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