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24 // 24 //
>> 25 //
>> 26 //
25 // G4BulirschStoer class implementation 27 // G4BulirschStoer class implementation
26 // Based on bulirsch_stoer.hpp from boost 28 // Based on bulirsch_stoer.hpp from boost
27 // 29 //
28 // Author: Dmitry Sorokin, Google Summer of Co << 30 // Author: Dmitry Sorokin - GSoC 2016
29 // ------------------------------------------- << 31 //
>> 32 ///////////////////////////////////////////////////////////////////////////////
30 33
31 #include "G4BulirschStoer.hh" 34 #include "G4BulirschStoer.hh"
32 35
33 #include "G4FieldUtils.hh" 36 #include "G4FieldUtils.hh"
34 37
35 namespace 38 namespace
36 { 39 {
37 constexpr G4double STEPFAC1 = 0.65; 40 constexpr G4double STEPFAC1 = 0.65;
38 constexpr G4double STEPFAC2 = 0.94; 41 constexpr G4double STEPFAC2 = 0.94;
39 constexpr G4double STEPFAC3 = 0.02; 42 constexpr G4double STEPFAC3 = 0.02;
40 constexpr G4double STEPFAC4 = 4.0; 43 constexpr G4double STEPFAC4 = 4.0;
41 constexpr G4double KFAC1 = 0.8; 44 constexpr G4double KFAC1 = 0.8;
42 constexpr G4double KFAC2 = 0.9; 45 constexpr G4double KFAC2 = 0.9;
43 constexpr G4double inv_STEPFAC1 = 1.0 / STEP 46 constexpr G4double inv_STEPFAC1 = 1.0 / STEPFAC1;
44 constexpr G4double inv_STEPFAC4 = 1.0 / STEP 47 constexpr G4double inv_STEPFAC4 = 1.0 / STEPFAC4;
45 } // namespace 48 } // namespace
46 49
47 G4BulirschStoer::G4BulirschStoer(G4EquationOfM 50 G4BulirschStoer::G4BulirschStoer(G4EquationOfMotion* equation,
48 G4int nvar, G 51 G4int nvar, G4double eps_rel, G4double max_dt)
49 : fnvar(nvar), m_eps_rel(eps_rel), m_midpoin << 52 : fnvar(nvar), m_eps_rel(eps_rel), m_midpoint(equation,nvar),
>> 53 m_last_step_rejected(false), m_first(true), m_dt_last(0.0), m_max_dt(max_dt)
50 { 54 {
51 /* initialize sequence of stage numbers and 55 /* initialize sequence of stage numbers and work */
52 56
53 for(G4int i = 0; i < m_k_max + 1; ++i) 57 for(G4int i = 0; i < m_k_max + 1; ++i)
54 { 58 {
55 m_interval_sequence[i] = 2 * (i + 1); 59 m_interval_sequence[i] = 2 * (i + 1);
56 if (i == 0) 60 if (i == 0)
57 { 61 {
58 m_cost[i] = m_interval_sequence[i]; 62 m_cost[i] = m_interval_sequence[i];
59 } 63 }
60 else 64 else
61 { 65 {
62 m_cost[i] = m_cost[i-1] + m_interval_seq 66 m_cost[i] = m_cost[i-1] + m_interval_sequence[i];
63 } 67 }
64 for(G4int k = 0; k < i; ++k) 68 for(G4int k = 0; k < i; ++k)
65 { 69 {
66 const G4double r = static_cast<G4double> 70 const G4double r = static_cast<G4double>(m_interval_sequence[i])
67 / static_cast<G4double> 71 / static_cast<G4double>(m_interval_sequence[k]);
68 m_coeff[i][k] = 1.0 / (r * r - 1.0); // 72 m_coeff[i][k] = 1.0 / (r * r - 1.0); // coefficients for extrapolation
69 } 73 }
70 74
71 // crude estimate of optimal order 75 // crude estimate of optimal order
72 m_current_k_opt = 4; 76 m_current_k_opt = 4;
73 77
74 // no calculation because log10 might not 78 // no calculation because log10 might not exist for value_type!
75 79
76 //const G4double logfact = -log10(std::max 80 //const G4double logfact = -log10(std::max(eps_rel, 1.0e-12)) * 0.6 + 0.5;
77 //m_current_k_opt = std::max(1., 81 //m_current_k_opt = std::max(1.,
78 // std::min(static_cast<G 82 // std::min(static_cast<G4double>(m_k_max-1), logfact));
79 } 83 }
80 } 84 }
81 85
82 G4BulirschStoer::step_result 86 G4BulirschStoer::step_result
83 G4BulirschStoer::try_step( const G4double in[] 87 G4BulirschStoer::try_step( const G4double in[], const G4double dxdt[],
84 G4double& t, G4doub 88 G4double& t, G4double out[], G4double& dt)
85 { 89 {
86 if(m_max_dt < dt) 90 if(m_max_dt < dt)
87 { 91 {
88 // given step size is bigger then max_dt s 92 // given step size is bigger then max_dt set limit and return fail
89 // 93 //
90 dt = m_max_dt; 94 dt = m_max_dt;
91 return step_result::fail; 95 return step_result::fail;
92 } 96 }
93 97
94 if (dt != m_dt_last) 98 if (dt != m_dt_last)
95 { 99 {
96 reset(); // step size changed from outside 100 reset(); // step size changed from outside -> reset
97 } 101 }
98 102
99 G4bool reject = true; 103 G4bool reject = true;
100 104
101 G4double new_h = dt; 105 G4double new_h = dt;
102 106
103 /* m_current_k_opt is the estimated current 107 /* m_current_k_opt is the estimated current optimal stage number */
104 108
105 for(G4int k = 0; k <= m_current_k_opt+1; ++k 109 for(G4int k = 0; k <= m_current_k_opt+1; ++k)
106 { 110 {
107 // the stage counts are stored in m_interv 111 // the stage counts are stored in m_interval_sequence
108 // 112 //
109 m_midpoint.SetSteps(m_interval_sequence[k] 113 m_midpoint.SetSteps(m_interval_sequence[k]);
110 if(k == 0) 114 if(k == 0)
111 { 115 {
112 m_midpoint.DoStep(in, dxdt, out, dt); 116 m_midpoint.DoStep(in, dxdt, out, dt);
113 /* the first step, nothing more to do */ 117 /* the first step, nothing more to do */
114 } 118 }
115 else 119 else
116 { 120 {
117 m_midpoint.DoStep(in, dxdt, m_table[k-1] 121 m_midpoint.DoStep(in, dxdt, m_table[k-1], dt);
118 extrapolate(k, out); 122 extrapolate(k, out);
119 // get error estimate 123 // get error estimate
120 for (G4int i = 0; i < fnvar; ++i) 124 for (G4int i = 0; i < fnvar; ++i)
121 { 125 {
122 m_err[i] = out[i] - m_table[0][i]; 126 m_err[i] = out[i] - m_table[0][i];
123 } 127 }
124 const G4double error = 128 const G4double error =
125 field_utils::relativeError(out, m_ 129 field_utils::relativeError(out, m_err, dt, m_eps_rel);
126 h_opt[k] = calc_h_opt(dt, error, k); 130 h_opt[k] = calc_h_opt(dt, error, k);
127 work[k] = static_cast<G4double>(m_cost[k 131 work[k] = static_cast<G4double>(m_cost[k]) / h_opt[k];
128 132
129 if( (k == m_current_k_opt-1) || m_first) 133 if( (k == m_current_k_opt-1) || m_first) // convergence before k_opt ?
130 { 134 {
131 if(error < 1.0) 135 if(error < 1.0)
132 { 136 {
133 // convergence 137 // convergence
134 reject = false; 138 reject = false;
135 if( (work[k] < KFAC2 * work[k-1]) || 139 if( (work[k] < KFAC2 * work[k-1]) || (m_current_k_opt <= 2) )
136 { 140 {
137 // leave order as is (except we we 141 // leave order as is (except we were in first round)
138 m_current_k_opt = std::min(m_k_max 142 m_current_k_opt = std::min(m_k_max - 1 , std::max(2 , k + 1));
139 new_h = h_opt[k]; 143 new_h = h_opt[k];
140 new_h *= static_cast<G4double>(m_c 144 new_h *= static_cast<G4double>(m_cost[k + 1])
141 / static_cast<G4double>(m_c 145 / static_cast<G4double>(m_cost[k]);
142 } 146 }
143 else 147 else
144 { 148 {
145 m_current_k_opt = std::min(m_k_max 149 m_current_k_opt = std::min(m_k_max - 1, std::max(2, k));
146 new_h = h_opt[k]; 150 new_h = h_opt[k];
147 } 151 }
148 break; 152 break;
149 } 153 }
150 if(should_reject(error , k) && !m_firs << 154 else if(should_reject(error , k) && !m_first)
151 { 155 {
152 reject = true; 156 reject = true;
153 new_h = h_opt[k]; 157 new_h = h_opt[k];
154 break; 158 break;
155 } 159 }
156 } 160 }
157 if(k == m_current_k_opt) // convergence 161 if(k == m_current_k_opt) // convergence at k_opt ?
158 { 162 {
159 if(error < 1.0) 163 if(error < 1.0)
160 { 164 {
161 // convergence 165 // convergence
162 reject = false; 166 reject = false;
163 if(work[k-1] < KFAC2 * work[k]) 167 if(work[k-1] < KFAC2 * work[k])
164 { 168 {
165 m_current_k_opt = std::max( 2 , m_ 169 m_current_k_opt = std::max( 2 , m_current_k_opt-1 );
166 new_h = h_opt[m_current_k_opt]; 170 new_h = h_opt[m_current_k_opt];
167 } 171 }
168 else if( (work[k] < KFAC2 * work[k-1 172 else if( (work[k] < KFAC2 * work[k-1]) && !m_last_step_rejected )
169 { 173 {
170 m_current_k_opt = std::min(m_k_max 174 m_current_k_opt = std::min(m_k_max - 1, m_current_k_opt + 1);
171 new_h = h_opt[k]; 175 new_h = h_opt[k];
172 new_h *= static_cast<G4double>(m_c 176 new_h *= static_cast<G4double>(m_cost[m_current_k_opt])
173 / static_cast<G4double>(m_c 177 / static_cast<G4double>(m_cost[k]);
174 } 178 }
175 else 179 else
176 { 180 {
177 new_h = h_opt[m_current_k_opt]; 181 new_h = h_opt[m_current_k_opt];
178 } 182 }
179 break; 183 break;
180 } 184 }
181 if(should_reject(error, k)) << 185 else if(should_reject(error, k))
182 { 186 {
183 reject = true; 187 reject = true;
184 new_h = h_opt[m_current_k_opt]; 188 new_h = h_opt[m_current_k_opt];
185 break; 189 break;
186 } 190 }
187 } 191 }
188 if(k == m_current_k_opt + 1) // converg 192 if(k == m_current_k_opt + 1) // convergence at k_opt+1 ?
189 { 193 {
190 if(error < 1.0) // convergence 194 if(error < 1.0) // convergence
191 { 195 {
192 reject = false; 196 reject = false;
193 if(work[k-2] < KFAC2 * work[k-1]) 197 if(work[k-2] < KFAC2 * work[k-1])
194 { 198 {
195 m_current_k_opt = std::max(2, m_cu 199 m_current_k_opt = std::max(2, m_current_k_opt - 1);
196 } 200 }
197 if((work[k] < KFAC2 * work[m_current 201 if((work[k] < KFAC2 * work[m_current_k_opt]) && !m_last_step_rejected)
198 { 202 {
199 m_current_k_opt = std::min(m_k_max 203 m_current_k_opt = std::min(m_k_max - 1 , k);
200 } 204 }
201 new_h = h_opt[m_current_k_opt]; 205 new_h = h_opt[m_current_k_opt];
202 } 206 }
203 else 207 else
204 { 208 {
205 reject = true; 209 reject = true;
206 new_h = h_opt[m_current_k_opt]; 210 new_h = h_opt[m_current_k_opt];
207 } 211 }
208 break; 212 break;
209 } 213 }
210 } 214 }
211 } 215 }
212 216
213 if(!reject) 217 if(!reject)
214 { 218 {
215 t += dt; 219 t += dt;
216 } 220 }
217 221
218 if(!m_last_step_rejected || new_h < dt) 222 if(!m_last_step_rejected || new_h < dt)
219 { 223 {
220 // limit step size 224 // limit step size
221 new_h = std::min(m_max_dt, new_h); 225 new_h = std::min(m_max_dt, new_h);
222 m_dt_last = new_h; 226 m_dt_last = new_h;
223 dt = new_h; 227 dt = new_h;
224 } 228 }
225 229
226 m_last_step_rejected = reject; 230 m_last_step_rejected = reject;
227 m_first = false; 231 m_first = false;
228 232
229 return reject ? step_result::fail : step_res 233 return reject ? step_result::fail : step_result::success;
230 } 234 }
231 235
232 void G4BulirschStoer::reset() 236 void G4BulirschStoer::reset()
233 { 237 {
234 m_first = true; 238 m_first = true;
235 m_last_step_rejected = false; 239 m_last_step_rejected = false;
236 } 240 }
237 241
238 void G4BulirschStoer::extrapolate(std::size_t << 242 void G4BulirschStoer::extrapolate(size_t k , G4double xest[])
239 { 243 {
240 /* polynomial extrapolation, see http://www. 244 /* polynomial extrapolation, see http://www.nr.com/webnotes/nr3web21.pdf
241 * uses the obtained intermediate results to 245 * uses the obtained intermediate results to extrapolate to dt->0 */
242 246
243 for(std::size_t j = k - 1 ; j > 0; --j) << 247 for(G4int j = k - 1 ; j > 0; --j)
244 { 248 {
245 for (G4int i = 0; i < fnvar; ++i) 249 for (G4int i = 0; i < fnvar; ++i)
246 { 250 {
247 m_table[j-1][i] = m_table[j][i] * (1. + 251 m_table[j-1][i] = m_table[j][i] * (1. + m_coeff[k][j])
248 - m_table[j-1][i] * m_co 252 - m_table[j-1][i] * m_coeff[k][j];
249 } 253 }
250 } 254 }
251 for (G4int i = 0; i < fnvar; ++i) 255 for (G4int i = 0; i < fnvar; ++i)
252 { 256 {
253 xest[i] = m_table[0][i] * (1. + m_coeff[k] 257 xest[i] = m_table[0][i] * (1. + m_coeff[k][0]) - xest[i] * m_coeff[k][0];
254 } 258 }
255 } 259 }
256 260
257 G4double 261 G4double
258 G4BulirschStoer::calc_h_opt(G4double h , G4dou << 262 G4BulirschStoer::calc_h_opt(G4double h , G4double error , size_t k) const
259 { 263 {
260 /* calculates the optimal step size for a gi 264 /* calculates the optimal step size for a given error and stage number */
261 265
262 const G4double expo = 1.0 / (2 * k + 1); 266 const G4double expo = 1.0 / (2 * k + 1);
263 const G4double facmin = std::pow(STEPFAC3, e 267 const G4double facmin = std::pow(STEPFAC3, expo);
264 G4double fac; 268 G4double fac;
265 269
266 G4double facminInv= 1.0 / facmin; 270 G4double facminInv= 1.0 / facmin;
267 if (error == 0.0) 271 if (error == 0.0)
268 { 272 {
269 fac = facminInv; 273 fac = facminInv;
270 } 274 }
271 else 275 else
272 { 276 {
273 fac = STEPFAC2 * std::pow(error * inv_STEP 277 fac = STEPFAC2 * std::pow(error * inv_STEPFAC1 , -expo);
274 fac = std::max(facmin * inv_STEPFAC4, std: 278 fac = std::max(facmin * inv_STEPFAC4, std::min( facminInv, fac));
275 } 279 }
276 280
277 return h * fac; 281 return h * fac;
278 } 282 }
279 283
280 //why is not used!!?? 284 //why is not used!!??
281 G4bool G4BulirschStoer::set_k_opt(std::size_t << 285 G4bool G4BulirschStoer::set_k_opt(size_t k, G4double& dt)
282 { 286 {
283 /* calculates the optimal stage number */ 287 /* calculates the optimal stage number */
284 288
285 if(k == 1) 289 if(k == 1)
286 { 290 {
287 m_current_k_opt = 2; 291 m_current_k_opt = 2;
288 return true; 292 return true;
289 } 293 }
290 if( (work[k-1] < KFAC1 * work[k]) || (k == m 294 if( (work[k-1] < KFAC1 * work[k]) || (k == m_k_max) ) // order decrease
291 { 295 {
292 m_current_k_opt = (G4int)k - 1; << 296 m_current_k_opt = k - 1;
293 dt = h_opt[ m_current_k_opt ]; 297 dt = h_opt[ m_current_k_opt ];
294 return true; 298 return true;
295 } 299 }
296 if( (work[k] < KFAC2 * work[k-1]) << 300 else if( (work[k] < KFAC2 * work[k-1])
297 || m_last_step_rejected || (k == m_k 301 || m_last_step_rejected || (k == m_k_max-1) )
298 { // same order - also do this if last step 302 { // same order - also do this if last step got rejected
299 m_current_k_opt = (G4int)k; << 303 m_current_k_opt = k;
300 dt = h_opt[m_current_k_opt]; 304 dt = h_opt[m_current_k_opt];
301 return true; 305 return true;
302 } 306 }
303 else { // order increase - only if last st 307 else { // order increase - only if last step was not rejected
304 m_current_k_opt = (G4int)k + 1; << 308 m_current_k_opt = k + 1;
305 dt = h_opt[m_current_k_opt - 1] * m_cost[m 309 dt = h_opt[m_current_k_opt - 1] * m_cost[m_current_k_opt]
306 / m_cost[m_current_k_opt - 1]; 310 / m_cost[m_current_k_opt - 1];
307 return true; 311 return true;
308 } 312 }
309 } 313 }
310 314
311 G4bool G4BulirschStoer::in_convergence_window( 315 G4bool G4BulirschStoer::in_convergence_window(G4int k) const
312 { 316 {
313 if( (k == m_current_k_opt - 1) && !m_last_st 317 if( (k == m_current_k_opt - 1) && !m_last_step_rejected )
314 { 318 {
315 return true; // decrease stepsize only if 319 return true; // decrease stepsize only if last step was not rejected
316 } 320 }
317 return (k == m_current_k_opt) || (k == m_cur 321 return (k == m_current_k_opt) || (k == m_current_k_opt + 1);
318 } 322 }
319 323
320 324
321 G4bool G4BulirschStoer::should_reject(G4double 325 G4bool G4BulirschStoer::should_reject(G4double error, G4int k) const
322 { 326 {
323 if(k == m_current_k_opt - 1) 327 if(k == m_current_k_opt - 1)
324 { 328 {
325 const auto d = G4double(m_interval_sequen << 329 const G4double d = G4double(m_interval_sequence[m_current_k_opt]
326 * m_interval_sequence[m_curr << 330 * m_interval_sequence[m_current_k_opt+1]);
327 const auto e = G4double(m_interval_sequen << 331 const G4double e = G4double(m_interval_sequence[0]);
328 const G4double e2 = e*e; 332 const G4double e2 = e*e;
329 // step will fail, criterion 17.3.17 in NR 333 // step will fail, criterion 17.3.17 in NR
330 return error * e2 * e2 > d * d; // was r 334 return error * e2 * e2 > d * d; // was return error > dOld * dOld; (where dOld= d/e; )
331 } 335 }
332 if(k == m_current_k_opt) << 336 else if(k == m_current_k_opt)
333 { 337 {
334 const auto d = G4double(m_interval_sequen << 338 const G4double d = G4double(m_interval_sequence[m_current_k_opt]);
335 const auto e = G4double(m_interval_sequen << 339 const G4double e = G4double(m_interval_sequence[0]);
336 return error * e * e > d * d; // was retu 340 return error * e * e > d * d; // was return error > dOld * dOld; (where dOld= d/e; )
337 } 341 }
338 else 342 else
339 { 343 {
340 return error > 1.0; 344 return error > 1.0;
341 } 345 }
342 } 346 }
343 347