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27 // $Id: G4PolynomialSolver.icc 67970 2013-03-13 10:10:06Z gcosmo $
29 // class G4PolynomialSolver
31 // 19.12.00 E.Medernach, First implementation
34 #define POLEPSILON 1e-12
35 #define POLINFINITY 9.0E99
36 #define ITERATION 12 // 20 But 8 is really enough for Newton with a good guess
38 template <class T, class F>
39 G4PolynomialSolver<T,F>::G4PolynomialSolver (T* typeF, F func, F deriv,
42 Precision = precision ;
43 FunctionClass = typeF ;
48 template <class T, class F>
49 G4PolynomialSolver<T,F>::~G4PolynomialSolver ()
53 template <class T, class F>
54 G4double G4PolynomialSolver<T,F>::solve(G4double IntervalMin,
57 return Newton(IntervalMin,IntervalMax);
61 /* If we want to be general this could work for any
62 polynomial of order more that 4 if we find the (ORDER + 1)
67 template <class T, class F>
69 G4PolynomialSolver<T,F>::BezierClipping(/*T* typeF,F func,F deriv,*/
70 G4double *IntervalMin,
71 G4double *IntervalMax)
73 /** BezierClipping is a clipping interval Newton method **/
74 /** It works by clipping the area where the polynomial is **/
76 G4double P[NBBEZIER][2],D[2];
77 G4double NewMin,NewMax;
79 G4int IntervalIsVoid = 1;
81 /*** Calculating Control Points ***/
82 /* We see the polynomial as a Bezier curve for some control points to find */
85 For 5 control points (polynomial of degree 4) this is:
87 0 p0 = F((*IntervalMin))
88 1/4 p1 = F((*IntervalMin)) + ((*IntervalMax) - (*IntervalMin))/4
90 2/4 p2 = 1/6 * (16*F(((*IntervalMax) + (*IntervalMin))/2)
91 - (p0 + 4*p1 + 4*p3 + p4))
92 3/4 p3 = F((*IntervalMax)) - ((*IntervalMax) - (*IntervalMin))/4
94 1 p4 = F((*IntervalMax))
97 /* x,y,z,dx,dy,dz are constant during searching */
99 D[0] = (FunctionClass->*Derivative)(*IntervalMin);
101 P[0][0] = (*IntervalMin);
102 P[0][1] = (FunctionClass->*Function)(*IntervalMin);
105 if (std::fabs(P[0][1]) < Precision) {
109 if (((*IntervalMax) - (*IntervalMin)) < POLEPSILON) {
113 P[1][0] = (*IntervalMin) + ((*IntervalMax) - (*IntervalMin))/4;
114 P[1][1] = P[0][1] + (((*IntervalMax) - (*IntervalMin))/4.0) * D[0];
116 D[1] = (FunctionClass->*Derivative)(*IntervalMax);
118 P[4][0] = (*IntervalMax);
119 P[4][1] = (FunctionClass->*Function)(*IntervalMax);
121 P[3][0] = (*IntervalMax) - ((*IntervalMax) - (*IntervalMin))/4;
122 P[3][1] = P[4][1] - ((*IntervalMax) - (*IntervalMin))/4 * D[1];
124 P[2][0] = ((*IntervalMax) + (*IntervalMin))/2;
125 P[2][1] = (16*(FunctionClass->*Function)(((*IntervalMax)+(*IntervalMin))/2)
126 - (P[0][1] + 4*P[1][1] + 4*P[3][1] + P[4][1]))/6 ;
129 G4double Intersection ;
132 NewMin = (*IntervalMax) ;
133 NewMax = (*IntervalMin) ;
138 /* there is an intersection only if each have different signs */
139 if (((P[j][1] > -Precision) && (P[i][1] < Precision)) ||
140 ((P[j][1] < Precision) && (P[i][1] > -Precision))) {
142 Intersection = P[j][0] - P[j][1]*((P[i][0] - P[j][0])/
143 (P[i][1] - P[j][1]));
144 if (Intersection < NewMin) {
145 NewMin = Intersection;
147 if (Intersection > NewMax) {
148 NewMax = Intersection;
154 if (IntervalIsVoid != 1) {
155 (*IntervalMax) = NewMax;
156 (*IntervalMin) = NewMin;
160 if (IntervalIsVoid == 1) {
167 template <class T, class F>
168 G4double G4PolynomialSolver<T,F>::Newton (G4double IntervalMin,
169 G4double IntervalMax)
171 /* So now we have a good guess and an interval where
172 if there are an intersection the root must be */
175 G4double Gradient = 0;
182 /* Reduce interval before applying Newton Method */
186 while ((NewtonIsSafe = BezierClipping(&IntervalMin,&IntervalMax)) == 0) ;
188 if (NewtonIsSafe == -1) {
193 Lambda = IntervalMin;
194 Value = (FunctionClass->*Function)(Lambda);
197 // while ((std::fabs(Value) > Precision)) {
200 Value = (FunctionClass->*Function)(Lambda);
202 Gradient = (FunctionClass->*Derivative)(Lambda);
204 Lambda = Lambda - Value/Gradient ;
206 if (std::fabs(Value) <= Precision) {