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25 //
26 // G4AnalyticalPolSolver class implementation
27 //
28 // Author: V.Grichine, 24.04.97
29 // --------------------------------------------------------------------
30
31 #include "globals.hh"
32 #include <complex>
33
34 #include "G4AnalyticalPolSolver.hh"
35
36 //////////////////////////////////////////////////////////////////////////////
37
38 G4AnalyticalPolSolver::G4AnalyticalPolSolver() { ; }
39
40 //////////////////////////////////////////////////////////////////////////////
41
42 G4AnalyticalPolSolver::~G4AnalyticalPolSolver() { ; }
43
44 //////////////////////////////////////////////////////////////////////////////
45 //
46 // Array r[3][5] p[5]
47 // Roots of poly p[0] x^2 + p[1] x+p[2]=0
48 //
49 // x = r[1][k] + i r[2][k]; k = 1, 2
50
51 G4int G4AnalyticalPolSolver::QuadRoots(G4double p[5], G4double r[3][5])
52 {
53 G4double b, c, d2, d;
54
55 b = -p[1] / p[0] / 2.;
56 c = p[2] / p[0];
57 d2 = b * b - c;
58
59 if(d2 >= 0.)
60 {
61 d = std::sqrt(d2);
62 r[1][1] = b - d;
63 r[1][2] = b + d;
64 r[2][1] = 0.;
65 r[2][2] = 0.;
66 }
67 else
68 {
69 d = std::sqrt(-d2);
70 r[2][1] = d;
71 r[2][2] = -d;
72 r[1][1] = b;
73 r[1][2] = b;
74 }
75
76 return 2;
77 }
78
79 //////////////////////////////////////////////////////////////////////////////
80 //
81 // Array r[3][5] p[5]
82 // Roots of poly p[0] x^3 + p[1] x^2...+p[3]=0
83 // x=r[1][k] + i r[2][k] k=1,...,3
84 // Assumes 0<arctan(x)<pi/2 for x>0
85
86 G4int G4AnalyticalPolSolver::CubicRoots(G4double p[5], G4double r[3][5])
87 {
88 G4double x, t, b, c, d;
89 G4int k;
90
91 if(p[0] != 1.)
92 {
93 for(k = 1; k < 4; k++)
94 {
95 p[k] = p[k] / p[0];
96 }
97 p[0] = 1.;
98 }
99 x = p[1] / 3.0;
100 t = x * p[1];
101 b = 0.5 * (x * (t / 1.5 - p[2]) + p[3]);
102 t = (t - p[2]) / 3.0;
103 c = t * t * t;
104 d = b * b - c;
105
106 if(d >= 0.)
107 {
108 d = std::pow((std::sqrt(d) + std::fabs(b)), 1.0 / 3.0);
109
110 if(d != 0.)
111 {
112 if(b > 0.)
113 {
114 b = -d;
115 }
116 else
117 {
118 b = d;
119 }
120 c = t / b;
121 }
122 d = std::sqrt(0.75) * (b - c);
123 r[2][2] = d;
124 b = b + c;
125 c = -0.5 * b - x;
126 r[1][2] = c;
127
128 if((b > 0. && x <= 0.) || (b < 0. && x > 0.))
129 {
130 r[1][1] = c;
131 r[2][1] = -d;
132 r[1][3] = b - x;
133 r[2][3] = 0;
134 }
135 else
136 {
137 r[1][1] = b - x;
138 r[2][1] = 0.;
139 r[1][3] = c;
140 r[2][3] = -d;
141 }
142 } // end of 2 equal or complex roots
143 else
144 {
145 if(b == 0.)
146 {
147 d = std::atan(1.0) / 1.5;
148 }
149 else
150 {
151 d = std::atan(std::sqrt(-d) / std::fabs(b)) / 3.0;
152 }
153
154 if(b < 0.)
155 {
156 b = std::sqrt(t) * 2.0;
157 }
158 else
159 {
160 b = -2.0 * std::sqrt(t);
161 }
162
163 c = std::cos(d) * b;
164 t = -std::sqrt(0.75) * std::sin(d) * b - 0.5 * c;
165 d = -t - c - x;
166 c = c - x;
167 t = t - x;
168
169 if(std::fabs(c) > std::fabs(t))
170 {
171 r[1][3] = c;
172 }
173 else
174 {
175 r[1][3] = t;
176 t = c;
177 }
178 if(std::fabs(d) > std::fabs(t))
179 {
180 r[1][2] = d;
181 }
182 else
183 {
184 r[1][2] = t;
185 t = d;
186 }
187 r[1][1] = t;
188
189 for(k = 1; k < 4; k++)
190 {
191 r[2][k] = 0.;
192 }
193 }
194 return 0;
195 }
196
197 //////////////////////////////////////////////////////////////////////////////
198 //
199 // Array r[3][5] p[5]
200 // Roots of poly p[0] x^4 + p[1] x^3...+p[4]=0
201 // x=r[1][k] + i r[2][k] k=1,...,4
202
203 G4int G4AnalyticalPolSolver::BiquadRoots(G4double p[5], G4double r[3][5])
204 {
205 G4double a, b, c, d, e;
206 G4int i, k, j;
207
208 if(p[0] != 1.0)
209 {
210 for(k = 1; k < 5; k++)
211 {
212 p[k] = p[k] / p[0];
213 }
214 p[0] = 1.;
215 }
216 e = 0.25 * p[1];
217 b = 2 * e;
218 c = b * b;
219 d = 0.75 * c;
220 b = p[3] + b * (c - p[2]);
221 a = p[2] - d;
222 c = p[4] + e * (e * a - p[3]);
223 a = a - d;
224
225 p[1] = 0.5 * a;
226 p[2] = (p[1] * p[1] - c) * 0.25;
227 p[3] = b * b / (-64.0);
228
229 if(p[3] < 0.)
230 {
231 CubicRoots(p, r);
232
233 for(k = 1; k < 4; k++)
234 {
235 if(r[2][k] == 0. && r[1][k] > 0)
236 {
237 d = r[1][k] * 4;
238 a = a + d;
239
240 if(a >= 0. && b >= 0.)
241 {
242 p[1] = std::sqrt(d);
243 }
244 else if(a <= 0. && b <= 0.)
245 {
246 p[1] = std::sqrt(d);
247 }
248 else
249 {
250 p[1] = -std::sqrt(d);
251 }
252
253 b = 0.5 * (a + b / p[1]);
254
255 p[2] = c / b;
256 QuadRoots(p, r);
257
258 for(i = 1; i < 3; i++)
259 {
260 for(j = 1; j < 3; j++)
261 {
262 r[j][i + 2] = r[j][i];
263 }
264 }
265 p[1] = -p[1];
266 p[2] = b;
267 QuadRoots(p, r);
268
269 for(i = 1; i < 5; i++)
270 {
271 r[1][i] = r[1][i] - e;
272 }
273
274 return 4;
275 }
276 }
277 }
278 if(p[2] < 0.)
279 {
280 b = std::sqrt(c);
281 d = b + b - a;
282 p[1] = 0.;
283
284 if(d > 0.)
285 {
286 p[1] = std::sqrt(d);
287 }
288 }
289 else
290 {
291 if(p[1] > 0.)
292 {
293 b = std::sqrt(p[2]) * 2.0 + p[1];
294 }
295 else
296 {
297 b = -std::sqrt(p[2]) * 2.0 + p[1];
298 }
299
300 if(b != 0.)
301 {
302 p[1] = 0;
303 }
304 else
305 {
306 for(k = 1; k < 5; k++)
307 {
308 r[1][k] = -e;
309 r[2][k] = 0;
310 }
311 return 0;
312 }
313 }
314
315 p[2] = c / b;
316 QuadRoots(p, r);
317
318 for(k = 1; k < 3; k++)
319 {
320 for(j = 1; j < 3; j++)
321 {
322 r[j][k + 2] = r[j][k];
323 }
324 }
325 p[1] = -p[1];
326 p[2] = b;
327 QuadRoots(p, r);
328
329 for(k = 1; k < 5; k++)
330 {
331 r[1][k] = r[1][k] - e;
332 }
333
334 return 4;
335 }
336
337 //////////////////////////////////////////////////////////////////////////////
338
339 G4int G4AnalyticalPolSolver::QuarticRoots(G4double p[5], G4double r[3][5])
340 {
341 G4double a0, a1, a2, a3, y1;
342 G4double R2, D2, E2, D, E, R = 0.;
343 G4double a, b, c, d, ds;
344
345 G4double reRoot[4];
346 G4int k;
347
348 for(k = 0; k < 4; k++)
349 {
350 reRoot[k] = DBL_MAX;
351 }
352
353 if(p[0] != 1.0)
354 {
355 for(k = 1; k < 5; k++)
356 {
357 p[k] = p[k] / p[0];
358 }
359 p[0] = 1.;
360 }
361 a3 = p[1];
362 a2 = p[2];
363 a1 = p[3];
364 a0 = p[4];
365
366 // resolvent cubic equation cofs:
367
368 p[1] = -a2;
369 p[2] = a1 * a3 - 4 * a0;
370 p[3] = 4 * a2 * a0 - a1 * a1 - a3 * a3 * a0;
371
372 CubicRoots(p, r);
373
374 for(k = 1; k < 4; k++)
375 {
376 if(r[2][k] == 0.) // find a real root
377 {
378 reRoot[k] = r[1][k];
379 }
380 else
381 {
382 reRoot[k] = DBL_MAX; // kInfinity;
383 }
384 }
385 y1 = DBL_MAX; // kInfinity;
386 for(k = 1; k < 4; k++)
387 {
388 if(reRoot[k] < y1)
389 {
390 y1 = reRoot[k];
391 }
392 }
393
394 R2 = 0.25 * a3 * a3 - a2 + y1;
395 b = 0.25 * (4 * a3 * a2 - 8 * a1 - a3 * a3 * a3);
396 c = 0.75 * a3 * a3 - 2 * a2;
397 a = c - R2;
398 d = 4 * y1 * y1 - 16 * a0;
399
400 if(R2 > 0.)
401 {
402 R = std::sqrt(R2);
403 D2 = a + b / R;
404 E2 = a - b / R;
405
406 if(D2 >= 0.)
407 {
408 D = std::sqrt(D2);
409 r[1][1] = -0.25 * a3 + 0.5 * R + 0.5 * D;
410 r[1][2] = -0.25 * a3 + 0.5 * R - 0.5 * D;
411 r[2][1] = 0.;
412 r[2][2] = 0.;
413 }
414 else
415 {
416 D = std::sqrt(-D2);
417 r[1][1] = -0.25 * a3 + 0.5 * R;
418 r[1][2] = -0.25 * a3 + 0.5 * R;
419 r[2][1] = 0.5 * D;
420 r[2][2] = -0.5 * D;
421 }
422 if(E2 >= 0.)
423 {
424 E = std::sqrt(E2);
425 r[1][3] = -0.25 * a3 - 0.5 * R + 0.5 * E;
426 r[1][4] = -0.25 * a3 - 0.5 * R - 0.5 * E;
427 r[2][3] = 0.;
428 r[2][4] = 0.;
429 }
430 else
431 {
432 E = std::sqrt(-E2);
433 r[1][3] = -0.25 * a3 - 0.5 * R;
434 r[1][4] = -0.25 * a3 - 0.5 * R;
435 r[2][3] = 0.5 * E;
436 r[2][4] = -0.5 * E;
437 }
438 }
439 else if(R2 < 0.)
440 {
441 R = std::sqrt(-R2);
442 G4complex CD2(a, -b / R);
443 G4complex CD = std::sqrt(CD2);
444
445 r[1][1] = -0.25 * a3 + 0.5 * real(CD);
446 r[1][2] = -0.25 * a3 - 0.5 * real(CD);
447 r[2][1] = 0.5 * R + 0.5 * imag(CD);
448 r[2][2] = 0.5 * R - 0.5 * imag(CD);
449 G4complex CE2(a, b / R);
450 G4complex CE = std::sqrt(CE2);
451
452 r[1][3] = -0.25 * a3 + 0.5 * real(CE);
453 r[1][4] = -0.25 * a3 - 0.5 * real(CE);
454 r[2][3] = -0.5 * R + 0.5 * imag(CE);
455 r[2][4] = -0.5 * R - 0.5 * imag(CE);
456 }
457 else // R2=0 case
458 {
459 if(d >= 0.)
460 {
461 D2 = c + std::sqrt(d);
462 E2 = c - std::sqrt(d);
463
464 if(D2 >= 0.)
465 {
466 D = std::sqrt(D2);
467 r[1][1] = -0.25 * a3 + 0.5 * R + 0.5 * D;
468 r[1][2] = -0.25 * a3 + 0.5 * R - 0.5 * D;
469 r[2][1] = 0.;
470 r[2][2] = 0.;
471 }
472 else
473 {
474 D = std::sqrt(-D2);
475 r[1][1] = -0.25 * a3 + 0.5 * R;
476 r[1][2] = -0.25 * a3 + 0.5 * R;
477 r[2][1] = 0.5 * D;
478 r[2][2] = -0.5 * D;
479 }
480 if(E2 >= 0.)
481 {
482 E = std::sqrt(E2);
483 r[1][3] = -0.25 * a3 - 0.5 * R + 0.5 * E;
484 r[1][4] = -0.25 * a3 - 0.5 * R - 0.5 * E;
485 r[2][3] = 0.;
486 r[2][4] = 0.;
487 }
488 else
489 {
490 E = std::sqrt(-E2);
491 r[1][3] = -0.25 * a3 - 0.5 * R;
492 r[1][4] = -0.25 * a3 - 0.5 * R;
493 r[2][3] = 0.5 * E;
494 r[2][4] = -0.5 * E;
495 }
496 }
497 else
498 {
499 ds = std::sqrt(-d);
500 G4complex CD2(c, ds);
501 G4complex CD = std::sqrt(CD2);
502
503 r[1][1] = -0.25 * a3 + 0.5 * real(CD);
504 r[1][2] = -0.25 * a3 - 0.5 * real(CD);
505 r[2][1] = 0.5 * R + 0.5 * imag(CD);
506 r[2][2] = 0.5 * R - 0.5 * imag(CD);
507
508 G4complex CE2(c, -ds);
509 G4complex CE = std::sqrt(CE2);
510
511 r[1][3] = -0.25 * a3 + 0.5 * real(CE);
512 r[1][4] = -0.25 * a3 - 0.5 * real(CE);
513 r[2][3] = -0.5 * R + 0.5 * imag(CE);
514 r[2][4] = -0.5 * R - 0.5 * imag(CE);
515 }
516 }
517 return 4;
518 }
519
520 //
521 //
522 //////////////////////////////////////////////////////////////////////////////
523