1. Jason McKesson
  2. GL Profile Suite

Source

GL Profile Suite / boost_1_51_0 / boost / math / tools / toms748_solve.hpp

  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
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
//  (C) Copyright John Maddock 2006.
//  Use, modification and distribution are subject to the
//  Boost Software License, Version 1.0. (See accompanying file
//  LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)

#ifndef BOOST_MATH_TOOLS_SOLVE_ROOT_HPP
#define BOOST_MATH_TOOLS_SOLVE_ROOT_HPP

#ifdef _MSC_VER
#pragma once
#endif

#include <boost/math/tools/precision.hpp>
#include <boost/math/policies/error_handling.hpp>
#include <boost/math/tools/config.hpp>
#include <boost/math/special_functions/sign.hpp>
#include <boost/cstdint.hpp>
#include <limits>

namespace boost{ namespace math{ namespace tools{

template <class T>
class eps_tolerance
{
public:
   eps_tolerance(unsigned bits)
   {
      BOOST_MATH_STD_USING
      eps = (std::max)(T(ldexp(1.0F, 1-bits)), T(4 * tools::epsilon<T>()));
   }
   bool operator()(const T& a, const T& b)
   {
      BOOST_MATH_STD_USING
      return fabs(a - b) <= (eps * (std::min)(fabs(a), fabs(b)));
   }
private:
   T eps;
};

struct equal_floor
{
   equal_floor(){}
   template <class T>
   bool operator()(const T& a, const T& b)
   {
      BOOST_MATH_STD_USING
      return floor(a) == floor(b);
   }
};

struct equal_ceil
{
   equal_ceil(){}
   template <class T>
   bool operator()(const T& a, const T& b)
   {
      BOOST_MATH_STD_USING
      return ceil(a) == ceil(b);
   }
};

struct equal_nearest_integer
{
   equal_nearest_integer(){}
   template <class T>
   bool operator()(const T& a, const T& b)
   {
      BOOST_MATH_STD_USING
      return floor(a + 0.5f) == floor(b + 0.5f);
   }
};

namespace detail{

template <class F, class T>
void bracket(F f, T& a, T& b, T c, T& fa, T& fb, T& d, T& fd)
{
   //
   // Given a point c inside the existing enclosing interval
   // [a, b] sets a = c if f(c) == 0, otherwise finds the new 
   // enclosing interval: either [a, c] or [c, b] and sets
   // d and fd to the point that has just been removed from
   // the interval.  In other words d is the third best guess
   // to the root.
   //
   BOOST_MATH_STD_USING  // For ADL of std math functions
   T tol = tools::epsilon<T>() * 2;
   //
   // If the interval [a,b] is very small, or if c is too close 
   // to one end of the interval then we need to adjust the
   // location of c accordingly:
   //
   if((b - a) < 2 * tol * a)
   {
      c = a + (b - a) / 2;
   }
   else if(c <= a + fabs(a) * tol)
   {
      c = a + fabs(a) * tol;
   }
   else if(c >= b - fabs(b) * tol)
   {
      c = b - fabs(a) * tol;
   }
   //
   // OK, lets invoke f(c):
   //
   T fc = f(c);
   //
   // if we have a zero then we have an exact solution to the root:
   //
   if(fc == 0)
   {
      a = c;
      fa = 0;
      d = 0;
      fd = 0;
      return;
   }
   //
   // Non-zero fc, update the interval:
   //
   if(boost::math::sign(fa) * boost::math::sign(fc) < 0)
   {
      d = b;
      fd = fb;
      b = c;
      fb = fc;
   }
   else
   {
      d = a;
      fd = fa;
      a = c;
      fa= fc;
   }
}

template <class T>
inline T safe_div(T num, T denom, T r)
{
   //
   // return num / denom without overflow,
   // return r if overflow would occur.
   //
   BOOST_MATH_STD_USING  // For ADL of std math functions

   if(fabs(denom) < 1)
   {
      if(fabs(denom * tools::max_value<T>()) <= fabs(num))
         return r;
   }
   return num / denom;
}

template <class T>
inline T secant_interpolate(const T& a, const T& b, const T& fa, const T& fb)
{
   //
   // Performs standard secant interpolation of [a,b] given
   // function evaluations f(a) and f(b).  Performs a bisection
   // if secant interpolation would leave us very close to either
   // a or b.  Rationale: we only call this function when at least
   // one other form of interpolation has already failed, so we know
   // that the function is unlikely to be smooth with a root very
   // close to a or b.
   //
   BOOST_MATH_STD_USING  // For ADL of std math functions

   T tol = tools::epsilon<T>() * 5;
   T c = a - (fa / (fb - fa)) * (b - a);
   if((c <= a + fabs(a) * tol) || (c >= b - fabs(b) * tol))
      return (a + b) / 2;
   return c;
}

template <class T>
T quadratic_interpolate(const T& a, const T& b, T const& d,
                           const T& fa, const T& fb, T const& fd, 
                           unsigned count)
{
   //
   // Performs quadratic interpolation to determine the next point,
   // takes count Newton steps to find the location of the
   // quadratic polynomial.
   //
   // Point d must lie outside of the interval [a,b], it is the third
   // best approximation to the root, after a and b.
   //
   // Note: this does not guarentee to find a root
   // inside [a, b], so we fall back to a secant step should
   // the result be out of range.
   //
   // Start by obtaining the coefficients of the quadratic polynomial:
   //
   T B = safe_div(T(fb - fa), T(b - a), tools::max_value<T>());
   T A = safe_div(T(fd - fb), T(d - b), tools::max_value<T>());
   A = safe_div(T(A - B), T(d - a), T(0));

   if(a == 0)
   {
      // failure to determine coefficients, try a secant step:
      return secant_interpolate(a, b, fa, fb);
   }
   //
   // Determine the starting point of the Newton steps:
   //
   T c;
   if(boost::math::sign(A) * boost::math::sign(fa) > 0)
   {
      c = a;
   }
   else
   {
      c = b;
   }
   //
   // Take the Newton steps:
   //
   for(unsigned i = 1; i <= count; ++i)
   {
      //c -= safe_div(B * c, (B + A * (2 * c - a - b)), 1 + c - a);
      c -= safe_div(T(fa+(B+A*(c-b))*(c-a)), T(B + A * (2 * c - a - b)), T(1 + c - a));
   }
   if((c <= a) || (c >= b))
   {
      // Oops, failure, try a secant step:
      c = secant_interpolate(a, b, fa, fb);
   }
   return c;
}

template <class T>
T cubic_interpolate(const T& a, const T& b, const T& d, 
                    const T& e, const T& fa, const T& fb, 
                    const T& fd, const T& fe)
{
   //
   // Uses inverse cubic interpolation of f(x) at points 
   // [a,b,d,e] to obtain an approximate root of f(x).
   // Points d and e lie outside the interval [a,b]
   // and are the third and forth best approximations
   // to the root that we have found so far.
   //
   // Note: this does not guarentee to find a root
   // inside [a, b], so we fall back to quadratic
   // interpolation in case of an erroneous result.
   //
   BOOST_MATH_INSTRUMENT_CODE(" a = " << a << " b = " << b
      << " d = " << d << " e = " << e << " fa = " << fa << " fb = " << fb 
      << " fd = " << fd << " fe = " << fe);
   T q11 = (d - e) * fd / (fe - fd);
   T q21 = (b - d) * fb / (fd - fb);
   T q31 = (a - b) * fa / (fb - fa);
   T d21 = (b - d) * fd / (fd - fb);
   T d31 = (a - b) * fb / (fb - fa);
   BOOST_MATH_INSTRUMENT_CODE(
      "q11 = " << q11 << " q21 = " << q21 << " q31 = " << q31
      << " d21 = " << d21 << " d31 = " << d31);
   T q22 = (d21 - q11) * fb / (fe - fb);
   T q32 = (d31 - q21) * fa / (fd - fa);
   T d32 = (d31 - q21) * fd / (fd - fa);
   T q33 = (d32 - q22) * fa / (fe - fa);
   T c = q31 + q32 + q33 + a;
   BOOST_MATH_INSTRUMENT_CODE(
      "q22 = " << q22 << " q32 = " << q32 << " d32 = " << d32
      << " q33 = " << q33 << " c = " << c);

   if((c <= a) || (c >= b))
   {
      // Out of bounds step, fall back to quadratic interpolation:
      c = quadratic_interpolate(a, b, d, fa, fb, fd, 3);
   BOOST_MATH_INSTRUMENT_CODE(
      "Out of bounds interpolation, falling back to quadratic interpolation. c = " << c);
   }

   return c;
}

} // namespace detail

template <class F, class T, class Tol, class Policy>
std::pair<T, T> toms748_solve(F f, const T& ax, const T& bx, const T& fax, const T& fbx, Tol tol, boost::uintmax_t& max_iter, const Policy& pol)
{
   //
   // Main entry point and logic for Toms Algorithm 748
   // root finder.
   //
   BOOST_MATH_STD_USING  // For ADL of std math functions

   static const char* function = "boost::math::tools::toms748_solve<%1%>";

   boost::uintmax_t count = max_iter;
   T a, b, fa, fb, c, u, fu, a0, b0, d, fd, e, fe;
   static const T mu = 0.5f;

   // initialise a, b and fa, fb:
   a = ax;
   b = bx;
   if(a >= b)
      policies::raise_domain_error(
         function, 
         "Parameters a and b out of order: a=%1%", a, pol);
   fa = fax;
   fb = fbx;

   if(tol(a, b) || (fa == 0) || (fb == 0))
   {
      max_iter = 0;
      if(fa == 0)
         b = a;
      else if(fb == 0)
         a = b;
      return std::make_pair(a, b);
   }

   if(boost::math::sign(fa) * boost::math::sign(fb) > 0)
      policies::raise_domain_error(
         function, 
         "Parameters a and b do not bracket the root: a=%1%", a, pol);
   // dummy value for fd, e and fe:
   fe = e = fd = 1e5F;

   if(fa != 0)
   {
      //
      // On the first step we take a secant step:
      //
      c = detail::secant_interpolate(a, b, fa, fb);
      detail::bracket(f, a, b, c, fa, fb, d, fd);
      --count;
      BOOST_MATH_INSTRUMENT_CODE(" a = " << a << " b = " << b);

      if(count && (fa != 0) && !tol(a, b))
      {
         //
         // On the second step we take a quadratic interpolation:
         //
         c = detail::quadratic_interpolate(a, b, d, fa, fb, fd, 2);
         e = d;
         fe = fd;
         detail::bracket(f, a, b, c, fa, fb, d, fd);
         --count;
         BOOST_MATH_INSTRUMENT_CODE(" a = " << a << " b = " << b);
      }
   }

   while(count && (fa != 0) && !tol(a, b))
   {
      // save our brackets:
      a0 = a;
      b0 = b;
      //
      // Starting with the third step taken
      // we can use either quadratic or cubic interpolation.
      // Cubic interpolation requires that all four function values
      // fa, fb, fd, and fe are distinct, should that not be the case
      // then variable prof will get set to true, and we'll end up
      // taking a quadratic step instead.
      //
      T min_diff = tools::min_value<T>() * 32;
      bool prof = (fabs(fa - fb) < min_diff) || (fabs(fa - fd) < min_diff) || (fabs(fa - fe) < min_diff) || (fabs(fb - fd) < min_diff) || (fabs(fb - fe) < min_diff) || (fabs(fd - fe) < min_diff);
      if(prof)
      {
         c = detail::quadratic_interpolate(a, b, d, fa, fb, fd, 2);
         BOOST_MATH_INSTRUMENT_CODE("Can't take cubic step!!!!");
      }
      else
      {
         c = detail::cubic_interpolate(a, b, d, e, fa, fb, fd, fe);
      }
      //
      // re-bracket, and check for termination:
      //
      e = d;
      fe = fd;
      detail::bracket(f, a, b, c, fa, fb, d, fd);
      if((0 == --count) || (fa == 0) || tol(a, b))
         break;
      BOOST_MATH_INSTRUMENT_CODE(" a = " << a << " b = " << b);
      //
      // Now another interpolated step:
      //
      prof = (fabs(fa - fb) < min_diff) || (fabs(fa - fd) < min_diff) || (fabs(fa - fe) < min_diff) || (fabs(fb - fd) < min_diff) || (fabs(fb - fe) < min_diff) || (fabs(fd - fe) < min_diff);
      if(prof)
      {
         c = detail::quadratic_interpolate(a, b, d, fa, fb, fd, 3);
         BOOST_MATH_INSTRUMENT_CODE("Can't take cubic step!!!!");
      }
      else
      {
         c = detail::cubic_interpolate(a, b, d, e, fa, fb, fd, fe);
      }
      //
      // Bracket again, and check termination condition, update e:
      //
      detail::bracket(f, a, b, c, fa, fb, d, fd);
      if((0 == --count) || (fa == 0) || tol(a, b))
         break;
      BOOST_MATH_INSTRUMENT_CODE(" a = " << a << " b = " << b);
      //
      // Now we take a double-length secant step:
      //
      if(fabs(fa) < fabs(fb))
      {
         u = a;
         fu = fa;
      }
      else
      {
         u = b;
         fu = fb;
      }
      c = u - 2 * (fu / (fb - fa)) * (b - a);
      if(fabs(c - u) > (b - a) / 2)
      {
         c = a + (b - a) / 2;
      }
      //
      // Bracket again, and check termination condition:
      //
      e = d;
      fe = fd;
      detail::bracket(f, a, b, c, fa, fb, d, fd);
      BOOST_MATH_INSTRUMENT_CODE(" a = " << a << " b = " << b);
      BOOST_MATH_INSTRUMENT_CODE(" tol = " << T((fabs(a) - fabs(b)) / fabs(a)));
      if((0 == --count) || (fa == 0) || tol(a, b))
         break;
      //
      // And finally... check to see if an additional bisection step is 
      // to be taken, we do this if we're not converging fast enough:
      //
      if((b - a) < mu * (b0 - a0))
         continue;
      //
      // bracket again on a bisection:
      //
      e = d;
      fe = fd;
      detail::bracket(f, a, b, T(a + (b - a) / 2), fa, fb, d, fd);
      --count;
      BOOST_MATH_INSTRUMENT_CODE("Not converging: Taking a bisection!!!!");
      BOOST_MATH_INSTRUMENT_CODE(" a = " << a << " b = " << b);
   } // while loop

   max_iter -= count;
   if(fa == 0)
   {
      b = a;
   }
   else if(fb == 0)
   {
      a = b;
   }
   return std::make_pair(a, b);
}

template <class F, class T, class Tol>
inline std::pair<T, T> toms748_solve(F f, const T& ax, const T& bx, const T& fax, const T& fbx, Tol tol, boost::uintmax_t& max_iter)
{
   return toms748_solve(f, ax, bx, fax, fbx, tol, max_iter, policies::policy<>());
}

template <class F, class T, class Tol, class Policy>
inline std::pair<T, T> toms748_solve(F f, const T& ax, const T& bx, Tol tol, boost::uintmax_t& max_iter, const Policy& pol)
{
   max_iter -= 2;
   std::pair<T, T> r = toms748_solve(f, ax, bx, f(ax), f(bx), tol, max_iter, pol);
   max_iter += 2;
   return r;
}

template <class F, class T, class Tol>
inline std::pair<T, T> toms748_solve(F f, const T& ax, const T& bx, Tol tol, boost::uintmax_t& max_iter)
{
   return toms748_solve(f, ax, bx, tol, max_iter, policies::policy<>());
}

template <class F, class T, class Tol, class Policy>
std::pair<T, T> bracket_and_solve_root(F f, const T& guess, T factor, bool rising, Tol tol, boost::uintmax_t& max_iter, const Policy& pol)
{
   BOOST_MATH_STD_USING
   static const char* function = "boost::math::tools::bracket_and_solve_root<%1%>";
   //
   // Set up inital brackets:
   //
   T a = guess;
   T b = a;
   T fa = f(a);
   T fb = fa;
   //
   // Set up invocation count:
   //
   boost::uintmax_t count = max_iter - 1;

   if((fa < 0) == (guess < 0 ? !rising : rising))
   {
      //
      // Zero is to the right of b, so walk upwards
      // until we find it:
      //
      while((boost::math::sign)(fb) == (boost::math::sign)(fa))
      {
         if(count == 0)
            policies::raise_evaluation_error(function, "Unable to bracket root, last nearest value was %1%", b, pol);
         //
         // Heuristic: every 20 iterations we double the growth factor in case the
         // initial guess was *really* bad !
         //
         if((max_iter - count) % 20 == 0)
            factor *= 2;
         //
         // Now go ahead and move our guess by "factor":
         //
         a = b;
         fa = fb;
         b *= factor;
         fb = f(b);
         --count;
         BOOST_MATH_INSTRUMENT_CODE("a = " << a << " b = " << b << " fa = " << fa << " fb = " << fb << " count = " << count);
      }
   }
   else
   {
      //
      // Zero is to the left of a, so walk downwards
      // until we find it:
      //
      while((boost::math::sign)(fb) == (boost::math::sign)(fa))
      {
         if(fabs(a) < tools::min_value<T>())
         {
            // Escape route just in case the answer is zero!
            max_iter -= count;
            max_iter += 1;
            return a > 0 ? std::make_pair(T(0), T(a)) : std::make_pair(T(a), T(0)); 
         }
         if(count == 0)
            policies::raise_evaluation_error(function, "Unable to bracket root, last nearest value was %1%", a, pol);
         //
         // Heuristic: every 20 iterations we double the growth factor in case the
         // initial guess was *really* bad !
         //
         if((max_iter - count) % 20 == 0)
            factor *= 2;
         //
         // Now go ahead and move are guess by "factor":
         //
         b = a;
         fb = fa;
         a /= factor;
         fa = f(a);
         --count;
         BOOST_MATH_INSTRUMENT_CODE("a = " << a << " b = " << b << " fa = " << fa << " fb = " << fb << " count = " << count);
      }
   }
   max_iter -= count;
   max_iter += 1;
   std::pair<T, T> r = toms748_solve(
      f, 
      (a < 0 ? b : a), 
      (a < 0 ? a : b), 
      (a < 0 ? fb : fa), 
      (a < 0 ? fa : fb), 
      tol, 
      count, 
      pol);
   max_iter += count;
   BOOST_MATH_INSTRUMENT_CODE("max_iter = " << max_iter << " count = " << count);
   return r;
}

template <class F, class T, class Tol>
inline std::pair<T, T> bracket_and_solve_root(F f, const T& guess, const T& factor, bool rising, Tol tol, boost::uintmax_t& max_iter)
{
   return bracket_and_solve_root(f, guess, factor, rising, tol, max_iter, policies::policy<>());
}

} // namespace tools
} // namespace math
} // namespace boost


#endif // BOOST_MATH_TOOLS_SOLVE_ROOT_HPP