1 | // |
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2 | // ******************************************************************** |
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3 | // * License and Disclaimer * |
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4 | // * * |
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5 | // * The Geant4 software is copyright of the Copyright Holders of * |
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6 | // * the Geant4 Collaboration. It is provided under the terms and * |
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7 | // * conditions of the Geant4 Software License, included in the file * |
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8 | // * LICENSE and available at http://cern.ch/geant4/license . These * |
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9 | // * include a list of copyright holders. * |
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10 | // * * |
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11 | // * Neither the authors of this software system, nor their employing * |
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12 | // * institutes,nor the agencies providing financial support for this * |
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13 | // * work make any representation or warranty, express or implied, * |
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14 | // * regarding this software system or assume any liability for its * |
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15 | // * use. Please see the license in the file LICENSE and URL above * |
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16 | // * for the full disclaimer and the limitation of liability. * |
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17 | // * * |
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18 | // * This code implementation is the result of the scientific and * |
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19 | // * technical work of the GEANT4 collaboration. * |
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20 | // * By using, copying, modifying or distributing the software (or * |
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21 | // * any work based on the software) you agree to acknowledge its * |
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22 | // * use in resulting scientific publications, and indicate your * |
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23 | // * acceptance of all terms of the Geant4 Software license. * |
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24 | // ******************************************************************** |
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25 | // |
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26 | // |
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27 | // $Id: G4AnalyticalPolSolver.cc,v 1.7 2007/11/13 17:35:06 gcosmo Exp $ |
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28 | // GEANT4 tag $Name: geant4-09-02-ref-02 $ |
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29 | // |
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30 | |
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31 | #include "globals.hh" |
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32 | #include <complex> |
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33 | |
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34 | #include "G4AnalyticalPolSolver.hh" |
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35 | |
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36 | ////////////////////////////////////////////////////////////////////////////// |
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37 | |
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38 | G4AnalyticalPolSolver::G4AnalyticalPolSolver() {;} |
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39 | |
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40 | ////////////////////////////////////////////////////////////////////////////// |
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41 | |
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42 | G4AnalyticalPolSolver::~G4AnalyticalPolSolver() {;} |
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43 | |
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44 | ////////////////////////////////////////////////////////////////////////////// |
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45 | // |
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46 | // Array r[3][5] p[5] |
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47 | // Roots of poly p[0] x^2 + p[1] x+p[2]=0 |
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48 | // |
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49 | // x = r[1][k] + i r[2][k]; k = 1, 2 |
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50 | |
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51 | G4int G4AnalyticalPolSolver::QuadRoots( G4double p[5], G4double r[3][5] ) |
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52 | { |
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53 | G4double b, c, d2, d; |
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54 | |
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55 | b = -p[1]/p[0]/2.; |
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56 | c = p[2]/p[0]; |
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57 | d2 = b*b - c; |
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58 | |
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59 | if( d2 >= 0. ) |
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60 | { |
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61 | d = std::sqrt(d2); |
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62 | r[1][1] = b - d; |
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63 | r[1][2] = b + d; |
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64 | r[2][1] = 0.; |
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65 | r[2][2] = 0.; |
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66 | } |
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67 | else |
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68 | { |
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69 | d = std::sqrt(-d2); |
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70 | r[2][1] = d; |
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71 | r[2][2] = -d; |
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72 | r[1][1] = b; |
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73 | r[1][2] = b; |
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74 | } |
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75 | |
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76 | return 2; |
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77 | } |
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78 | |
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79 | ////////////////////////////////////////////////////////////////////////////// |
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80 | // |
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81 | // Array r[3][5] p[5] |
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82 | // Roots of poly p[0] x^3 + p[1] x^2...+p[3]=0 |
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83 | // x=r[1][k] + i r[2][k] k=1,...,3 |
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84 | // Assumes 0<arctan(x)<pi/2 for x>0 |
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85 | |
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86 | G4int G4AnalyticalPolSolver::CubicRoots( G4double p[5], G4double r[3][5] ) |
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87 | { |
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88 | G4double x,t,b,c,d; |
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89 | G4int k; |
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90 | |
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91 | if( p[0] != 1. ) |
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92 | { |
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93 | for(k = 1; k < 4; k++ ) { p[k] = p[k]/p[0]; } |
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94 | p[0] = 1.; |
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95 | } |
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96 | x = p[1]/3.0; |
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97 | t = x*p[1]; |
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98 | b = 0.5*( x*( t/1.5 - p[2] ) + p[3] ); |
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99 | t = ( t - p[2] )/3.0; |
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100 | c = t*t*t; |
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101 | d = b*b - c; |
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102 | |
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103 | if( d >= 0. ) |
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104 | { |
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105 | d = std::pow( (std::sqrt(d) + std::fabs(b) ), 1.0/3.0 ); |
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106 | |
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107 | if( d != 0. ) |
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108 | { |
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109 | if( b > 0. ) { b = -d; } |
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110 | else { b = d; } |
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111 | c = t/b; |
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112 | } |
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113 | d = std::sqrt(0.75)*(b - c); |
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114 | r[2][2] = d; |
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115 | b = b + c; |
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116 | c = -0.5*b-x; |
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117 | r[1][2] = c; |
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118 | |
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119 | if( ( b > 0. && x <= 0. ) || ( b < 0. && x > 0. ) ) |
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120 | { |
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121 | r[1][1] = c; |
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122 | r[2][1] = -d; |
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123 | r[1][3] = b - x; |
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124 | r[2][3] = 0; |
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125 | } |
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126 | else |
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127 | { |
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128 | r[1][1] = b - x; |
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129 | r[2][1] = 0.; |
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130 | r[1][3] = c; |
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131 | r[2][3] = -d; |
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132 | } |
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133 | } // end of 2 equal or complex roots |
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134 | else |
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135 | { |
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136 | if( b == 0. ) { d = std::atan(1.0)/1.5; } |
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137 | else { d = std::atan( std::sqrt(-d)/std::fabs(b) )/3.0; } |
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138 | |
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139 | if( b < 0. ) { b = std::sqrt(t)*2.0; } |
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140 | else { b = -2.0*std::sqrt(t); } |
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141 | |
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142 | c = std::cos(d)*b; |
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143 | t = -std::sqrt(0.75)*std::sin(d)*b - 0.5*c; |
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144 | d = -t - c - x; |
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145 | c = c - x; |
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146 | t = t - x; |
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147 | |
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148 | if( std::fabs(c) > std::fabs(t) ) { r[1][3] = c; } |
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149 | else |
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150 | { |
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151 | r[1][3] = t; |
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152 | t = c; |
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153 | } |
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154 | if( std::fabs(d) > std::fabs(t) ) { r[1][2] = d; } |
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155 | else |
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156 | { |
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157 | r[1][2] = t; |
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158 | t = d; |
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159 | } |
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160 | r[1][1] = t; |
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161 | |
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162 | for(k = 1; k < 4; k++ ) { r[2][k] = 0.; } |
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163 | } |
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164 | return 0; |
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165 | } |
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166 | |
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167 | ////////////////////////////////////////////////////////////////////////////// |
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168 | // |
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169 | // Array r[3][5] p[5] |
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170 | // Roots of poly p[0] x^4 + p[1] x^3...+p[4]=0 |
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171 | // x=r[1][k] + i r[2][k] k=1,...,4 |
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172 | |
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173 | G4int G4AnalyticalPolSolver::BiquadRoots( G4double p[5], G4double r[3][5] ) |
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174 | { |
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175 | G4double a, b, c, d, e; |
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176 | G4int i, k, j, noRoots; |
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177 | |
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178 | if(p[0] != 1.0) |
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179 | { |
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180 | for( k = 1; k < 5; k++) { p[k] = p[k]/p[0]; } |
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181 | p[0] = 1.; |
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182 | } |
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183 | e = 0.25*p[1]; |
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184 | b = 2*e; |
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185 | c = b*b; |
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186 | d = 0.75*c; |
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187 | b = p[3] + b*( c - p[2] ); |
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188 | a = p[2] - d; |
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189 | c = p[4] + e*( e*a - p[3] ); |
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190 | a = a - d; |
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191 | |
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192 | p[1] = 0.5*a; |
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193 | p[2] = (p[1]*p[1]-c)*0.25; |
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194 | p[3] = b*b/(-64.0); |
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195 | |
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196 | if( p[3] < 0. ) |
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197 | { |
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198 | noRoots = CubicRoots(p,r); |
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199 | |
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200 | for( k = 1; k < 4; k++ ) |
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201 | { |
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202 | if( r[2][k] == 0. && r[1][k] > 0 ) |
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203 | { |
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204 | d = r[1][k]*4; |
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205 | a = a + d; |
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206 | |
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207 | if ( a >= 0. && b >= 0.) { p[1] = std::sqrt(d); } |
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208 | else if( a <= 0. && b <= 0.) { p[1] = std::sqrt(d); } |
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209 | else { p[1] = -std::sqrt(d); } |
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210 | |
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211 | b = 0.5*( a + b/p[1] ); |
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212 | |
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213 | p[2] = c/b; |
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214 | noRoots = QuadRoots(p,r); |
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215 | |
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216 | for( i = 1; i < 3; i++ ) |
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217 | { |
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218 | for( j = 1; j < 3; j++ ) { r[j][i+2] = r[j][i]; } |
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219 | } |
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220 | p[1] = -p[1]; |
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221 | p[2] = b; |
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222 | noRoots = QuadRoots(p,r); |
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223 | |
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224 | for( i = 1; i < 5; i++ ) { r[1][i] = r[1][i] - e; } |
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225 | |
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226 | return 4; |
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227 | } |
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228 | } |
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229 | } |
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230 | if( p[2] < 0. ) |
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231 | { |
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232 | b = std::sqrt(c); |
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233 | d = b + b - a; |
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234 | p[1] = 0.; |
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235 | |
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236 | if( d > 0. ) { p[1] = std::sqrt(d); } |
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237 | } |
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238 | else |
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239 | { |
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240 | if( p[1] > 0.) { b = std::sqrt(p[2])*2.0 + p[1]; } |
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241 | else { b = -std::sqrt(p[2])*2.0 + p[1]; } |
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242 | |
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243 | if( b != 0.) { p[1] = 0; } |
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244 | else |
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245 | { |
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246 | for(k = 1; k < 5; k++ ) |
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247 | { |
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248 | r[1][k] = -e; |
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249 | r[2][k] = 0; |
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250 | } |
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251 | return 0; |
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252 | } |
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253 | } |
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254 | |
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255 | p[2] = c/b; |
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256 | noRoots = QuadRoots(p,r); |
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257 | |
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258 | for( k = 1; k < 3; k++ ) |
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259 | { |
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260 | for( j = 1; j < 3; j++ ) { r[j][k+2] = r[j][k]; } |
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261 | } |
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262 | p[1] = -p[1]; |
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263 | p[2] = b; |
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264 | noRoots = QuadRoots(p,r); |
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265 | |
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266 | for( k = 1; k < 5; k++ ) { r[1][k] = r[1][k] - e; } |
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267 | |
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268 | return 4; |
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269 | } |
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270 | |
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271 | ////////////////////////////////////////////////////////////////////////////// |
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272 | |
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273 | G4int G4AnalyticalPolSolver::QuarticRoots( G4double p[5], G4double r[3][5]) |
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274 | { |
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275 | G4double a0, a1, a2, a3, y1; |
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276 | G4double R2, D2, E2, D, E, R = 0.; |
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277 | G4double a, b, c, d, ds; |
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278 | |
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279 | G4double reRoot[4]; |
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280 | G4int k, noRoots, noReRoots = 0; |
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281 | |
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282 | for( k = 0; k < 4; k++ ) { reRoot[k] = DBL_MAX; } |
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283 | |
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284 | if( p[0] != 1.0 ) |
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285 | { |
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286 | for( k = 1; k < 5; k++) { p[k] = p[k]/p[0]; } |
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287 | p[0] = 1.; |
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288 | } |
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289 | a3 = p[1]; |
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290 | a2 = p[2]; |
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291 | a1 = p[3]; |
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292 | a0 = p[4]; |
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293 | |
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294 | // resolvent cubic equation cofs: |
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295 | |
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296 | p[1] = -a2; |
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297 | p[2] = a1*a3 - 4*a0; |
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298 | p[3] = 4*a2*a0 - a1*a1 - a3*a3*a0; |
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299 | |
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300 | noRoots = CubicRoots(p,r); |
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301 | |
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302 | for( k = 1; k < 4; k++ ) |
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303 | { |
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304 | if( r[2][k] == 0. ) // find a real root |
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305 | { |
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306 | noReRoots++; |
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307 | reRoot[k] = r[1][k]; |
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308 | } |
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309 | else reRoot[k] = DBL_MAX; // kInfinity; |
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310 | } |
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311 | y1 = DBL_MAX; // kInfinity; |
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312 | for( k = 1; k < 4; k++ ) |
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313 | { |
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314 | if ( reRoot[k] < y1 ) { y1 = reRoot[k]; } |
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315 | } |
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316 | |
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317 | R2 = 0.25*a3*a3 - a2 + y1; |
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318 | b = 0.25*(4*a3*a2 - 8*a1 - a3*a3*a3); |
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319 | c = 0.75*a3*a3 - 2*a2; |
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320 | a = c - R2; |
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321 | d = 4*y1*y1 - 16*a0; |
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322 | |
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323 | if( R2 > 0.) |
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324 | { |
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325 | R = std::sqrt(R2); |
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326 | D2 = a + b/R; |
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327 | E2 = a - b/R; |
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328 | |
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329 | if( D2 >= 0. ) |
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330 | { |
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331 | D = std::sqrt(D2); |
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332 | r[1][1] = -0.25*a3 + 0.5*R + 0.5*D; |
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333 | r[1][2] = -0.25*a3 + 0.5*R - 0.5*D; |
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334 | r[2][1] = 0.; |
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335 | r[2][2] = 0.; |
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336 | } |
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337 | else |
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338 | { |
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339 | D = std::sqrt(-D2); |
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340 | r[1][1] = -0.25*a3 + 0.5*R; |
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341 | r[1][2] = -0.25*a3 + 0.5*R; |
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342 | r[2][1] = 0.5*D; |
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343 | r[2][2] = -0.5*D; |
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344 | } |
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345 | if( E2 >= 0. ) |
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346 | { |
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347 | E = std::sqrt(E2); |
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348 | r[1][3] = -0.25*a3 - 0.5*R + 0.5*E; |
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349 | r[1][4] = -0.25*a3 - 0.5*R - 0.5*E; |
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350 | r[2][3] = 0.; |
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351 | r[2][4] = 0.; |
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352 | } |
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353 | else |
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354 | { |
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355 | E = std::sqrt(-E2); |
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356 | r[1][3] = -0.25*a3 - 0.5*R; |
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357 | r[1][4] = -0.25*a3 - 0.5*R; |
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358 | r[2][3] = 0.5*E; |
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359 | r[2][4] = -0.5*E; |
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360 | } |
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361 | } |
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362 | else if( R2 < 0.) |
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363 | { |
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364 | R = std::sqrt(-R2); |
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365 | G4complex CD2(a,-b/R); |
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366 | G4complex CD = std::sqrt(CD2); |
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367 | |
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368 | r[1][1] = -0.25*a3 + 0.5*real(CD); |
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369 | r[1][2] = -0.25*a3 - 0.5*real(CD); |
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370 | r[2][1] = 0.5*R + 0.5*imag(CD); |
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371 | r[2][2] = 0.5*R - 0.5*imag(CD); |
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372 | G4complex CE2(a,b/R); |
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373 | G4complex CE = std::sqrt(CE2); |
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374 | |
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375 | r[1][3] = -0.25*a3 + 0.5*real(CE); |
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376 | r[1][4] = -0.25*a3 - 0.5*real(CE); |
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377 | r[2][3] = -0.5*R + 0.5*imag(CE); |
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378 | r[2][4] = -0.5*R - 0.5*imag(CE); |
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379 | } |
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380 | else // R2=0 case |
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381 | { |
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382 | if(d >= 0.) |
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383 | { |
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384 | D2 = c + std::sqrt(d); |
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385 | E2 = c - std::sqrt(d); |
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386 | |
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387 | if( D2 >= 0. ) |
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388 | { |
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389 | D = std::sqrt(D2); |
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390 | r[1][1] = -0.25*a3 + 0.5*R + 0.5*D; |
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391 | r[1][2] = -0.25*a3 + 0.5*R - 0.5*D; |
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392 | r[2][1] = 0.; |
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393 | r[2][2] = 0.; |
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394 | } |
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395 | else |
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396 | { |
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397 | D = std::sqrt(-D2); |
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398 | r[1][1] = -0.25*a3 + 0.5*R; |
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399 | r[1][2] = -0.25*a3 + 0.5*R; |
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400 | r[2][1] = 0.5*D; |
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401 | r[2][2] = -0.5*D; |
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402 | } |
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403 | if( E2 >= 0. ) |
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404 | { |
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405 | E = std::sqrt(E2); |
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406 | r[1][3] = -0.25*a3 - 0.5*R + 0.5*E; |
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407 | r[1][4] = -0.25*a3 - 0.5*R - 0.5*E; |
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408 | r[2][3] = 0.; |
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409 | r[2][4] = 0.; |
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410 | } |
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411 | else |
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412 | { |
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413 | E = std::sqrt(-E2); |
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414 | r[1][3] = -0.25*a3 - 0.5*R; |
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415 | r[1][4] = -0.25*a3 - 0.5*R; |
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416 | r[2][3] = 0.5*E; |
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417 | r[2][4] = -0.5*E; |
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418 | } |
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419 | } |
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420 | else |
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421 | { |
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422 | ds = std::sqrt(-d); |
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423 | G4complex CD2(c,ds); |
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424 | G4complex CD = std::sqrt(CD2); |
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425 | |
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426 | r[1][1] = -0.25*a3 + 0.5*real(CD); |
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427 | r[1][2] = -0.25*a3 - 0.5*real(CD); |
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428 | r[2][1] = 0.5*R + 0.5*imag(CD); |
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429 | r[2][2] = 0.5*R - 0.5*imag(CD); |
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430 | |
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431 | G4complex CE2(c,-ds); |
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432 | G4complex CE = std::sqrt(CE2); |
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433 | |
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434 | r[1][3] = -0.25*a3 + 0.5*real(CE); |
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435 | r[1][4] = -0.25*a3 - 0.5*real(CE); |
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436 | r[2][3] = -0.5*R + 0.5*imag(CE); |
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437 | r[2][4] = -0.5*R - 0.5*imag(CE); |
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438 | } |
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439 | } |
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440 | return 4; |
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441 | } |
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442 | |
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443 | // |
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444 | // |
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445 | ////////////////////////////////////////////////////////////////////////////// |
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