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25 // 25 //
26 // G4PolynomialSolver inline methods implement <<
27 // 26 //
28 // Author: E.Medernach, 19.12.2000 - First imp << 27 //
29 // ------------------------------------------- << 28 // class G4PolynomialSolver
>> 29 //
>> 30 // 19.12.00 E.Medernach, First implementation
>> 31 //
30 32
31 #define POLEPSILON 1e-12 << 33 #define POLEPSILON 1e-12
32 #define POLINFINITY 9.0E99 << 34 #define POLINFINITY 9.0E99
33 #define ITERATION 12 // 20 But 8 is really en << 35 #define ITERATION 12 // 20 But 8 is really enough for Newton with a good guess
34 36
35 template <class T, class F> 37 template <class T, class F>
36 G4PolynomialSolver<T, F>::G4PolynomialSolver(T << 38 G4PolynomialSolver<T,F>::G4PolynomialSolver (T* typeF, F func, F deriv,
37 G 39 G4double precision)
38 { 40 {
39 Precision = precision; << 41 Precision = precision ;
40 FunctionClass = typeF; << 42 FunctionClass = typeF ;
41 Function = func; << 43 Function = func ;
42 Derivative = deriv; << 44 Derivative = deriv ;
43 } 45 }
44 46
45 template <class T, class F> 47 template <class T, class F>
46 G4PolynomialSolver<T, F>::~G4PolynomialSolver( << 48 G4PolynomialSolver<T,F>::~G4PolynomialSolver ()
47 {} << 49 {
>> 50 }
48 51
49 template <class T, class F> 52 template <class T, class F>
50 G4double G4PolynomialSolver<T, F>::solve(G4dou << 53 G4double G4PolynomialSolver<T,F>::solve(G4double IntervalMin,
51 G4dou << 54 G4double IntervalMax)
52 { 55 {
53 return Newton(IntervalMin, IntervalMax); << 56 return Newton(IntervalMin,IntervalMax);
54 } 57 }
55 58
>> 59
56 /* If we want to be general this could work fo 60 /* If we want to be general this could work for any
57 polynomial of order more that 4 if we find 61 polynomial of order more that 4 if we find the (ORDER + 1)
58 control points 62 control points
59 */ 63 */
60 #define NBBEZIER 5 64 #define NBBEZIER 5
61 65
62 template <class T, class F> 66 template <class T, class F>
63 G4int G4PolynomialSolver<T, F>::BezierClipping << 67 G4int
64 << 68 G4PolynomialSolver<T,F>::BezierClipping(/*T* typeF,F func,F deriv,*/
65 << 69 G4double *IntervalMin,
>> 70 G4double *IntervalMax)
66 { 71 {
67 /** BezierClipping is a clipping interval Ne 72 /** BezierClipping is a clipping interval Newton method **/
68 /** It works by clipping the area where the 73 /** It works by clipping the area where the polynomial is **/
69 74
70 G4double P[NBBEZIER][2], D[2]; << 75 G4double P[NBBEZIER][2],D[2];
71 G4double NewMin, NewMax; << 76 G4double NewMin,NewMax;
72 77
73 G4int IntervalIsVoid = 1; 78 G4int IntervalIsVoid = 1;
74 << 79
75 /*** Calculating Control Points ***/ 80 /*** Calculating Control Points ***/
76 /* We see the polynomial as a Bezier curve f 81 /* We see the polynomial as a Bezier curve for some control points to find */
77 82
78 /* 83 /*
79 For 5 control points (polynomial of degree 84 For 5 control points (polynomial of degree 4) this is:
80 << 85
81 0 p0 = F((*IntervalMin)) 86 0 p0 = F((*IntervalMin))
82 1/4 p1 = F((*IntervalMin)) + ((*Interval 87 1/4 p1 = F((*IntervalMin)) + ((*IntervalMax) - (*IntervalMin))/4
83 * F'((*IntervalMin)) 88 * F'((*IntervalMin))
84 2/4 p2 = 1/6 * (16*F(((*IntervalMax) + ( 89 2/4 p2 = 1/6 * (16*F(((*IntervalMax) + (*IntervalMin))/2)
85 - (p0 + 4*p1 + 4*p3 + p4 << 90 - (p0 + 4*p1 + 4*p3 + p4))
86 3/4 p3 = F((*IntervalMax)) - ((*Interval 91 3/4 p3 = F((*IntervalMax)) - ((*IntervalMax) - (*IntervalMin))/4
87 * F'((*IntervalMax)) 92 * F'((*IntervalMax))
88 1 p4 = F((*IntervalMax)) 93 1 p4 = F((*IntervalMax))
89 */ << 94 */
90 95
91 /* x,y,z,dx,dy,dz are constant during search 96 /* x,y,z,dx,dy,dz are constant during searching */
92 97
93 D[0] = (FunctionClass->*Derivative)(*Interva 98 D[0] = (FunctionClass->*Derivative)(*IntervalMin);
94 << 99
95 P[0][0] = (*IntervalMin); 100 P[0][0] = (*IntervalMin);
96 P[0][1] = (FunctionClass->*Function)(*Interv 101 P[0][1] = (FunctionClass->*Function)(*IntervalMin);
>> 102
97 103
98 if(std::fabs(P[0][1]) < Precision) << 104 if (std::fabs(P[0][1]) < Precision) {
99 { <<
100 return 1; 105 return 1;
101 } 106 }
102 << 107
103 if(((*IntervalMax) - (*IntervalMin)) < POLEP << 108 if (((*IntervalMax) - (*IntervalMin)) < POLEPSILON) {
104 { <<
105 return 1; 109 return 1;
106 } 110 }
107 111
108 P[1][0] = (*IntervalMin) + ((*IntervalMax) - << 112 P[1][0] = (*IntervalMin) + ((*IntervalMax) - (*IntervalMin))/4;
109 P[1][1] = P[0][1] + (((*IntervalMax) - (*Int << 113 P[1][1] = P[0][1] + (((*IntervalMax) - (*IntervalMin))/4.0) * D[0];
110 114
111 D[1] = (FunctionClass->*Derivative)(*Interva 115 D[1] = (FunctionClass->*Derivative)(*IntervalMax);
112 116
113 P[4][0] = (*IntervalMax); 117 P[4][0] = (*IntervalMax);
114 P[4][1] = (FunctionClass->*Function)(*Interv 118 P[4][1] = (FunctionClass->*Function)(*IntervalMax);
>> 119
>> 120 P[3][0] = (*IntervalMax) - ((*IntervalMax) - (*IntervalMin))/4;
>> 121 P[3][1] = P[4][1] - ((*IntervalMax) - (*IntervalMin))/4 * D[1];
>> 122
>> 123 P[2][0] = ((*IntervalMax) + (*IntervalMin))/2;
>> 124 P[2][1] = (16*(FunctionClass->*Function)(((*IntervalMax)+(*IntervalMin))/2)
>> 125 - (P[0][1] + 4*P[1][1] + 4*P[3][1] + P[4][1]))/6 ;
>> 126
>> 127 {
>> 128 G4double Intersection ;
>> 129 G4int i,j;
>> 130
>> 131 NewMin = (*IntervalMax) ;
>> 132 NewMax = (*IntervalMin) ;
>> 133
>> 134 for (i=0;i<5;i++)
>> 135 for (j=i+1;j<5;j++)
>> 136 {
>> 137 /* there is an intersection only if each have different signs */
>> 138 if (((P[j][1] > -Precision) && (P[i][1] < Precision)) ||
>> 139 ((P[j][1] < Precision) && (P[i][1] > -Precision))) {
>> 140 IntervalIsVoid = 0;
>> 141 Intersection = P[j][0] - P[j][1]*((P[i][0] - P[j][0])/
>> 142 (P[i][1] - P[j][1]));
>> 143 if (Intersection < NewMin) {
>> 144 NewMin = Intersection;
>> 145 }
>> 146 if (Intersection > NewMax) {
>> 147 NewMax = Intersection;
>> 148 }
>> 149 }
>> 150 }
115 151
116 P[3][0] = (*IntervalMax) - ((*IntervalMax) - << 152
117 P[3][1] = P[4][1] - ((*IntervalMax) - (*Inte << 153 if (IntervalIsVoid != 1) {
118 <<
119 P[2][0] = ((*IntervalMax) + (*IntervalMin)) <<
120 P[2][1] = <<
121 (16 * (FunctionClass->*Function)(((*Interv <<
122 (P[0][1] + 4 * P[1][1] + 4 * P[3][1] + P[ <<
123 6; <<
124 <<
125 { <<
126 G4double Intersection; <<
127 G4int i, j; <<
128 <<
129 NewMin = (*IntervalMax); <<
130 NewMax = (*IntervalMin); <<
131 <<
132 for(i = 0; i < 5; ++i) <<
133 for(j = i + 1; j < 5; ++j) <<
134 { <<
135 /* there is an intersection only if ea <<
136 if(((P[j][1] > -Precision) && (P[i][1] <<
137 ((P[j][1] < Precision) && (P[i][1] <<
138 { <<
139 IntervalIsVoid = 0; <<
140 Intersection = <<
141 P[j][0] - P[j][1] * ((P[i][0] - P[ <<
142 if(Intersection < NewMin) <<
143 { <<
144 NewMin = Intersection; <<
145 } <<
146 if(Intersection > NewMax) <<
147 { <<
148 NewMax = Intersection; <<
149 } <<
150 } <<
151 } <<
152 <<
153 if(IntervalIsVoid != 1) <<
154 { <<
155 (*IntervalMax) = NewMax; 154 (*IntervalMax) = NewMax;
156 (*IntervalMin) = NewMin; 155 (*IntervalMin) = NewMin;
157 } 156 }
158 } 157 }
159 << 158
160 if(IntervalIsVoid == 1) << 159 if (IntervalIsVoid == 1) {
161 { <<
162 return -1; 160 return -1;
163 } 161 }
164 << 162
165 return 0; 163 return 0;
166 } 164 }
167 165
168 template <class T, class F> 166 template <class T, class F>
169 G4double G4PolynomialSolver<T, F>::Newton(G4do << 167 G4double G4PolynomialSolver<T,F>::Newton (G4double IntervalMin,
170 G4do 168 G4double IntervalMax)
171 { 169 {
172 /* So now we have a good guess and an interv 170 /* So now we have a good guess and an interval where
173 if there are an intersection the root mus 171 if there are an intersection the root must be */
174 172
175 G4double Value = 0; << 173 G4double Value = 0;
176 G4double Gradient = 0; 174 G4double Gradient = 0;
177 G4double Lambda; << 175 G4double Lambda ;
178 <<
179 G4int i = 0; <<
180 G4int j = 0; <<
181 176
>> 177 G4int i=0;
>> 178 G4int j=0;
>> 179
>> 180
182 /* Reduce interval before applying Newton Me 181 /* Reduce interval before applying Newton Method */
183 { 182 {
184 G4int NewtonIsSafe; << 183 G4int NewtonIsSafe ;
185 184
186 while((NewtonIsSafe = BezierClipping(&Inte << 185 while ((NewtonIsSafe = BezierClipping(&IntervalMin,&IntervalMax)) == 0) ;
187 ; <<
188 186
189 if(NewtonIsSafe == -1) << 187 if (NewtonIsSafe == -1) {
190 { <<
191 return POLINFINITY; 188 return POLINFINITY;
192 } 189 }
193 } 190 }
194 191
195 Lambda = IntervalMin; 192 Lambda = IntervalMin;
196 Value = (FunctionClass->*Function)(Lambda); << 193 Value = (FunctionClass->*Function)(Lambda);
>> 194
197 195
198 // while ((std::fabs(Value) > Precision)) { 196 // while ((std::fabs(Value) > Precision)) {
199 while(j != -1) << 197 while (j != -1) {
200 { << 198
201 Value = (FunctionClass->*Function)(Lambda) 199 Value = (FunctionClass->*Function)(Lambda);
202 200
203 Gradient = (FunctionClass->*Derivative)(La 201 Gradient = (FunctionClass->*Derivative)(Lambda);
204 202
205 Lambda = Lambda - Value / Gradient; << 203 Lambda = Lambda - Value/Gradient ;
206 204
207 if(std::fabs(Value) <= Precision) << 205 if (std::fabs(Value) <= Precision) {
208 { << 206 j ++;
209 ++j; << 207 if (j == 2) {
210 if(j == 2) << 208 j = -1;
211 { << 209 }
212 j = -1; << 210 } else {
213 } << 211 i ++;
214 } << 212
215 else << 213 if (i > ITERATION)
216 { << 214 return POLINFINITY;
217 ++i; << 215 }
218 <<
219 if(i > ITERATION) <<
220 return POLINFINITY; <<
221 } <<
222 } 216 }
223 return Lambda; << 217 return Lambda ;
224 } 218 }
225 219