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25 // 25 //
26 // G4GaussLegendreQ class implementation <<
27 // 26 //
28 // Author: V.Grichine, 13.05.1997 << 27 //
29 // ------------------------------------------- <<
30 28
31 #include "G4GaussLegendreQ.hh" 29 #include "G4GaussLegendreQ.hh"
32 #include "G4PhysicalConstants.hh" 30 #include "G4PhysicalConstants.hh"
33 31
34 G4GaussLegendreQ::G4GaussLegendreQ(function pF << 32 G4GaussLegendreQ::G4GaussLegendreQ( function pFunction )
35 : G4VGaussianQuadrature(pFunction) << 33 : G4VGaussianQuadrature(pFunction)
36 {} << 34 {
>> 35 }
37 36
38 // ------------------------------------------- 37 // --------------------------------------------------------------------------
39 // 38 //
40 // Constructor for GaussLegendre quadrature me 39 // Constructor for GaussLegendre quadrature method. The value nLegendre sets
41 // the accuracy required, i.e the number of po 40 // the accuracy required, i.e the number of points where the function pFunction
42 // will be evaluated during integration. The c 41 // will be evaluated during integration. The constructor creates the arrays for
43 // abscissas and weights that are used in Gaus << 42 // abscissas and weights that are used in Gauss-Legendre quadrature method.
44 // The values a and b are the limits of integr 43 // The values a and b are the limits of integration of the pFunction.
45 // nLegendre MUST BE EVEN !!! 44 // nLegendre MUST BE EVEN !!!
46 45
47 G4GaussLegendreQ::G4GaussLegendreQ(function pF << 46 G4GaussLegendreQ::G4GaussLegendreQ( function pFunction,
48 : G4VGaussianQuadrature(pFunction) << 47 G4int nLegendre )
>> 48 : G4VGaussianQuadrature(pFunction)
49 { 49 {
50 const G4double tolerance = 1.6e-10; << 50 const G4double tolerance = 1.6e-10 ;
51 G4int k = nLegendre; << 51 G4int k = nLegendre ;
52 fNumber = (nLegendre + 1) / << 52 fNumber = (nLegendre + 1)/2 ;
53 if(2 * fNumber != k) << 53 if(2*fNumber != k)
54 { << 54 {
55 G4Exception("G4GaussLegendreQ::G4GaussLege << 55 G4Exception("G4GaussLegendreQ::G4GaussLegendreQ()", "InvalidCall",
56 FatalException, "Invalid nLege << 56 FatalException, "Invalid nLegendre argument !") ;
57 } << 57 }
58 G4double newton0 = 0.0, newton1 = 0.0, temp1 << 58 G4double newton0=0.0, newton1=0.0,
59 temp = 0.0; << 59 temp1=0.0, temp2=0.0, temp3=0.0, temp=0.0 ;
60 << 60
61 fAbscissa = new G4double[fNumber]; << 61 fAbscissa = new G4double[fNumber] ;
62 fWeight = new G4double[fNumber]; << 62 fWeight = new G4double[fNumber] ;
63 << 63
64 for(G4int i = 1; i <= fNumber; ++i) // Loop << 64 for(G4int i=1;i<=fNumber;i++) // Loop over the desired roots
65 { << 65 {
66 newton0 = std::cos(pi * (i - 0.25) / (k + << 66 newton0 = std::cos(pi*(i - 0.25)/(k + 0.5)) ; // Initial root
67 do << 67 do // approximation
68 { << 68 { // loop of Newton's method
69 temp1 = 1.0; << 69 temp1 = 1.0 ;
70 temp2 = 0.0; << 70 temp2 = 0.0 ;
71 for(G4int j = 1; j <= k; ++j) << 71 for(G4int j=1;j<=k;j++)
72 { << 72 {
73 temp3 = temp2; << 73 temp3 = temp2 ;
74 temp2 = temp1; << 74 temp2 = temp1 ;
75 temp1 = ((2.0 * j - 1.0) * newton0 * t << 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
76 } 80 }
77 temp = k * (newton0 * temp1 - temp2) << 81 while(std::fabs(newton0 - newton1) > tolerance) ;
78 newton1 = newton0; << 82
79 newton0 = newton1 - temp1 / temp; // Ne << 83 fAbscissa[fNumber-i] = newton0 ;
80 } while(std::fabs(newton0 - newton1) > tol << 84 fWeight[fNumber-i] = 2.0/((1.0 - newton0*newton0)*temp*temp) ;
81 << 85 }
82 fAbscissa[fNumber - i] = newton0; <<
83 fWeight[fNumber - i] = 2.0 / ((1.0 - new <<
84 } <<
85 } 86 }
86 87
87 // ------------------------------------------- 88 // --------------------------------------------------------------------------
88 // 89 //
89 // Returns the integral of the function to be 90 // Returns the integral of the function to be pointed by fFunction between a
90 // and b, by 2*fNumber point Gauss-Legendre in 91 // and b, by 2*fNumber point Gauss-Legendre integration: the function is
91 // evaluated exactly 2*fNumber times at interi 92 // evaluated exactly 2*fNumber times at interior points in the range of
92 // integration. Since the weights and abscissa 93 // integration. Since the weights and abscissas are, in this case, symmetric
93 // around the midpoint of the range of integra 94 // around the midpoint of the range of integration, there are actually only
94 // fNumber distinct values of each. 95 // fNumber distinct values of each.
95 96
96 G4double G4GaussLegendreQ::Integral(G4double a << 97 G4double
>> 98 G4GaussLegendreQ::Integral(G4double a, G4double b) const
97 { 99 {
98 G4double xMean = 0.5 * (a + b), xDiff = 0.5 << 100 G4double xMean = 0.5*(a + b),
99 dx = 0.0; << 101 xDiff = 0.5*(b - a),
100 for(G4int i = 0; i < fNumber; ++i) << 102 integral = 0.0, dx = 0.0 ;
101 { << 103 for(G4int i=0;i<fNumber;i++)
102 dx = xDiff * fAbscissa[i]; << 104 {
103 integral += fWeight[i] * (fFunction(xMean << 105 dx = xDiff*fAbscissa[i] ;
104 } << 106 integral += fWeight[i]*(fFunction(xMean + dx) + fFunction(xMean - dx)) ;
105 return integral *= xDiff; << 107 }
>> 108 return integral *= xDiff ;
106 } 109 }
107 110
108 // ------------------------------------------- 111 // --------------------------------------------------------------------------
109 // 112 //
110 // Returns the integral of the function to be 113 // Returns the integral of the function to be pointed by fFunction between a
111 // and b, by ten point Gauss-Legendre integrat 114 // and b, by ten point Gauss-Legendre integration: the function is evaluated
112 // exactly ten times at interior points in the 115 // exactly ten times at interior points in the range of integration. Since the
113 // weights and abscissas are, in this case, sy 116 // weights and abscissas are, in this case, symmetric around the midpoint of
114 // the range of integration, there are actuall 117 // the range of integration, there are actually only five distinct values of
115 // each. 118 // each.
116 119
117 G4double G4GaussLegendreQ::QuickIntegral(G4dou << 120 G4double
>> 121 G4GaussLegendreQ::QuickIntegral(G4double a, G4double b) const
118 { 122 {
119 // From Abramowitz M., Stegan I.A. 1964 , Ha << 123 // From Abramowitz M., Stegan I.A. 1964 , Handbook of Math... , p. 916
120 124
121 static const G4double abscissa[] = { 0.14887 << 125 static const G4double abscissa[] = { 0.148874338981631, 0.433395394129247,
122 0.67940 << 126 0.679409568299024, 0.865063366688985,
123 0.97390 << 127 0.973906528517172 } ;
124 << 128
125 static const G4double weight[] = { 0.2955242 << 129 static const G4double weight[] = { 0.295524224714753, 0.269266719309996,
126 0.2190863 << 130 0.219086362515982, 0.149451349150581,
127 0.0666713 << 131 0.066671344308688 } ;
128 G4double xMean = 0.5 * (a + b), xDiff = 0.5 << 132 G4double xMean = 0.5*(a + b),
129 dx = 0.0; << 133 xDiff = 0.5*(b - a),
130 for(G4int i = 0; i < 5; ++i) << 134 integral = 0.0, dx = 0.0 ;
131 { << 135 for(G4int i=0;i<5;i++)
132 dx = xDiff * abscissa[i]; << 136 {
133 integral += weight[i] * (fFunction(xMean + << 137 dx = xDiff*abscissa[i] ;
134 } << 138 integral += weight[i]*(fFunction(xMean + dx) + fFunction(xMean - dx)) ;
135 return integral *= xDiff; << 139 }
>> 140 return integral *= xDiff ;
136 } 141 }
137 142
138 // ------------------------------------------- 143 // -------------------------------------------------------------------------
139 // 144 //
140 // Returns the integral of the function to be 145 // Returns the integral of the function to be pointed by fFunction between a
141 // and b, by 96 point Gauss-Legendre integrati 146 // and b, by 96 point Gauss-Legendre integration: the function is evaluated
142 // exactly ten times at interior points in the 147 // exactly ten times at interior points in the range of integration. Since the
143 // weights and abscissas are, in this case, sy 148 // weights and abscissas are, in this case, symmetric around the midpoint of
144 // the range of integration, there are actuall 149 // the range of integration, there are actually only five distinct values of
145 // each. 150 // each.
146 151
147 G4double G4GaussLegendreQ::AccurateIntegral(G4 << 152 G4double
148 { << 153 G4GaussLegendreQ::AccurateIntegral(G4double a, G4double b) const
149 // From Abramowitz M., Stegan I.A. 1964 , Ha << 154 {
150 << 155 // From Abramowitz M., Stegan I.A. 1964 , Handbook of Math... , p. 919
151 static const G4double abscissa[] = { << 156
152 0.016276744849602969579, 0.048812985136049 << 157 static const
153 0.081297495464425558994, 0.113695850110665 << 158 G4double abscissa[] = {
154 0.145973714654896941989, 0.178096882367618 << 159 0.016276744849602969579, 0.048812985136049731112,
155 << 160 0.081297495464425558994, 0.113695850110665920911,
156 0.210031310460567203603, 0.241743156163840 << 161 0.145973714654896941989, 0.178096882367618602759, // 6
157 0.273198812591049141487, 0.304364944354496 << 162
158 0.335208522892625422616, 0.365696861472313 << 163 0.210031310460567203603, 0.241743156163840012328,
159 << 164 0.273198812591049141487, 0.304364944354496353024,
160 0.395797649828908603285, 0.425478988407300 << 165 0.335208522892625422616, 0.365696861472313635031, // 12
161 0.454709422167743008636, 0.483457973920596 << 166
162 0.511694177154667673586, 0.539388108324357 << 167 0.395797649828908603285, 0.425478988407300545365,
163 << 168 0.454709422167743008636, 0.483457973920596359768,
164 0.566510418561397168404, 0.593032364777572 << 169 0.511694177154667673586, 0.539388108324357436227, // 18
165 0.618925840125468570386, 0.644163403784967 << 170
166 0.668718310043916153953, 0.692564536642171 << 171 0.566510418561397168404, 0.593032364777572080684,
167 << 172 0.618925840125468570386, 0.644163403784967106798,
168 0.715676812348967626225, 0.738030643744400 << 173 0.668718310043916153953, 0.692564536642171561344, // 24
169 0.759602341176647498703, 0.780369043867433 << 174
170 0.800308744139140817229, 0.819400310737931 << 175 0.715676812348967626225, 0.738030643744400132851,
171 << 176 0.759602341176647498703, 0.780369043867433217604,
172 0.837623511228187121494, 0.854959033434601 << 177 0.800308744139140817229, 0.819400310737931675539, // 30
173 0.871388505909296502874, 0.886894517402420 << 178
174 0.901460635315852341319, 0.915071423120898 << 179 0.837623511228187121494, 0.854959033434601455463,
175 << 180 0.871388505909296502874, 0.886894517402420416057,
176 0.927712456722308690965, 0.939370339752755 << 181 0.901460635315852341319, 0.915071423120898074206, // 36
177 0.950032717784437635756, 0.959688291448742 << 182
178 0.968326828463264212174, 0.975939174585136 << 183 0.927712456722308690965, 0.939370339752755216932,
179 << 184 0.950032717784437635756, 0.959688291448742539300,
180 0.982517263563014677447, 0.988054126329623 << 185 0.968326828463264212174, 0.975939174585136466453, // 42
181 0.992543900323762624572, 0.995981842987209 << 186
182 0.998364375863181677724, 0.999689503883230 << 187 0.982517263563014677447, 0.988054126329623799481,
183 }; << 188 0.992543900323762624572, 0.995981842987209290650,
184 << 189 0.998364375863181677724, 0.999689503883230766828 // 48
185 static const G4double weight[] = { << 190 } ;
186 0.032550614492363166242, 0.032516118713868 << 191
187 0.032447163714064269364, 0.032343822568575 << 192 static const
188 0.032206204794030250669, 0.032034456231992 << 193 G4double weight[] = {
189 << 194 0.032550614492363166242, 0.032516118713868835987,
190 0.031828758894411006535, 0.031589330770727 << 195 0.032447163714064269364, 0.032343822568575928429,
191 0.031316425596862355813, 0.031010332586313 << 196 0.032206204794030250669, 0.032034456231992663218, // 6
192 0.030671376123669149014, 0.030299915420827 << 197
193 << 198 0.031828758894411006535, 0.031589330770727168558,
194 0.029896344136328385984, 0.029461089958167 << 199 0.031316425596862355813, 0.031010332586313837423,
195 0.028994614150555236543, 0.028497411065085 << 200 0.030671376123669149014, 0.030299915420827593794, // 12
196 0.027970007616848334440, 0.027412962726029 << 201
197 << 202 0.029896344136328385984, 0.029461089958167905970,
198 0.026826866725591762198, 0.026212340735672 << 203 0.028994614150555236543, 0.028497411065085385646,
199 0.025570036005349361499, 0.024900633222483 << 204 0.027970007616848334440, 0.027412962726029242823, // 18
200 0.024204841792364691282, 0.023483399085926 << 205
201 << 206 0.026826866725591762198, 0.026212340735672413913,
202 0.022737069658329374001, 0.021966644438744 << 207 0.025570036005349361499, 0.024900633222483610288,
203 0.021172939892191298988, 0.020356797154333 << 208 0.024204841792364691282, 0.023483399085926219842, // 24
204 0.019519081140145022410, 0.018660679627411 << 209
205 << 210 0.022737069658329374001, 0.021966644438744349195,
206 0.017782502316045260838, 0.016885479864245 << 211 0.021172939892191298988, 0.020356797154333324595,
207 0.015970562902562291381, 0.015038721026994 << 212 0.019519081140145022410, 0.018660679627411467385, // 30
208 0.014090941772314860916, 0.013128229566961 << 213
209 << 214 0.017782502316045260838, 0.016885479864245172450,
210 0.012151604671088319635, 0.011162102099838 << 215 0.015970562902562291381, 0.015038721026994938006,
211 0.010160770535008415758, 0.009148671230783 << 216 0.014090941772314860916, 0.013128229566961572637, // 36
212 0.008126876925698759217, 0.007096470791153 << 217
213 << 218 0.012151604671088319635, 0.011162102099838498591,
214 0.006058545504235961683, 0.005014202742927 << 219 0.010160770535008415758, 0.009148671230783386633,
215 0.003964554338444686674, 0.002910731817934 << 220 0.008126876925698759217, 0.007096470791153865269, // 42
216 0.001853960788946921732, 0.000796792065552 << 221
217 }; << 222 0.006058545504235961683, 0.005014202742927517693,
218 G4double xMean = 0.5 * (a + b), xDiff = 0.5 << 223 0.003964554338444686674, 0.002910731817934946408,
219 dx = 0.0; << 224 0.001853960788946921732, 0.000796792065552012429 // 48
220 for(G4int i = 0; i < 48; ++i) << 225 } ;
221 { << 226 G4double xMean = 0.5*(a + b),
222 dx = xDiff * abscissa[i]; << 227 xDiff = 0.5*(b - a),
223 integral += weight[i] * (fFunction(xMean + << 228 integral = 0.0, dx = 0.0 ;
224 } << 229 for(G4int i=0;i<48;i++)
225 return integral *= xDiff; << 230 {
>> 231 dx = xDiff*abscissa[i] ;
>> 232 integral += weight[i]*(fFunction(xMean + dx) + fFunction(xMean - dx)) ;
>> 233 }
>> 234 return integral *= xDiff ;
226 } 235 }
227 236