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