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Basics.cc
1 // Basics.cc is a part of the PYTHIA event generator.
2 // Copyright (C) 2012 Torbjorn Sjostrand.
3 // PYTHIA is licenced under the GNU GPL version 2, see COPYING for details.
4 // Please respect the MCnet Guidelines, see GUIDELINES for details.
5 
6 // Function definitions (not found in the header) for the Rndm, Vec4,
7 // RotBstMatrix and Hist classes, and some related global functions.
8 
9 #include "Basics.h"
10 
11 // Access time information.
12 #include <ctime>
13 
14 namespace Pythia8 {
15 
16 //==========================================================================
17 
18 // Rndm class.
19 // This class handles random number generation according to the
20 // Marsaglia-Zaman-Tsang algorithm
21 
22 //--------------------------------------------------------------------------
23 
24 // Constants: could be changed here if desired, but normally should not.
25 // These are of technical nature, as described for each.
26 
27 // The default seed, i.e. the Marsaglia-Zaman random number sequence.
28 const int Rndm::DEFAULTSEED = 19780503;
29 
30 //--------------------------------------------------------------------------
31 
32 // Method to pass in pointer for external random number generation.
33 
34 bool Rndm::rndmEnginePtr( RndmEngine* rndmEngPtrIn) {
35 
36  // Save pointer.
37  if (rndmEngPtrIn == 0) return false;
38  rndmEngPtr = rndmEngPtrIn;
39  useExternalRndm = true;
40 
41  // Done.
42  return true;
43 
44 }
45 
46 //--------------------------------------------------------------------------
47 
48 // Initialize, normally at construction or in first call.
49 
50 void Rndm::init(int seedIn) {
51 
52  // Pick seed in convenient way. Assure it to be non-negative.
53  int seed = seedIn;
54  if (seedIn < 0) seed = DEFAULTSEED;
55  else if (seedIn == 0) seed = int(time(0));
56  if (seed < 0) seed = -seed;
57 
58  // Unpack seed.
59  int ij = (seed/30082) % 31329;
60  int kl = seed % 30082;
61  int i = (ij/177) % 177 + 2;
62  int j = ij % 177 + 2;
63  int k = (kl/169) % 178 + 1;
64  int l = kl % 169;
65 
66  // Initialize random number array.
67  for (int ii = 0; ii < 97; ++ii) {
68  double s = 0.;
69  double t = 0.5;
70  for (int jj = 0; jj < 48; ++jj) {
71  int m = (( (i*j)%179 )*k) % 179;
72  i = j;
73  j = k;
74  k = m;
75  l = (53*l+1) % 169;
76  if ( (l*m) % 64 >= 32) s += t;
77  t *= 0.5;
78  }
79  u[ii] = s;
80  }
81 
82  // Initialize other variables.
83  double twom24 = 1.;
84  for (int i24 = 0; i24 < 24; ++i24) twom24 *= 0.5;
85  c = 362436. * twom24;
86  cd = 7654321. * twom24;
87  cm = 16777213. * twom24;
88  i97 = 96;
89  j97 = 32;
90 
91  // Finished.
92  initRndm = true;
93  seedSave = seed;
94  sequence = 0;
95 
96 }
97 
98 //--------------------------------------------------------------------------
99 
100 // Generate next random number uniformly between 0 and 1.
101 
102 double Rndm::flat() {
103 
104  // Use external random number generator if such has been linked.
105  if (useExternalRndm) return rndmEngPtr->flat();
106 
107  // Ensure that already initialized.
108  if (!initRndm) init(DEFAULTSEED);
109 
110  // Find next random number and update saved state.
111  ++sequence;
112  double uni;
113  do {
114  uni = u[i97] - u[j97];
115  if (uni < 0.) uni += 1.;
116  u[i97] = uni;
117  if (--i97 < 0) i97 = 96;
118  if (--j97 < 0) j97 = 96;
119  c -= cd;
120  if (c < 0.) c += cm;
121  uni -= c;
122  if(uni < 0.) uni += 1.;
123  } while (uni <= 0. || uni >= 1.);
124  return uni;
125 
126 }
127 
128 //--------------------------------------------------------------------------
129 
130 // Pick one option among vector of (positive) probabilities.
131 
132 int Rndm::pick(const vector<double>& prob) {
133 
134  double work = 0.;
135  for (int i = 0; i < int(prob.size()); ++i) work += prob[i];
136  work *= flat();
137  int index = -1;
138  do work -= prob[++index];
139  while (work > 0. && index < int(prob.size()));
140  return index;
141 
142 }
143 
144 //--------------------------------------------------------------------------
145 
146 // Save current state of the random number generator to a binary file.
147 
148 bool Rndm::dumpState(string fileName) {
149 
150  // Open file as output stream.
151  const char* fn = fileName.c_str();
152  ofstream ofs(fn, ios::binary);
153 
154  if (!ofs.good()) {
155  cout << " Rndm::dumpState: could not open output file" << endl;
156  return false;
157  }
158 
159  // Write the state of the generator on the file.
160  ofs.write((char *) &seedSave, sizeof(int));
161  ofs.write((char *) &sequence, sizeof(long));
162  ofs.write((char *) &i97, sizeof(int));
163  ofs.write((char *) &j97, sizeof(int));
164  ofs.write((char *) &c, sizeof(double));
165  ofs.write((char *) &cd, sizeof(double));
166  ofs.write((char *) &cm, sizeof(double));
167  ofs.write((char *) &u, sizeof(double) * 97);
168 
169  // Write confirmation on cout.
170  cout << " PYTHIA Rndm::dumpState: seed = " << seedSave
171  << ", sequence no = " << sequence << endl;
172  return true;
173 
174 }
175 
176 //--------------------------------------------------------------------------
177 
178 // Read in the state of the random number generator from a binary file.
179 
180 bool Rndm::readState(string fileName) {
181 
182  // Open file as input stream.
183  const char* fn = fileName.c_str();
184  ifstream ifs(fn, ios::binary);
185 
186  if (!ifs.good()) {
187  cout << " Rndm::readState: could not open input file" << endl;
188  return false;
189  }
190 
191  // Read the state of the generator from the file.
192  ifs.read((char *) &seedSave, sizeof(int));
193  ifs.read((char *) &sequence, sizeof(long));
194  ifs.read((char *) &i97, sizeof(int));
195  ifs.read((char *) &j97, sizeof(int));
196  ifs.read((char *) &c, sizeof(double));
197  ifs.read((char *) &cd, sizeof(double));
198  ifs.read((char *) &cm, sizeof(double));
199  ifs.read((char *) &u, sizeof(double) *97);
200 
201  // Write confirmation on cout.
202  cout << " PYTHIA Rndm::readState: seed " << seedSave
203  << ", sequence no = " << sequence << endl;
204  return true;
205 
206 }
207 
208 //==========================================================================
209 
210 // Vec4 class.
211 // This class implements four-vectors, in energy-momentum space.
212 // (But could also be used to hold space-time four-vectors.)
213 
214 //--------------------------------------------------------------------------
215 
216 // Constants: could be changed here if desired, but normally should not.
217 // These are of technical nature, as described for each.
218 
219 // Small number to avoid division by zero.
220 const double Vec4::TINY = 1e-20;
221 
222 //--------------------------------------------------------------------------
223 
224 // Rotation (simple).
225 
226 void Vec4::rot(double thetaIn, double phiIn) {
227 
228  double cthe = cos(thetaIn);
229  double sthe = sin(thetaIn);
230  double cphi = cos(phiIn);
231  double sphi = sin(phiIn);
232  double tmpx = cthe * cphi * xx - sphi * yy + sthe * cphi * zz;
233  double tmpy = cthe * sphi * xx + cphi * yy + sthe * sphi * zz;
234  double tmpz = -sthe * xx + cthe * zz;
235  xx = tmpx;
236  yy = tmpy;
237  zz = tmpz;
238 
239 }
240 
241 //--------------------------------------------------------------------------
242 
243 // Azimuthal rotation phi around an arbitrary axis (nz, ny, nz).
244 
245 void Vec4::rotaxis(double phiIn, double nx, double ny, double nz) {
246 
247  double norm = 1./sqrt(nx*nx + ny*ny + nz*nz);
248  nx *= norm;
249  ny *= norm;
250  nz *= norm;
251  double cphi = cos(phiIn);
252  double sphi = sin(phiIn);
253  double comb = (nx * xx + ny * yy + nz * zz) * (1. - cphi);
254  double tmpx = cphi * xx + comb * nx + sphi * (ny * zz - nz * yy);
255  double tmpy = cphi * yy + comb * ny + sphi * (nz * xx - nx * zz);
256  double tmpz = cphi * zz + comb * nz + sphi * (nx * yy - ny * xx);
257  xx = tmpx;
258  yy = tmpy;
259  zz = tmpz;
260 
261 }
262 
263 //--------------------------------------------------------------------------
264 
265 // Azimuthal rotation phi around an arbitrary (3-vector component of) axis.
266 
267 void Vec4::rotaxis(double phiIn, const Vec4& n) {
268 
269  double nx = n.xx;
270  double ny = n.yy;
271  double nz = n.zz;
272  double norm = 1./sqrt(nx*nx + ny*ny + nz*nz);
273  nx *= norm;
274  ny *=norm;
275  nz *=norm;
276  double cphi = cos(phiIn);
277  double sphi = sin(phiIn);
278  double comb = (nx * xx + ny * yy + nz * zz) * (1. - cphi);
279  double tmpx = cphi * xx + comb * nx + sphi * (ny * zz - nz * yy);
280  double tmpy = cphi * yy + comb * ny + sphi * (nz * xx - nx * zz);
281  double tmpz = cphi * zz + comb * nz + sphi * (nx * yy - ny * xx);
282  xx = tmpx;
283  yy = tmpy;
284  zz = tmpz;
285 
286 }
287 
288 //--------------------------------------------------------------------------
289 
290 // Boost (simple).
291 
292 void Vec4::bst(double betaX, double betaY, double betaZ) {
293 
294  double beta2 = betaX*betaX + betaY*betaY + betaZ*betaZ;
295  double gamma = 1. / sqrt(1. - beta2);
296  double prod1 = betaX * xx + betaY * yy + betaZ * zz;
297  double prod2 = gamma * (gamma * prod1 / (1. + gamma) + tt);
298  xx += prod2 * betaX;
299  yy += prod2 * betaY;
300  zz += prod2 * betaZ;
301  tt = gamma * (tt + prod1);
302 
303 }
304 
305 //--------------------------------------------------------------------------
306 
307 // Boost (simple, given gamma).
308 
309 void Vec4::bst(double betaX, double betaY, double betaZ, double gamma) {
310 
311  double prod1 = betaX * xx + betaY * yy + betaZ * zz;
312  double prod2 = gamma * (gamma * prod1 / (1. + gamma) + tt);
313  xx += prod2 * betaX;
314  yy += prod2 * betaY;
315  zz += prod2 * betaZ;
316  tt = gamma * (tt + prod1);
317 
318 }
319 
320 //--------------------------------------------------------------------------
321 
322 // Boost given by a Vec4 p.
323 
324 void Vec4::bst(const Vec4& pIn) {
325 
326  double betaX = pIn.xx / pIn.tt;
327  double betaY = pIn.yy / pIn.tt;
328  double betaZ = pIn.zz / pIn.tt;
329  double beta2 = betaX*betaX + betaY*betaY + betaZ*betaZ;
330  double gamma = 1. / sqrt(1. - beta2);
331  double prod1 = betaX * xx + betaY * yy + betaZ * zz;
332  double prod2 = gamma * (gamma * prod1 / (1. + gamma) + tt);
333  xx += prod2 * betaX;
334  yy += prod2 * betaY;
335  zz += prod2 * betaZ;
336  tt = gamma * (tt + prod1);
337 
338 }
339 
340 //--------------------------------------------------------------------------
341 
342 // Boost given by a Vec4 p and double m.
343 
344 void Vec4::bst(const Vec4& pIn, double mIn) {
345 
346  double betaX = pIn.xx / pIn.tt;
347  double betaY = pIn.yy / pIn.tt;
348  double betaZ = pIn.zz / pIn.tt;
349  double gamma = pIn.tt / mIn;
350  double prod1 = betaX * xx + betaY * yy + betaZ * zz;
351  double prod2 = gamma * (gamma * prod1 / (1. + gamma) + tt);
352  xx += prod2 * betaX;
353  yy += prod2 * betaY;
354  zz += prod2 * betaZ;
355  tt = gamma * (tt + prod1);
356 
357 }
358 
359 //--------------------------------------------------------------------------
360 
361 // Boost given by a Vec4 p; boost in opposite direction.
362 
363 void Vec4::bstback(const Vec4& pIn) {
364 
365  double betaX = -pIn.xx / pIn.tt;
366  double betaY = -pIn.yy / pIn.tt;
367  double betaZ = -pIn.zz / pIn.tt;
368  double beta2 = betaX*betaX + betaY*betaY + betaZ*betaZ;
369  double gamma = 1. / sqrt(1. - beta2);
370  double prod1 = betaX * xx + betaY * yy + betaZ * zz;
371  double prod2 = gamma * (gamma * prod1 / (1. + gamma) + tt);
372  xx += prod2 * betaX;
373  yy += prod2 * betaY;
374  zz += prod2 * betaZ;
375  tt = gamma * (tt + prod1);
376 
377 }
378 
379 //--------------------------------------------------------------------------
380 
381 // Boost given by a Vec4 p and double m; boost in opposite direction.
382 
383 void Vec4::bstback(const Vec4& pIn, double mIn) {
384 
385  double betaX = -pIn.xx / pIn.tt;
386  double betaY = -pIn.yy / pIn.tt;
387  double betaZ = -pIn.zz / pIn.tt;
388  double gamma = pIn.tt / mIn;
389  double prod1 = betaX * xx + betaY * yy + betaZ * zz;
390  double prod2 = gamma * (gamma * prod1 / (1. + gamma) + tt);
391  xx += prod2 * betaX;
392  yy += prod2 * betaY;
393  zz += prod2 * betaZ;
394  tt = gamma * (tt + prod1);
395 
396 }
397 
398 //--------------------------------------------------------------------------
399 
400 // Arbitrary combination of rotations and boosts defined by 4 * 4 matrix.
401 
402 void Vec4::rotbst(const RotBstMatrix& M) {
403 
404  double x = xx; double y = yy; double z = zz; double t = tt;
405  tt = M.M[0][0] * t + M.M[0][1] * x + M.M[0][2] * y + M.M[0][3] * z;
406  xx = M.M[1][0] * t + M.M[1][1] * x + M.M[1][2] * y + M.M[1][3] * z;
407  yy = M.M[2][0] * t + M.M[2][1] * x + M.M[2][2] * y + M.M[2][3] * z;
408  zz = M.M[3][0] * t + M.M[3][1] * x + M.M[3][2] * y + M.M[3][3] * z;
409 
410 }
411 
412 //--------------------------------------------------------------------------
413 
414 // The invariant mass of two four-vectors.
415 
416 double m(const Vec4& v1, const Vec4& v2) {
417  double m2 = pow2(v1.tt + v2.tt) - pow2(v1.xx + v2.xx)
418  - pow2(v1.yy + v2.yy) - pow2(v1.zz + v2.zz);
419  return (m2 > 0.) ? sqrt(m2) : 0.;
420 }
421 
422 //--------------------------------------------------------------------------
423 
424 // The squared invariant mass of two four-vectors.
425 
426 double m2(const Vec4& v1, const Vec4& v2) {
427  double m2 = pow2(v1.tt + v2.tt) - pow2(v1.xx + v2.xx)
428  - pow2(v1.yy + v2.yy) - pow2(v1.zz + v2.zz);
429  return m2;
430 }
431 
432 //--------------------------------------------------------------------------
433 
434 // The scalar product of two three-vectors.
435 
436 double dot3(const Vec4& v1, const Vec4& v2) {
437  return v1.xx*v2.xx + v1.yy*v2.yy + v1.zz*v2.zz;
438 }
439 
440 //--------------------------------------------------------------------------
441 
442 // The cross product of two three-vectors.
443 
444 Vec4 cross3(const Vec4& v1, const Vec4& v2) {
445  Vec4 v;
446  v.xx = v1.yy * v2.zz - v1.zz * v2.yy;
447  v.yy = v1.zz * v2.xx - v1.xx * v2.zz;
448  v.zz = v1.xx * v2.yy - v1.yy * v2.xx; return v;
449 }
450 
451 //--------------------------------------------------------------------------
452 
453 // Opening angle between two three-vectors.
454 
455 double theta(const Vec4& v1, const Vec4& v2) {
456  double cthe = (v1.xx * v2.xx + v1.yy * v2.yy + v1.zz * v2.zz)
457  / sqrt( (v1.xx*v1.xx + v1.yy*v1.yy + v1.zz*v1.zz)
458  * (v2.xx*v2.xx + v2.yy*v2.yy + v2.zz*v2.zz) );
459  cthe = max(-1., min(1., cthe));
460  return acos(cthe);
461 }
462 
463 //--------------------------------------------------------------------------
464 
465 // Cosine of the opening angle between two three-vectors.
466 
467 double costheta(const Vec4& v1, const Vec4& v2) {
468  double cthe = (v1.xx * v2.xx + v1.yy * v2.yy + v1.zz * v2.zz)
469  / sqrt( (v1.xx*v1.xx + v1.yy*v1.yy + v1.zz*v1.zz)
470  * (v2.xx*v2.xx + v2.yy*v2.yy + v2.zz*v2.zz) );
471  cthe = max(-1., min(1., cthe));
472  return cthe;
473 }
474 
475 //--------------------------------------------------------------------------
476 
477 // Azimuthal angle between two three-vectors.
478 
479 double phi(const Vec4& v1, const Vec4& v2) {
480  double cphi = (v1.xx * v2.xx + v1.yy * v2.yy) / sqrt( max( Vec4::TINY,
481  (v1.xx*v1.xx + v1.yy*v1.yy) * (v2.xx*v2.xx + v2.yy*v2.yy) ));
482  cphi = max(-1., min(1., cphi));
483  return acos(cphi);
484 }
485 
486 //--------------------------------------------------------------------------
487 
488 // Cosine of the azimuthal angle between two three-vectors.
489 
490 double cosphi(const Vec4& v1, const Vec4& v2) {
491  double cphi = (v1.xx * v2.xx + v1.yy * v2.yy) / sqrt( max( Vec4::TINY,
492  (v1.xx*v1.xx + v1.yy*v1.yy) * (v2.xx*v2.xx + v2.yy*v2.yy) ));
493  cphi = max(-1., min(1., cphi));
494  return cphi;
495 }
496 
497 //--------------------------------------------------------------------------
498 
499 // Azimuthal angle between two three-vectors around a third.
500 
501 double phi(const Vec4& v1, const Vec4& v2, const Vec4& n) {
502  double nx = n.xx; double ny = n.yy; double nz = n.zz;
503  double norm = 1. / sqrt(nx*nx + ny*ny + nz*nz);
504  nx *= norm; ny *=norm; nz *=norm;
505  double v1s = v1.xx * v1.xx + v1.yy * v1.yy + v1.zz * v1.zz;
506  double v2s = v2.xx * v2.xx + v2.yy * v2.yy + v2.zz * v2.zz;
507  double v1v2 = v1.xx * v2.xx + v1.yy * v2.yy + v1.zz * v2.zz;
508  double v1n = v1.xx * nx + v1.yy * ny + v1.zz * nz;
509  double v2n = v2.xx * nx + v2.yy * ny + v2.zz * nz;
510  double cphi = (v1v2 - v1n * v2n) / sqrt( max( Vec4::TINY,
511  (v1s - v1n*v1n) * (v2s - v2n*v2n) ));
512  cphi = max(-1., min(1., cphi));
513  return acos(cphi);
514 }
515 
516 //--------------------------------------------------------------------------
517 
518 // Cosine of the azimuthal angle between two three-vectors around a third.
519 
520 double cosphi(const Vec4& v1, const Vec4& v2, const Vec4& n) {
521  double nx = n.xx; double ny = n.yy; double nz = n.zz;
522  double norm = 1. / sqrt(nx*nx + ny*ny + nz*nz);
523  nx *= norm; ny *=norm; nz *=norm;
524  double v1s = v1.xx * v1.xx + v1.yy * v1.yy + v1.zz * v1.zz;
525  double v2s = v2.xx * v2.xx + v2.yy * v2.yy + v2.zz * v2.zz;
526  double v1v2 = v1.xx * v2.xx + v1.yy * v2.yy + v1.zz * v2.zz;
527  double v1n = v1.xx * nx + v1.yy * ny + v1.zz * nz;
528  double v2n = v2.xx * nx + v2.yy * ny + v2.zz * nz;
529  double cphi = (v1v2 - v1n * v2n) / sqrt( max( Vec4::TINY,
530  (v1s - v1n*v1n) * (v2s - v2n*v2n) ));
531  cphi = max(-1., min(1., cphi));
532  return cphi;
533 }
534 
535 //--------------------------------------------------------------------------
536 
537 // Print a four-vector: also operator overloading with friend.
538 
539 ostream& operator<<(ostream& os, const Vec4& v) {
540  os << fixed << setprecision(3) << " " << setw(9) << v.xx << " "
541  << setw(9) << v.yy << " " << setw(9) << v.zz << " " << setw(9)
542  << v.tt << " (" << setw(9) << v.mCalc() << ")\n";
543  return os;
544 }
545 
546 //==========================================================================
547 
548 // RotBstMatrix class.
549 // This class implements 4 * 4 matrices that encode an arbitrary combination
550 // of rotations and boosts, that can be applied to Vec4 four-vectors.
551 
552 //--------------------------------------------------------------------------
553 
554 // Constants: could be changed here if desired, but normally should not.
555 // These are of technical nature, as described for each.
556 
557 // Small number to avoid division by zero.
558 const double RotBstMatrix::TINY = 1e-20;
559 
560 //--------------------------------------------------------------------------
561 
562 // Rotate by polar angle theta and azimuthal angle phi.
563 
564 void RotBstMatrix::rot(double theta, double phi) {
565 
566  // Set up rotation matrix.
567  double cthe = cos(theta);
568  double sthe = sin(theta);
569  double cphi = cos(phi);
570  double sphi = sin(phi);
571  double Mrot[4][4] = {
572  {1., 0., 0., 0.},
573  {0., cthe * cphi, - sphi, sthe * cphi},
574  {0., cthe * sphi, cphi, sthe * sphi},
575  {0., -sthe, 0., cthe } };
576 
577  // Rotate current matrix accordingly.
578  double Mtmp[4][4];
579  for (int i = 0; i < 4; ++i)
580  for (int j = 0; j < 4; ++j)
581  Mtmp[i][j] = M[i][j];
582  for (int i = 0; i < 4; ++i)
583  for (int j = 0; j < 4; ++j)
584  M[i][j] = Mrot[i][0] * Mtmp[0][j] + Mrot[i][1] * Mtmp[1][j]
585  + Mrot[i][2] * Mtmp[2][j] + Mrot[i][3] * Mtmp[3][j];
586 
587 }
588 
589 //--------------------------------------------------------------------------
590 
591 // Rotate so that vector originally along z axis becomes parallel with p.
592 
593 void RotBstMatrix::rot(const Vec4& p) {
594 
595  double theta = p.theta();
596  double phi = p.phi();
597  rot(0., -phi);
598  rot(theta, phi);
599 
600 }
601 
602 //--------------------------------------------------------------------------
603 
604 // Boost with velocity vector (betaX, betaY, betaZ).
605 
606 void RotBstMatrix::bst(double betaX, double betaY, double betaZ) {
607 
608  // Set up boost matrix.
609  double gm = 1. / sqrt( max( TINY, 1. - betaX*betaX - betaY*betaY
610  - betaZ*betaZ ) );
611  double gf = gm*gm / (1. + gm);
612  double Mbst[4][4] = {
613  { gm, gm*betaX, gm*betaY, gm*betaZ },
614  { gm*betaX, 1. + gf*betaX*betaX, gf*betaX*betaY, gf*betaX*betaZ },
615  { gm*betaY, gf*betaY*betaX, 1. + gf*betaY*betaY, gf*betaY*betaZ },
616  { gm*betaZ, gf*betaZ*betaX, gf*betaZ*betaY, 1. + gf*betaZ*betaZ } };
617 
618  // Boost current matrix correspondingly.
619  double Mtmp[4][4];
620  for (int i = 0; i < 4; ++i)
621  for (int j = 0; j < 4; ++j)
622  Mtmp[i][j] = M[i][j];
623  for (int i = 0; i < 4; ++i)
624  for (int j = 0; j < 4; ++j)
625  M[i][j] = Mbst[i][0] * Mtmp[0][j] + Mbst[i][1] * Mtmp[1][j]
626  + Mbst[i][2] * Mtmp[2][j] + Mbst[i][3] * Mtmp[3][j];
627 
628 }
629 
630 //--------------------------------------------------------------------------
631 
632 // Boost so that vector originally at rest obtains same velocity as p.
633 
634 void RotBstMatrix::bst(const Vec4& p) {
635  double betaX = p.px() / p.e();
636  double betaY = p.py() / p.e();
637  double betaZ = p.pz() / p.e();
638  bst(betaX, betaY, betaZ);
639 }
640 
641 //--------------------------------------------------------------------------
642 
643 // Boost so vector originally with same velocity as p is brought to rest.
644 
645 void RotBstMatrix::bstback(const Vec4& p) {
646  double betaX = -p.px() / p.e();
647  double betaY = -p.py() / p.e();
648  double betaZ = -p.pz() / p.e();
649  bst(betaX, betaY, betaZ);
650 }
651 
652 //--------------------------------------------------------------------------
653 
654 // Boost that transforms p1 to p2, where p1^2 = p2^2 is assumed.
655 
656 void RotBstMatrix::bst(const Vec4& p1, const Vec4& p2) {
657  double eSum = p1.e() + p2.e();
658  double betaX = (p2.px() - p1.px()) / eSum;
659  double betaY = (p2.py() - p1.py()) / eSum;
660  double betaZ = (p2.pz() - p1.pz()) / eSum;
661  double fac = 2. / (1. + betaX*betaX + betaY*betaY + betaZ*betaZ);
662  betaX *= fac; betaY *= fac; betaZ *= fac;
663  bst(betaX, betaY, betaZ);
664 }
665 
666 //--------------------------------------------------------------------------
667 
668 // Boost and rotation that transforms from p1 and p2
669 // to their rest frame with p1 along +z axis.
670 
671 void RotBstMatrix::toCMframe(const Vec4& p1, const Vec4& p2) {
672  Vec4 pSum = p1 + p2;
673  Vec4 dir = p1;
674  dir.bstback(pSum);
675  double theta = dir.theta();
676  double phi = dir.phi();
677  bstback(pSum);
678  rot(0., -phi);
679  rot(-theta, phi);
680 }
681 
682 //--------------------------------------------------------------------------
683 
684 // Rotation and boost that transforms from rest frame of p1 and p2
685 // with p1 along +z axis to actual frame of p1 and p2. (Inverse of above.)
686 
687 void RotBstMatrix::fromCMframe(const Vec4& p1, const Vec4& p2) {
688  Vec4 pSum = p1 + p2;
689  Vec4 dir = p1;
690  dir.bstback(pSum);
691  double theta = dir.theta();
692  double phi = dir.phi();
693  rot(0., -phi);
694  rot(theta, phi);
695  bst(pSum);
696 }
697 
698 //--------------------------------------------------------------------------
699 
700 // Combine existing rotation/boost matrix with another one.
701 
702 void RotBstMatrix::rotbst(const RotBstMatrix& Mrb) {
703  double Mtmp[4][4];
704  for (int i = 0; i < 4; ++i)
705  for (int j = 0; j < 4; ++j)
706  Mtmp[i][j] = M[i][j];
707  for (int i = 0; i < 4; ++i)
708  for (int j = 0; j < 4; ++j)
709  M[i][j] = Mrb.M[i][0] * Mtmp[0][j] + Mrb.M[i][1] * Mtmp[1][j]
710  + Mrb.M[i][2] * Mtmp[2][j] + Mrb.M[i][3] * Mtmp[3][j];
711 }
712 
713 //--------------------------------------------------------------------------
714 
715 // Invert the rotation and boost.
716 
717 void RotBstMatrix::invert() {
718  double Mtmp[4][4];
719  for (int i = 0; i < 4; ++i)
720  for (int j = 0; j < 4; ++j)
721  Mtmp[i][j] = M[i][j];
722  for (int i = 0; i < 4; ++i)
723  for (int j = 0; j < 4; ++j)
724  M[i][j] = ( (i == 0 && j > 0) || (i > 0 && j == 0) )
725  ? - Mtmp[j][i] : Mtmp[j][i];
726 }
727 
728 //--------------------------------------------------------------------------
729 
730 // Reset to diagonal matrix.
731 
732 void RotBstMatrix::reset() {
733  for (int i = 0; i < 4; ++i)
734  for (int j = 0; j < 4; ++j)
735  M[i][j] = (i==j) ? 1. : 0.;
736 }
737 
738 //--------------------------------------------------------------------------
739 
740 // Crude estimate deviation from unit matrix.
741 
742 double RotBstMatrix::deviation() const {
743  double devSum = 0.;
744  for (int i = 0; i < 4; ++i)
745  for (int j = 0; j < 4; ++j)
746  devSum += (i==j) ? abs(M[i][j] - 1.) : abs(M[i][j]);
747  return devSum;
748 }
749 
750 //--------------------------------------------------------------------------
751 
752 // Print a rotation and boost matrix: operator overloading with friend.
753 
754 ostream& operator<<(ostream& os, const RotBstMatrix& M) {
755  os << fixed << setprecision(5) << " Rotation/boost matrix: \n";
756  for (int i = 0; i <4; ++i)
757  os << setw(10) << M.M[i][0] << setw(10) << M.M[i][1]
758  << setw(10) << M.M[i][2] << setw(10) << M.M[i][3] << "\n";
759  return os;
760 }
761 
762 //==========================================================================
763 
764 // Hist class.
765 // This class handles a single histogram at a time
766 // (or a vector of histograms).
767 
768 //--------------------------------------------------------------------------
769 
770 // Constants: could be changed here if desired, but normally should not.
771 // These are of technical nature, as described for each.
772 
773 // Maximum number of bins in a histogram.
774 const int Hist::NBINMAX = 1000;
775 
776 // Maximum number of columns that can be printed for a histogram.
777 const int Hist::NCOLMAX = 100;
778 
779 // Maximum number of lines a histogram can use at output.
780 const int Hist::NLINES = 30;
781 
782 // Tolerance in deviation of xMin and xMax between two histograms.
783 const double Hist::TOLERANCE = 0.001;
784 
785 // Small number to avoid division by zero.
786 const double Hist::TINY = 1e-20;
787 
788 // When minbin/maxbin < SMALLFRAC the y scale goes down to zero.
789 const double Hist::SMALLFRAC = 0.1;
790 
791 // Constants for printout: fixed steps on y scale; filling characters.
792 const double DYAC[] = {0.04, 0.05, 0.06, 0.08, 0.10,
793  0.12, 0.15, 0.20, 0.25, 0.30};
794 const char NUMBER[] = {'0', '1', '2', '3', '4', '5',
795  '6', '7', '8', '9', 'X' };
796 
797 //--------------------------------------------------------------------------
798 
799 // Book a histogram.
800 
801 void Hist::book(string titleIn, int nBinIn, double xMinIn,
802  double xMaxIn) {
803 
804  title = titleIn;
805  nBin = nBinIn;
806  if (nBinIn < 1) nBin = 1;
807  if (nBinIn > NBINMAX) nBin = NBINMAX;
808  xMin = xMinIn;
809  xMax = xMaxIn;
810  dx = (xMax - xMin)/nBin;
811  res.resize(nBin);
812  null();
813 
814 }
815 
816 //--------------------------------------------------------------------------
817 
818 // Reset bin contents.
819 
820 void Hist::null() {
821 
822  nFill = 0;
823  under = 0.;
824  inside = 0.;
825  over = 0.;
826  for (int ix = 0; ix < nBin; ++ix) res[ix] = 0.;
827 
828 }
829 
830 //--------------------------------------------------------------------------
831 
832 // Fill bin with weight.
833 
834 void Hist::fill(double x, double w) {
835 
836  ++nFill;
837  int iBin = int(floor((x - xMin)/dx));
838  if (iBin < 0) under += w;
839  else if (iBin >= nBin) over += w;
840  else {inside += w; res[iBin] += w; }
841 
842 }
843 
844 //--------------------------------------------------------------------------
845 
846 // Print a histogram: also operator overloading with friend.
847 
848 ostream& operator<<(ostream& os, const Hist& h) {
849 
850  // Do not print empty histograms.
851  if (h.nFill <= 0) return os;
852 
853  // Write time and title.
854  time_t t = time(0);
855  char date[18];
856  strftime(date,18,"%Y-%m-%d %H:%M",localtime(&t));
857  os << "\n\n " << date << " " << h.title << "\n\n";
858 
859  // Group bins, where required, to make printout have fewer columns.
860  int nGroup = 1 + (h.nBin - 1) / Hist::NCOLMAX;
861  int nCol = 1 + (h.nBin - 1) / nGroup;
862  vector<double> resCol(nCol);
863  for (int iCol = 0; iCol < nCol; ++iCol) {
864  resCol[iCol] = 0.;
865  for (int ix = nGroup * iCol; ix < min( h.nBin, nGroup * (iCol + 1)); ++ix)
866  resCol[iCol] += h.res[ix];
867  }
868 
869  // Find minimum and maximum bin content.
870  double yMin = resCol[0];
871  double yMax = resCol[0];
872  for (int iCol = 1; iCol < nCol; ++iCol) {
873  if (resCol[iCol] < yMin) yMin = resCol[iCol];
874  if (resCol[iCol] > yMax) yMax = resCol[iCol];
875  }
876 
877  // Determine scale and step size for y axis.
878  if (yMax - yMin > Hist::NLINES * DYAC[0] * 1e-9) {
879  if (yMin > 0. && yMin < Hist::SMALLFRAC * yMax) yMin = 0.;
880  if (yMax < 0. && yMax > Hist::SMALLFRAC * yMin) yMax = 0.;
881  int iPowY = int(floor( log10(yMax - yMin) ));
882  if (yMax - yMin < Hist::NLINES * DYAC[0] * pow(10.,iPowY))
883  iPowY = iPowY - 1;
884  if (yMax - yMin > Hist::NLINES * DYAC[9] * pow(10.,iPowY))
885  iPowY = iPowY + 1;
886  double nLinePow = Hist::NLINES * pow(10.,iPowY);
887  double delY = DYAC[0];
888  for (int idel = 0; idel < 9; ++idel)
889  if (yMax - yMin >= nLinePow * DYAC[idel]) delY = DYAC[idel+1];
890  double dy = delY * pow(10.,iPowY);
891 
892  // Convert bin contents to integer form; fractional fill in top row.
893  vector<int> row(nCol);
894  vector<int> frac(nCol);
895  for (int iCol = 0; iCol < nCol ; ++iCol) {
896  double cta = abs(resCol[iCol]) / dy;
897  row[iCol] = int(cta + 0.95);
898  if(resCol[iCol] < 0.) row[iCol] = - row[iCol];
899  frac[iCol] = int(10. * (cta + 1.05 - floor(cta + 0.95)));
900  }
901  int rowMin = int(abs(yMin)/dy + 0.95);
902  if ( yMin < 0) rowMin = - rowMin;
903  int rowMax = int(abs(yMax)/dy + 0.95);
904  if ( yMax < 0) rowMax = - rowMax;
905 
906  // Print histogram row by row.
907  os << fixed << setprecision(2);
908  for (int iRow = rowMax; iRow >= rowMin; iRow--) if (iRow != 0) {
909  os << " " << setw(10) << iRow*delY << "*10^"
910  << setw(2) << iPowY << " ";
911  for (int iCol = 0; iCol < nCol ; ++iCol) {
912  if (iRow == row[iCol]) os << NUMBER[frac[iCol]];
913  else if (iRow * (row[iCol] - iRow) > 0) os << NUMBER[10];
914  else os << " ";
915  } os << "\n";
916  } os << "\n";
917 
918  // Print sign and value of bin contents
919  double maxim = log10(max(yMax, -yMin));
920  int iPowBin = int(floor(maxim + 0.0001));
921  os << " Contents ";
922  for (int iCol = 0; iCol < nCol ; ++iCol) {
923  if (resCol[iCol] < - pow(10., iPowBin - 4)) os << "-";
924  else os << " ";
925  row[iCol] = int(abs(resCol[iCol]) * pow(10., 3 - iPowBin) + 0.5);
926  } os << "\n";
927  for (int iRow = 3; iRow >= 0; iRow--) {
928  os << " *10^" << setw(2) << iPowBin + iRow - 3 << " ";
929  int mask = int( pow(10., iRow) + 0.5);
930  for (int iCol = 0; iCol < nCol ; ++iCol) {
931  os << NUMBER[(row[iCol] / mask) % 10];
932  } os << "\n";
933  } os << "\n";
934 
935  // Print sign and value of lower bin edge.
936  maxim = log10( max( -h.xMin, h.xMax - h.dx));
937  int iPowExp = int(floor(maxim + 0.0001));
938  os << " Low edge ";
939  for (int iCol = 0; iCol < nCol ; ++iCol) {
940  if (h.xMin + iCol * nGroup * h.dx < - pow(10., iPowExp - 3)) os << "-";
941  else os << " ";
942  row[iCol] = int(abs(h.xMin + iCol * nGroup * h.dx)
943  * pow(10., 2 - iPowExp) + 0.5);
944  } os << "\n";
945  for (int iRow = 2; iRow >= 0; iRow--) {
946  os << " *10^" << setw(2) << iPowExp + iRow - 2 << " ";
947  int mask = int( pow(10., iRow) + 0.5);
948  for (int iCol = 0; iCol < nCol ; ++iCol)
949  os << NUMBER[(row[iCol] / mask) % 10];
950  os << "\n";
951  } os << "\n";
952 
953  // Print explanation if histogram cannot be shown.
954  } else os << " Histogram not shown since lowest value" << scientific
955  << setprecision(4) << setw(12) << yMin << " and highest value"
956  << setw(12) << yMax << " are too close \n \n";
957 
958  // Calculate and print statistics.
959  double cSum = 0.;
960  double cxSum = 0.;
961  double cxxSum = 0.;
962  for (int ix = 0; ix < h.nBin ; ++ix) {
963  double cta = abs(h.res[ix]);
964  double x = h.xMin + (ix + 0.5) * h.dx;
965  cSum = cSum + cta;
966  cxSum = cxSum + cta * x;
967  cxxSum = cxxSum + cta * x * x;
968  }
969  double xmean = cxSum / max(cSum, Hist::TINY);
970  double rms = sqrtpos( cxxSum / max(cSum, Hist::TINY) - xmean*xmean );
971  os << scientific << setprecision(4)
972  << " Entries =" << setw(12) << h.nFill
973  << " Mean =" << setw(12) << xmean
974  << " Underflow =" << setw(12) << h.under
975  << " Low edge =" << setw(12) << h.xMin << "\n"
976  << " All chan =" << setw(12) << h.inside
977  << " Rms =" << setw(12) << rms
978  << " Overflow =" << setw(12) << h.over
979  << " High edge =" << setw(12) << h.xMax << endl;
980  return os;
981 }
982 
983 //--------------------------------------------------------------------------
984 
985 // Print histogram contents as a table (e.g. for Gnuplot).
986 
987 void Hist::table(ostream& os) const {
988 
989  // Print histogram vector bin by bin, with mean x as first column.
990  os << scientific << setprecision(4);
991  for (int ix = 0; ix < nBin; ++ix)
992  os << setw(12) << xMin + (ix + 0.5) * dx
993  << setw(12) << res[ix] << "\n";
994 
995 }
996 
997 //--------------------------------------------------------------------------
998 
999 // Print a table out of two histograms with same x axis (e.g. for Gnuplot).
1000 
1001 void table(const Hist& h1, const Hist& h2, ostream& os) {
1002 
1003  // Require histogram x axes to agree.
1004  if (h1.nBin != h2.nBin || abs(h1.xMin - h2.xMin) > Hist::TOLERANCE * h1.dx
1005  || abs(h1.xMax - h2.xMax) > Hist::TOLERANCE * h1.dx) return;
1006 
1007  // Print histogram vectors bin by bin, with mean x as first column.
1008  os << scientific << setprecision(4);
1009  for (int ix = 0; ix < h1.nBin; ++ix)
1010  os << setw(12) << h1.xMin + (ix + 0.5) * h1.dx
1011  << setw(12) << h1.res[ix] << setw(12) << h2.res[ix] << "\n";
1012 
1013 }
1014 
1015 void table(const Hist& h1, const Hist& h2, string fileName) {
1016  ofstream streamName(fileName.c_str());
1017  table( h1, h2, streamName);
1018 }
1019 
1020 //--------------------------------------------------------------------------
1021 
1022 // Get content of specific bin.
1023 // Special values are bin 0 for underflow and bin nBin+1 for overflow.
1024 // All other bins outside proper histogram range return 0.
1025 
1026 double Hist::getBinContent(int iBin) {
1027 
1028  if (iBin > 0 && iBin <= nBin) return res[iBin - 1];
1029  else if (iBin == 0) return under;
1030  else if (iBin == nBin + 1) return over;
1031  else return 0.;
1032 
1033 }
1034 
1035 //--------------------------------------------------------------------------
1036 
1037 // Check whether another histogram has same size and limits.
1038 
1039 bool Hist::sameSize(const Hist& h) const {
1040 
1041  if (nBin == h.nBin && abs(xMin - h.xMin) < TOLERANCE * dx &&
1042  abs(xMax - h.xMax) < TOLERANCE * dx) return true;
1043  else return false;
1044 
1045 }
1046 
1047 //--------------------------------------------------------------------------
1048 
1049 // Take 10-logarithm or natural logarithm of contents bin by bin.
1050 
1051 void Hist::takeLog(bool tenLog) {
1052 
1053  // Find smallest positive bin content, and put min a bit below.
1054  double yMin = 1e20;
1055  for (int ix = 0; ix < nBin; ++ix)
1056  if (res[ix] > 1e-20 && res[ix] < yMin ) yMin = res[ix];
1057  yMin *= 0.8;
1058 
1059  // Take 10-logarithm bin by bin, but ensure positivity.
1060  if (tenLog) {
1061  for (int ix = 0; ix < nBin; ++ix)
1062  res[ix] = log10( max( yMin, res[ix]) );
1063  under = log10( max( yMin, under) );
1064  inside = log10( max( yMin, inside) );
1065  over = log10( max( yMin, over) );
1066 
1067  // Take natural logarithm bin by bin, but ensure positivity.
1068  } else {
1069  for (int ix = 0; ix < nBin; ++ix)
1070  res[ix] = log( max( yMin, res[ix]) );
1071  under = log( max( yMin, under) );
1072  inside = log( max( yMin, inside) );
1073  over = log( max( yMin, over) );
1074  }
1075 
1076 }
1077 
1078 //--------------------------------------------------------------------------
1079 
1080 // Take square root of contents bin by bin; set 0 for negative content.
1081 
1082 void Hist::takeSqrt() {
1083 
1084  for (int ix = 0; ix < nBin; ++ix) res[ix] = sqrtpos(res[ix]);
1085  under = sqrtpos(under);
1086  inside = sqrtpos(inside);
1087  over = sqrtpos(over);
1088 
1089 }
1090 
1091 //--------------------------------------------------------------------------
1092 
1093 // Add histogram to existing one.
1094 
1095 Hist& Hist::operator+=(const Hist& h) {
1096  if (!sameSize(h)) return *this;
1097  nFill += h.nFill;
1098  under += h.under;
1099  inside += h.inside;
1100  over += h.over;
1101  for (int ix = 0; ix < nBin; ++ix) res[ix] += h.res[ix];
1102  return *this;
1103 }
1104 
1105 //--------------------------------------------------------------------------
1106 
1107 // Subtract histogram from existing one.
1108 
1109 Hist& Hist::operator-=(const Hist& h) {
1110  if (!sameSize(h)) return *this;
1111  nFill += h.nFill;
1112  under -= h.under;
1113  inside -= h.inside;
1114  over -= h.over;
1115  for (int ix = 0; ix < nBin; ++ix) res[ix] -= h.res[ix];
1116  return *this;
1117 }
1118 
1119 //--------------------------------------------------------------------------
1120 
1121 // Multiply existing histogram by another one.
1122 
1123 Hist& Hist::operator*=(const Hist& h) {
1124  if (!sameSize(h)) return *this;
1125  nFill += h.nFill;
1126  under *= h.under;
1127  inside *= h.inside;
1128  over *= h.over;
1129  for (int ix = 0; ix < nBin; ++ix) res[ix] *= h.res[ix];
1130  return *this;
1131 }
1132 
1133 //--------------------------------------------------------------------------
1134 
1135 // Divide existing histogram by another one.
1136 
1137 Hist& Hist::operator/=(const Hist& h) {
1138  if (!sameSize(h)) return *this;
1139  nFill += h.nFill;
1140  under = (abs(h.under) < Hist::TINY) ? 0. : under/h.under;
1141  inside = (abs(h.inside) < Hist::TINY) ? 0. : inside/h.inside;
1142  over = (abs(h.over) < Hist::TINY) ? 0. : over/h.over;
1143  for (int ix = 0; ix < nBin; ++ix)
1144  res[ix] = (abs(h.res[ix]) < Hist::TINY) ? 0. : res[ix]/h.res[ix];
1145  return *this;
1146 }
1147 
1148 //--------------------------------------------------------------------------
1149 
1150 // Add constant offset to histogram.
1151 
1152 Hist& Hist::operator+=(double f) {
1153  under += f;
1154  inside += nBin * f;
1155  over += f;
1156  for (int ix = 0; ix < nBin; ++ix) res[ix] += f;
1157  return *this;
1158 }
1159 
1160 //--------------------------------------------------------------------------
1161 
1162 // Subtract constant offset from histogram.
1163 
1164 Hist& Hist::operator-=(double f) {
1165  under -= f;
1166  inside -= nBin * f;
1167  over -= f;
1168  for (int ix = 0; ix < nBin; ++ix) res[ix] -= f;
1169  return *this;
1170 }
1171 
1172 //--------------------------------------------------------------------------
1173 
1174 // Multiply histogram by constant
1175 
1176 Hist& Hist::operator*=(double f) {
1177  under *= f;
1178  inside *= f;
1179  over *= f;
1180  for (int ix = 0; ix < nBin; ++ix) res[ix] *= f;
1181  return *this;
1182 }
1183 
1184 //--------------------------------------------------------------------------
1185 
1186 // Divide histogram by constant
1187 
1188 Hist& Hist::operator/=(double f) {
1189  under /= f;
1190  inside /= f;
1191  over /= f;
1192  for (int ix = 0; ix < nBin; ++ix) res[ix] /= f;
1193  return *this;
1194 }
1195 
1196 //--------------------------------------------------------------------------
1197 
1198 // Implementation of operator overloading with friends.
1199 
1200 Hist operator+(double f, const Hist& h1) {
1201  Hist h = h1; return h += f;}
1202 
1203 Hist operator+(const Hist& h1, double f) {
1204  Hist h = h1; return h += f;}
1205 
1206 Hist operator+(const Hist& h1, const Hist& h2) {
1207  Hist h = h1; return h += h2;}
1208 
1209 Hist operator-(double f, const Hist& h1) {
1210  Hist h = h1;
1211  h.under = f - h1.under;
1212  h.inside = h1.nBin * f - h1.inside;
1213  h.over = f - h1.over;
1214  for (int ix = 0; ix < h1.nBin; ++ix) h.res[ix] = f - h1.res[ix];
1215  return h;}
1216 
1217 Hist operator-(const Hist& h1, double f) {
1218  Hist h = h1; return h -= f;}
1219 
1220 Hist operator-(const Hist& h1, const Hist& h2) {
1221  Hist h = h1; return h -= h2;}
1222 
1223 Hist operator*(double f, const Hist& h1) {
1224  Hist h = h1; return h *= f;}
1225 
1226 Hist operator*(const Hist& h1, double f) {
1227  Hist h = h1; return h *= f;}
1228 
1229 Hist operator*(const Hist& h1, const Hist& h2) {
1230  Hist h = h1; return h *= h2;}
1231 
1232 Hist operator/(double f, const Hist& h1) {
1233  Hist h = h1;
1234  h.under = (abs(h1.under) < Hist::TINY) ? 0. : f/h1.under;
1235  h.inside = (abs(h1.inside) < Hist::TINY) ? 0. : f/h1.inside;
1236  h.over = (abs(h1.over) < Hist::TINY) ? 0. : f/h1.over;
1237  for (int ix = 0; ix < h1.nBin; ++ix)
1238  h.res[ix] = (abs(h1.res[ix]) < Hist::TINY) ? 0. : f/h1.res[ix];
1239  return h;
1240 }
1241 
1242 Hist operator/(const Hist& h1, double f) {
1243  Hist h = h1; return h /= f;}
1244 
1245 Hist operator/(const Hist& h1, const Hist& h2) {
1246  Hist h = h1; return h /= h2;}
1247 
1248 //==========================================================================
1249 
1250 } // end namespace Pythia8
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