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25 //
26 // G4GaussLegendreQ class implementation
27 //
28 // Author: V.Grichine, 13.05.1997
29 // --------------------------------------------------------------------
30
31 #include "G4GaussLegendreQ.hh"
32 #include "G4PhysicalConstants.hh"
33
34 G4GaussLegendreQ::G4GaussLegendreQ(function pFunction)
35 : G4VGaussianQuadrature(pFunction)
36 {}
37
38 // --------------------------------------------------------------------------
39 //
40 // Constructor for GaussLegendre quadrature method. The value nLegendre sets
41 // the accuracy required, i.e the number of points where the function pFunction
42 // will be evaluated during integration. The constructor creates the arrays for
43 // abscissas and weights that are used in Gauss-Legendre quadrature method.
44 // The values a and b are the limits of integration of the pFunction.
45 // nLegendre MUST BE EVEN !!!
46
47 G4GaussLegendreQ::G4GaussLegendreQ(function pFunction, G4int nLegendre)
48 : G4VGaussianQuadrature(pFunction)
49 {
50 const G4double tolerance = 1.6e-10;
51 G4int k = nLegendre;
52 fNumber = (nLegendre + 1) / 2;
53 if(2 * fNumber != k)
54 {
55 G4Exception("G4GaussLegendreQ::G4GaussLegendreQ()", "InvalidCall",
56 FatalException, "Invalid nLegendre argument !");
57 }
58 G4double newton0 = 0.0, newton1 = 0.0, temp1 = 0.0, temp2 = 0.0, temp3 = 0.0,
59 temp = 0.0;
60
61 fAbscissa = new G4double[fNumber];
62 fWeight = new G4double[fNumber];
63
64 for(G4int i = 1; i <= fNumber; ++i) // Loop over the desired roots
65 {
66 newton0 = std::cos(pi * (i - 0.25) / (k + 0.5)); // Initial root
67 do // approximation
68 { // loop of Newton's method
69 temp1 = 1.0;
70 temp2 = 0.0;
71 for(G4int j = 1; j <= k; ++j)
72 {
73 temp3 = temp2;
74 temp2 = temp1;
75 temp1 = ((2.0 * j - 1.0) * newton0 * temp2 - (j - 1.0) * temp3) / j;
76 }
77 temp = k * (newton0 * temp1 - temp2) / (newton0 * newton0 - 1.0);
78 newton1 = newton0;
79 newton0 = newton1 - temp1 / temp; // Newton's method
80 } while(std::fabs(newton0 - newton1) > tolerance);
81
82 fAbscissa[fNumber - i] = newton0;
83 fWeight[fNumber - i] = 2.0 / ((1.0 - newton0 * newton0) * temp * temp);
84 }
85 }
86
87 // --------------------------------------------------------------------------
88 //
89 // Returns the integral of the function to be pointed by fFunction between a
90 // and b, by 2*fNumber point Gauss-Legendre integration: the function is
91 // evaluated exactly 2*fNumber times at interior points in the range of
92 // integration. Since the weights and abscissas are, in this case, symmetric
93 // around the midpoint of the range of integration, there are actually only
94 // fNumber distinct values of each.
95
96 G4double G4GaussLegendreQ::Integral(G4double a, G4double b) const
97 {
98 G4double xMean = 0.5 * (a + b), xDiff = 0.5 * (b - a), integral = 0.0,
99 dx = 0.0;
100 for(G4int i = 0; i < fNumber; ++i)
101 {
102 dx = xDiff * fAbscissa[i];
103 integral += fWeight[i] * (fFunction(xMean + dx) + fFunction(xMean - dx));
104 }
105 return integral *= xDiff;
106 }
107
108 // --------------------------------------------------------------------------
109 //
110 // Returns the integral of the function to be pointed by fFunction between a
111 // and b, by ten point Gauss-Legendre integration: the function is evaluated
112 // exactly ten times at interior points in the range of integration. Since the
113 // weights and abscissas are, in this case, symmetric around the midpoint of
114 // the range of integration, there are actually only five distinct values of
115 // each.
116
117 G4double G4GaussLegendreQ::QuickIntegral(G4double a, G4double b) const
118 {
119 // From Abramowitz M., Stegan I.A. 1964 , Handbook of Math... , p. 916
120
121 static const G4double abscissa[] = { 0.148874338981631, 0.433395394129247,
122 0.679409568299024, 0.865063366688985,
123 0.973906528517172 };
124
125 static const G4double weight[] = { 0.295524224714753, 0.269266719309996,
126 0.219086362515982, 0.149451349150581,
127 0.066671344308688 };
128 G4double xMean = 0.5 * (a + b), xDiff = 0.5 * (b - a), integral = 0.0,
129 dx = 0.0;
130 for(G4int i = 0; i < 5; ++i)
131 {
132 dx = xDiff * abscissa[i];
133 integral += weight[i] * (fFunction(xMean + dx) + fFunction(xMean - dx));
134 }
135 return integral *= xDiff;
136 }
137
138 // -------------------------------------------------------------------------
139 //
140 // Returns the integral of the function to be pointed by fFunction between a
141 // and b, by 96 point Gauss-Legendre integration: the function is evaluated
142 // exactly ten times at interior points in the range of integration. Since the
143 // weights and abscissas are, in this case, symmetric around the midpoint of
144 // the range of integration, there are actually only five distinct values of
145 // each.
146
147 G4double G4GaussLegendreQ::AccurateIntegral(G4double a, G4double b) const
148 {
149 // From Abramowitz M., Stegan I.A. 1964 , Handbook of Math... , p. 919
150
151 static const G4double abscissa[] = {
152 0.016276744849602969579, 0.048812985136049731112,
153 0.081297495464425558994, 0.113695850110665920911,
154 0.145973714654896941989, 0.178096882367618602759, // 6
155
156 0.210031310460567203603, 0.241743156163840012328,
157 0.273198812591049141487, 0.304364944354496353024,
158 0.335208522892625422616, 0.365696861472313635031, // 12
159
160 0.395797649828908603285, 0.425478988407300545365,
161 0.454709422167743008636, 0.483457973920596359768,
162 0.511694177154667673586, 0.539388108324357436227, // 18
163
164 0.566510418561397168404, 0.593032364777572080684,
165 0.618925840125468570386, 0.644163403784967106798,
166 0.668718310043916153953, 0.692564536642171561344, // 24
167
168 0.715676812348967626225, 0.738030643744400132851,
169 0.759602341176647498703, 0.780369043867433217604,
170 0.800308744139140817229, 0.819400310737931675539, // 30
171
172 0.837623511228187121494, 0.854959033434601455463,
173 0.871388505909296502874, 0.886894517402420416057,
174 0.901460635315852341319, 0.915071423120898074206, // 36
175
176 0.927712456722308690965, 0.939370339752755216932,
177 0.950032717784437635756, 0.959688291448742539300,
178 0.968326828463264212174, 0.975939174585136466453, // 42
179
180 0.982517263563014677447, 0.988054126329623799481,
181 0.992543900323762624572, 0.995981842987209290650,
182 0.998364375863181677724, 0.999689503883230766828 // 48
183 };
184
185 static const G4double weight[] = {
186 0.032550614492363166242, 0.032516118713868835987,
187 0.032447163714064269364, 0.032343822568575928429,
188 0.032206204794030250669, 0.032034456231992663218, // 6
189
190 0.031828758894411006535, 0.031589330770727168558,
191 0.031316425596862355813, 0.031010332586313837423,
192 0.030671376123669149014, 0.030299915420827593794, // 12
193
194 0.029896344136328385984, 0.029461089958167905970,
195 0.028994614150555236543, 0.028497411065085385646,
196 0.027970007616848334440, 0.027412962726029242823, // 18
197
198 0.026826866725591762198, 0.026212340735672413913,
199 0.025570036005349361499, 0.024900633222483610288,
200 0.024204841792364691282, 0.023483399085926219842, // 24
201
202 0.022737069658329374001, 0.021966644438744349195,
203 0.021172939892191298988, 0.020356797154333324595,
204 0.019519081140145022410, 0.018660679627411467385, // 30
205
206 0.017782502316045260838, 0.016885479864245172450,
207 0.015970562902562291381, 0.015038721026994938006,
208 0.014090941772314860916, 0.013128229566961572637, // 36
209
210 0.012151604671088319635, 0.011162102099838498591,
211 0.010160770535008415758, 0.009148671230783386633,
212 0.008126876925698759217, 0.007096470791153865269, // 42
213
214 0.006058545504235961683, 0.005014202742927517693,
215 0.003964554338444686674, 0.002910731817934946408,
216 0.001853960788946921732, 0.000796792065552012429 // 48
217 };
218 G4double xMean = 0.5 * (a + b), xDiff = 0.5 * (b - a), integral = 0.0,
219 dx = 0.0;
220 for(G4int i = 0; i < 48; ++i)
221 {
222 dx = xDiff * abscissa[i];
223 integral += weight[i] * (fFunction(xMean + dx) + fFunction(xMean - dx));
224 }
225 return integral *= xDiff;
226 }
227