>
1563 Namespace: vigra
1564 */
1565 inline double loggamma(double x)
1566 {
1567 return detail::GammaImpl<double>::loggamma(x);
1568 }
1569
1570
1571 namespace detail {
1572
1573 // both f1 and f2 are unsigned here
1574 template<class FPT>
1575 inline
1576 FPT safeFloatDivision( FPT f1, FPT f2 )
1577 {
1578 return f2 < NumericTraits<FPT>::one() && f1 > f2 * NumericTraits<FPT>::max()
1579 ? NumericTraits<FPT>::max()
1580 : (f2 > NumericTraits<FPT>::one() && f1 < f2 * NumericTraits<FPT>::smallestPositive()) ||
1581 f1 == NumericTraits<FPT>::zero()
1582 ? NumericTraits<FPT>::zero()
1583 : f1/f2;
1584 }
1585
1586 } // namespace detail
1587
1588 /** \brief Tolerance based floating-point comparison.
1589
1590 Check whether two floating point numbers are equal within the given tolerance.
1591 This is useful because floating point numbers that should be equal in theory are
1592 rarely exactly equal in practice. If the tolerance \a epsilon is not given,
1593 twice the machine epsilon is used.
1594
1595 <b>\#include</b> <vigra/mathutil.hxx><br>
1596 Namespace: vigra
1597 */
1598 template <class T1, class T2>
1599 bool
1600 closeAtTolerance(T1 l, T2 r, typename PromoteTraits<T1, T2>::Promote epsilon)
1601 {
1602 typedef typename PromoteTraits<T1, T2>::Promote T;
1603 if(l == 0.0)
1604 return VIGRA_CSTD::fabs(r) <= epsilon;
1605 if(r == 0.0)
1606 return VIGRA_CSTD::fabs(l) <= epsilon;
1607 T diff = VIGRA_CSTD::fabs( l - r );
1608 T d1 = detail::safeFloatDivision<T>( diff, VIGRA_CSTD::fabs( r ) );
1609 T d2 = detail::safeFloatDivision<T>( diff, VIGRA_CSTD::fabs( l ) );
1610
1611 return (d1 <= epsilon && d2 <= epsilon);
1612 }
1613
1614 template <class T1, class T2>
1615 inline bool closeAtTolerance(T1 l, T2 r)
1616 {
1617 typedef typename PromoteTraits<T1, T2>::Promote T;
1618 return closeAtTolerance(l, r, T(2.0) * NumericTraits<T>::epsilon());
1619 }
1620
1621 //@}
1622
1623 } // namespace vigra
1624
1625 #endif /* VIGRA_MATHUTIL_HXX */
vigra::ellipticIntegralF
double ellipticIntegralF(double x, double k)
The incomplete elliptic integral of the first kind.
Definition: mathutil.hxx:817
vigra::sin_pi
REAL sin_pi(REAL x)
sin(pi*x).
Definition: mathutil.hxx:1166
vigra::atan2
FixedPoint16< 2, OverflowHandling > atan2(FixedPoint16< IntBits, OverflowHandling > y, FixedPoint16< IntBits, OverflowHandling > x)
Arctangent. Accuracy better than 1/3 degree (9 significant bits).
Definition: fixedpoint.hxx:1654
vigra::norm
NormTraits< T >::NormType norm(T const &t)
The norm of a numerical object.
Definition: mathutil.hxx:679
vigra::arg
R arg(const FFTWComplex< R > &a)
pahse
Definition: fftw3.hxx:1009
vigra::sign
T1 sign(T1 t1, T2 t2)
The binary sign function.
Definition: mathutil.hxx:587
vigra::linalg::sin
linalg::TemporaryMatrix< T > sin(MultiArrayView< 2, T, C > const &v)
vigra::symmetric3x3Eigenvalues
void symmetric3x3Eigenvalues(T a00, T a01, T a02, T a11, T a12, T a22, T *r0, T *r1, T *r2)
Compute the eigenvalues of a 3x3 real symmetric matrix.
Definition: mathutil.hxx:716
vigra::linalg::exp
linalg::TemporaryMatrix< T > exp(MultiArrayView< 2, T, C > const &v)
vigra::power
V power(const V &x)
Exponentiation to a positive integer power by squaring.
Definition: mathutil.hxx:389
vigra::round
int round(FixedPoint< IntBits, FracBits > v)
rounding to the nearest integer.
Definition: fixedpoint.hxx:683
vigra::signi
int signi(T t)
The integer sign function.
Definition: mathutil.hxx:570
vigra
Definition: accessor.hxx:43
vigra::hypot
FixedPoint16< IntBits, OverflowHandling > hypot(FixedPoint16< IntBits, OverflowHandling > v1, FixedPoint16< IntBits, OverflowHandling > v2)
Length of hypotenuse.
Definition: fixedpoint.hxx:1640
vigra::odd
bool odd(int t)
Check if an integer is odd.
vigra::cos_pi
REAL cos_pi(REAL x)
cos(pi*x).
Definition: mathutil.hxx:1204
vigra::noncentralChi2
double noncentralChi2(unsigned int degreesOfFreedom, double noncentrality, double arg, double accuracy=1e-7)
Non-central chi square distribution.
Definition: mathutil.hxx:1035
vigra::noncentralChi2CDFApprox
double noncentralChi2CDFApprox(unsigned int degreesOfFreedom, double noncentrality, double arg)
Cumulative non-central chi square distribution (approximate).
Definition: mathutil.hxx:1073
vigra::log2i
Int32 log2i(UInt32 x)
Compute the base-2 logarithm of an integer.
Definition: mathutil.hxx:323
vigra::sq
NumericTraits< T >::Promote sq(T t)
The square function.
Definition: mathutil.hxx:344
vigra::noncentralChi2CDF
double noncentralChi2CDF(unsigned int degreesOfFreedom, double noncentrality, double arg, double accuracy=1e-7)
Cumulative non-central chi square distribution.
Definition: mathutil.hxx:1053
vigra::sum
NumericTraits< V >::Promote sum(TinyVectorBase< V, SIZE, D1, D2 > const &l)
sum of the vector&#39;s elements
Definition: tinyvector.hxx:1871
vigra::Int32
detail::SelectIntegerType< 32, detail::SignedIntTypes >::type Int32
32-bit signed int
Definition: sized_int.hxx:175
vigra::legendre
REAL legendre(unsigned int l, REAL x)
Legendre polynomial.
Definition: mathutil.hxx:1151
vigra::squaredNorm
NormTraits< T >::SquaredNormType squaredNorm(T const &t)
The squared norm of a numerical object.
vigra::symmetric2x2Eigenvalues
void symmetric2x2Eigenvalues(T a00, T a01, T a11, T *r0, T *r1)
Compute the eigenvalues of a 2x2 real symmetric matrix.
Definition: mathutil.hxx:696
vigra::even
bool even(int t)
Check if an integer is even.
vigra::chi2CDF
double chi2CDF(unsigned int degreesOfFreedom, double arg, double accuracy=1e-7)
Cumulative chi square distribution.
Definition: mathutil.hxx:1019
vigra::gamma
double gamma(double x)
The gamma function.
Definition: mathutil.hxx:1549
vigra::linalg::log
linalg::TemporaryMatrix< T > log(MultiArrayView< 2, T, C > const &v)
vigra::loggamma
double loggamma(double x)
The natural logarithm of the gamma function.
Definition: mathutil.hxx:1565
vigra::UInt32
detail::SelectIntegerType< 32, detail::UnsignedIntTypes >::type UInt32
32-bit unsigned int
Definition: sized_int.hxx:183
vigra::abs
FFTWComplex< R >::NormType abs(const FFTWComplex< R > &a)
absolute value (= magnitude)
Definition: fftw3.hxx:1002
vigra::sqrti
Int32 sqrti(Int32 v)
Signed integer square root.
Definition: mathutil.hxx:500
vigra::floorPower2
UInt32 floorPower2(UInt32 x)
Round down to the nearest power of 2.
Definition: mathutil.hxx:279
vigra::chi2
double chi2(unsigned int degreesOfFreedom, double arg, double accuracy=1e-7)
Chi square distribution.
Definition: mathutil.hxx:1004
vigra::ceilPower2
UInt32 ceilPower2(UInt32 x)
Round up to the nearest power of 2.
Definition: mathutil.hxx:257
vigra::roundi
int roundi(double t)
Round and cast to integer.
Definition: mathutil.hxx:240
vigra::ceil
int ceil(FixedPoint< IntBits, FracBits > v)
rounding up.
Definition: fixedpoint.hxx:675
vigra::linalg::cos
linalg::TemporaryMatrix< T > cos(MultiArrayView< 2, T, C > const &v)
vigra::floor
int floor(FixedPoint< IntBits, FracBits > v)
rounding down.
Definition: fixedpoint.hxx:667
vigra::ellipticIntegralE
double ellipticIntegralE(double x, double k)
The incomplete elliptic integral of the second kind.
Definition: mathutil.hxx:841
vigra::sqrt
SquareRootTraits< FixedPoint< IntBits, FracBits > >::SquareRootResult sqrt(FixedPoint< IntBits, FracBits > v)
square root.
Definition: fixedpoint.hxx:616

© Ullrich Köthe (ullrich.koethe@iwr.uni-heidelberg.de) Heidelberg Collaboratory for Image Processing, University of Heidelberg, Germany | html generated using doxygen and Python vigra 1.10.0