[833] | 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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[850] | 26 | // $Id: G4JTPolynomialSolver.cc,v 1.7 2008/03/13 09:35:57 gcosmo Exp $ |
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[1058] | 27 | // GEANT4 tag $Name: geant4-09-02-ref-02 $ |
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[833] | 28 | // |
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| 29 | // -------------------------------------------------------------------- |
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| 30 | // GEANT 4 class source file |
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| 31 | // |
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| 32 | // G4JTPolynomialSolver |
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| 33 | // |
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| 34 | // Implementation based on Jenkins-Traub algorithm. |
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| 35 | // -------------------------------------------------------------------- |
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| 36 | |
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| 37 | #include "G4JTPolynomialSolver.hh" |
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| 38 | |
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| 39 | const G4double G4JTPolynomialSolver::base = 2; |
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| 40 | const G4double G4JTPolynomialSolver::eta = DBL_EPSILON; |
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| 41 | const G4double G4JTPolynomialSolver::infin = DBL_MAX; |
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| 42 | const G4double G4JTPolynomialSolver::smalno = DBL_MIN; |
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| 43 | const G4double G4JTPolynomialSolver::are = DBL_EPSILON; |
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| 44 | const G4double G4JTPolynomialSolver::mre = DBL_EPSILON; |
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| 45 | const G4double G4JTPolynomialSolver::lo = DBL_MIN/DBL_EPSILON ; |
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| 46 | |
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| 47 | G4JTPolynomialSolver::G4JTPolynomialSolver() |
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| 48 | { |
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| 49 | } |
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| 50 | |
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| 51 | G4JTPolynomialSolver::~G4JTPolynomialSolver() |
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| 52 | { |
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| 53 | } |
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| 54 | |
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| 55 | G4int G4JTPolynomialSolver::FindRoots(G4double *op, G4int degr, |
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| 56 | G4double *zeror, G4double *zeroi) |
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| 57 | { |
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| 58 | G4double t=0.0, aa=0.0, bb=0.0, cc=0.0, factor=1.0; |
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| 59 | G4double max=0.0, min=infin, xxx=0.0, x=0.0, sc=0.0, bnd=0.0; |
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| 60 | G4double xm=0.0, ff=0.0, df=0.0, dx=0.0; |
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| 61 | G4int cnt=0, nz=0, i=0, j=0, jj=0, l=0, nm1=0, zerok=0; |
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| 62 | |
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| 63 | // Initialization of constants for shift rotation. |
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| 64 | // |
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| 65 | G4double xx = std::sqrt(0.5); |
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| 66 | G4double yy = -xx, |
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| 67 | rot = 94.0*deg; |
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| 68 | G4double cosr = std::cos(rot), |
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| 69 | sinr = std::sin(rot); |
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| 70 | n = degr; |
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| 71 | |
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| 72 | // Algorithm fails if the leading coefficient is zero. |
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| 73 | // |
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| 74 | if (!(op[0] != 0.0)) { return -1; } |
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| 75 | |
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| 76 | // Remove the zeros at the origin, if any. |
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| 77 | // |
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| 78 | while (!(op[n] != 0.0)) |
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| 79 | { |
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| 80 | j = degr - n; |
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| 81 | zeror[j] = 0.0; |
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| 82 | zeroi[j] = 0.0; |
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| 83 | n--; |
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| 84 | } |
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| 85 | if (n < 1) { return -1; } |
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| 86 | |
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| 87 | // Allocate buffers here |
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| 88 | // |
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| 89 | std::vector<G4double> temp(degr+1) ; |
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| 90 | std::vector<G4double> pt(degr+1) ; |
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| 91 | |
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| 92 | p.assign(degr+1,0) ; |
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| 93 | qp.assign(degr+1,0) ; |
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| 94 | k.assign(degr+1,0) ; |
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| 95 | qk.assign(degr+1,0) ; |
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| 96 | svk.assign(degr+1,0) ; |
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| 97 | |
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| 98 | // Make a copy of the coefficients. |
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| 99 | // |
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| 100 | for (i=0;i<=n;i++) |
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| 101 | { p[i] = op[i]; } |
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| 102 | |
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| 103 | do |
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| 104 | { |
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| 105 | if (n == 1) // Start the algorithm for one zero. |
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| 106 | { |
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| 107 | zeror[degr-1] = -p[1]/p[0]; |
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| 108 | zeroi[degr-1] = 0.0; |
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| 109 | n -= 1; |
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| 110 | return degr - n ; |
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| 111 | } |
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| 112 | if (n == 2) // Calculate the final zero or pair of zeros. |
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| 113 | { |
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| 114 | Quadratic(p[0],p[1],p[2],&zeror[degr-2],&zeroi[degr-2], |
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| 115 | &zeror[degr-1],&zeroi[degr-1]); |
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| 116 | n -= 2; |
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| 117 | return degr - n ; |
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| 118 | } |
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| 119 | |
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| 120 | // Find largest and smallest moduli of coefficients. |
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| 121 | // |
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| 122 | max = 0.0; |
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| 123 | min = infin; |
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| 124 | for (i=0;i<=n;i++) |
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| 125 | { |
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| 126 | x = std::fabs(p[i]); |
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| 127 | if (x > max) { max = x; } |
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| 128 | if (x != 0.0 && x < min) { min = x; } |
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| 129 | } |
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| 130 | |
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| 131 | // Scale if there are large or very small coefficients. |
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| 132 | // Computes a scale factor to multiply the coefficients of the |
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| 133 | // polynomial. The scaling is done to avoid overflow and to |
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| 134 | // avoid undetected underflow interfering with the convergence |
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| 135 | // criterion. The factor is a power of the base. |
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| 136 | // |
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| 137 | sc = lo/min; |
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| 138 | |
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| 139 | if ( ((sc <= 1.0) && (max >= 10.0)) |
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| 140 | || ((sc > 1.0) && (infin/sc >= max)) |
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| 141 | || ((infin/sc >= max) && (max >= 10)) ) |
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| 142 | { |
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| 143 | if (!( sc != 0.0 )) |
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| 144 | { sc = smalno ; } |
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| 145 | l = (G4int)(std::log(sc)/std::log(base) + 0.5); |
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| 146 | factor = std::pow(base*1.0,l); |
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| 147 | if (factor != 1.0) |
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| 148 | { |
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| 149 | for (i=0;i<=n;i++) |
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| 150 | { p[i] = factor*p[i]; } // Scale polynomial. |
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| 151 | } |
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| 152 | } |
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| 153 | |
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| 154 | // Compute lower bound on moduli of roots. |
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| 155 | // |
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| 156 | for (i=0;i<=n;i++) |
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| 157 | { |
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| 158 | pt[i] = (std::fabs(p[i])); |
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| 159 | } |
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| 160 | pt[n] = - pt[n]; |
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| 161 | |
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| 162 | // Compute upper estimate of bound. |
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| 163 | // |
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| 164 | x = std::exp((std::log(-pt[n])-std::log(pt[0])) / (G4double)n); |
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| 165 | |
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| 166 | // If Newton step at the origin is better, use it. |
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| 167 | // |
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| 168 | if (pt[n-1] != 0.0) |
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| 169 | { |
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| 170 | xm = -pt[n]/pt[n-1]; |
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| 171 | if (xm < x) { x = xm; } |
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| 172 | } |
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| 173 | |
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| 174 | // Chop the interval (0,x) until ff <= 0 |
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| 175 | // |
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| 176 | while (1) |
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| 177 | { |
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| 178 | xm = x*0.1; |
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| 179 | ff = pt[0]; |
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| 180 | for (i=1;i<=n;i++) |
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| 181 | { ff = ff*xm + pt[i]; } |
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| 182 | if (ff <= 0.0) { break; } |
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| 183 | x = xm; |
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| 184 | } |
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| 185 | dx = x; |
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| 186 | |
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| 187 | // Do Newton interation until x converges to two decimal places. |
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| 188 | // |
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| 189 | while (std::fabs(dx/x) > 0.005) |
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| 190 | { |
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| 191 | ff = pt[0]; |
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| 192 | df = ff; |
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| 193 | for (i=1;i<n;i++) |
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| 194 | { |
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| 195 | ff = ff*x + pt[i]; |
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| 196 | df = df*x + ff; |
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| 197 | } |
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| 198 | ff = ff*x + pt[n]; |
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| 199 | dx = ff/df; |
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| 200 | x -= dx; |
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| 201 | } |
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| 202 | bnd = x; |
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| 203 | |
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| 204 | // Compute the derivative as the initial k polynomial |
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| 205 | // and do 5 steps with no shift. |
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| 206 | // |
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| 207 | nm1 = n - 1; |
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| 208 | for (i=1;i<n;i++) |
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| 209 | { k[i] = (G4double)(n-i)*p[i]/(G4double)n; } |
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| 210 | k[0] = p[0]; |
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| 211 | aa = p[n]; |
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| 212 | bb = p[n-1]; |
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| 213 | zerok = (k[n-1] == 0); |
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| 214 | for(jj=0;jj<5;jj++) |
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| 215 | { |
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| 216 | cc = k[n-1]; |
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| 217 | if (!zerok) // Use a scaled form of recurrence if k at 0 is nonzero. |
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| 218 | { |
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| 219 | // Use a scaled form of recurrence if value of k at 0 is nonzero. |
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| 220 | // |
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| 221 | t = -aa/cc; |
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| 222 | for (i=0;i<nm1;i++) |
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| 223 | { |
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| 224 | j = n-i-1; |
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| 225 | k[j] = t*k[j-1]+p[j]; |
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| 226 | } |
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| 227 | k[0] = p[0]; |
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| 228 | zerok = (std::fabs(k[n-1]) <= std::fabs(bb)*eta*10.0); |
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| 229 | } |
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| 230 | else // Use unscaled form of recurrence. |
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| 231 | { |
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| 232 | for (i=0;i<nm1;i++) |
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| 233 | { |
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| 234 | j = n-i-1; |
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| 235 | k[j] = k[j-1]; |
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| 236 | } |
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| 237 | k[0] = 0.0; |
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| 238 | zerok = (!(k[n-1] != 0.0)); |
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| 239 | } |
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| 240 | } |
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| 241 | |
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| 242 | // Save k for restarts with new shifts. |
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| 243 | // |
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| 244 | for (i=0;i<n;i++) |
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| 245 | { temp[i] = k[i]; } |
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| 246 | |
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| 247 | // Loop to select the quadratic corresponding to each new shift. |
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| 248 | // |
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| 249 | for (cnt = 0;cnt < 20;cnt++) |
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| 250 | { |
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| 251 | // Quadratic corresponds to a double shift to a |
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| 252 | // non-real point and its complex conjugate. The point |
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| 253 | // has modulus bnd and amplitude rotated by 94 degrees |
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| 254 | // from the previous shift. |
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| 255 | // |
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| 256 | xxx = cosr*xx - sinr*yy; |
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| 257 | yy = sinr*xx + cosr*yy; |
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| 258 | xx = xxx; |
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| 259 | sr = bnd*xx; |
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| 260 | si = bnd*yy; |
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| 261 | u = -2.0 * sr; |
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| 262 | v = bnd; |
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| 263 | ComputeFixedShiftPolynomial(20*(cnt+1),&nz); |
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| 264 | if (nz != 0) |
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| 265 | { |
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| 266 | // The second stage jumps directly to one of the third |
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| 267 | // stage iterations and returns here if successful. |
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| 268 | // Deflate the polynomial, store the zero or zeros and |
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| 269 | // return to the main algorithm. |
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| 270 | // |
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| 271 | j = degr - n; |
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| 272 | zeror[j] = szr; |
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| 273 | zeroi[j] = szi; |
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| 274 | n -= nz; |
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| 275 | for (i=0;i<=n;i++) |
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| 276 | { p[i] = qp[i]; } |
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| 277 | if (nz != 1) |
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| 278 | { |
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| 279 | zeror[j+1] = lzr; |
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| 280 | zeroi[j+1] = lzi; |
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| 281 | } |
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| 282 | break; |
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| 283 | } |
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| 284 | else |
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| 285 | { |
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| 286 | // If the iteration is unsuccessful another quadratic |
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| 287 | // is chosen after restoring k. |
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| 288 | // |
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| 289 | for (i=0;i<n;i++) |
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| 290 | { |
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| 291 | k[i] = temp[i]; |
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| 292 | } |
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| 293 | } |
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| 294 | } |
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| 295 | } |
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| 296 | while (nz != 0); // End of initial DO loop |
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| 297 | |
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| 298 | // Return with failure if no convergence with 20 shifts. |
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| 299 | // |
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| 300 | return degr - n; |
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| 301 | } |
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| 302 | |
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| 303 | void G4JTPolynomialSolver::ComputeFixedShiftPolynomial(G4int l2, G4int *nz) |
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| 304 | { |
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| 305 | // Computes up to L2 fixed shift k-polynomials, testing for convergence |
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| 306 | // in the linear or quadratic case. Initiates one of the variable shift |
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| 307 | // iterations and returns with the number of zeros found. |
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| 308 | |
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| 309 | G4double svu=0.0, svv=0.0, ui=0.0, vi=0.0, xs=0.0; |
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| 310 | G4double betas=0.25, betav=0.25, oss=sr, ovv=v, |
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| 311 | ss=0.0, vv=0.0, ts=1.0, tv=1.0; |
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| 312 | G4double ots=0.0, otv=0.0; |
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| 313 | G4double tvv=1.0, tss=1.0; |
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| 314 | G4int type=0, i=0, j=0, iflag=0, vpass=0, spass=0, vtry=0, stry=0; |
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| 315 | |
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| 316 | *nz = 0; |
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| 317 | |
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| 318 | // Evaluate polynomial by synthetic division. |
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| 319 | // |
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| 320 | QuadraticSyntheticDivision(n,&u,&v,p,qp,&a,&b); |
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| 321 | ComputeScalarFactors(&type); |
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| 322 | for (j=0;j<l2;j++) |
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| 323 | { |
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| 324 | // Calculate next k polynomial and estimate v. |
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| 325 | // |
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| 326 | ComputeNextPolynomial(&type); |
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| 327 | ComputeScalarFactors(&type); |
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| 328 | ComputeNewEstimate(type,&ui,&vi); |
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| 329 | vv = vi; |
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| 330 | |
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| 331 | // Estimate xs. |
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| 332 | // |
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| 333 | ss = 0.0; |
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| 334 | if (k[n-1] != 0.0) { ss = -p[n]/k[n-1]; } |
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| 335 | tv = 1.0; |
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| 336 | ts = 1.0; |
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| 337 | if (j == 0 || type == 3) |
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| 338 | { |
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| 339 | ovv = vv; |
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| 340 | oss = ss; |
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| 341 | otv = tv; |
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| 342 | ots = ts; |
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| 343 | continue; |
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| 344 | } |
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| 345 | |
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| 346 | // Compute relative measures of convergence of xs and v sequences. |
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| 347 | // |
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| 348 | if (vv != 0.0) { tv = std::fabs((vv-ovv)/vv); } |
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| 349 | if (ss != 0.0) { ts = std::fabs((ss-oss)/ss); } |
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| 350 | |
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| 351 | // If decreasing, multiply two most recent convergence measures. |
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| 352 | tvv = 1.0; |
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| 353 | if (tv < otv) { tvv = tv*otv; } |
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| 354 | tss = 1.0; |
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| 355 | if (ts < ots) { tss = ts*ots; } |
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| 356 | |
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| 357 | // Compare with convergence criteria. |
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| 358 | vpass = (tvv < betav); |
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| 359 | spass = (tss < betas); |
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| 360 | if (!(spass || vpass)) |
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| 361 | { |
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| 362 | ovv = vv; |
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| 363 | oss = ss; |
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| 364 | otv = tv; |
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| 365 | ots = ts; |
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| 366 | continue; |
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| 367 | } |
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| 368 | |
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| 369 | // At least one sequence has passed the convergence test. |
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| 370 | // Store variables before iterating. |
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| 371 | // |
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| 372 | svu = u; |
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| 373 | svv = v; |
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| 374 | for (i=0;i<n;i++) |
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| 375 | { |
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| 376 | svk[i] = k[i]; |
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| 377 | } |
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| 378 | xs = ss; |
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| 379 | |
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| 380 | // Choose iteration according to the fastest converging sequence. |
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| 381 | // |
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| 382 | vtry = 0; |
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| 383 | stry = 0; |
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| 384 | if ((spass && (!vpass)) || (tss < tvv)) |
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| 385 | { |
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| 386 | RealPolynomialIteration(&xs,nz,&iflag); |
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| 387 | if (*nz > 0) { return; } |
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| 388 | |
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| 389 | // Linear iteration has failed. Flag that it has been |
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| 390 | // tried and decrease the convergence criterion. |
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| 391 | // |
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| 392 | stry = 1; |
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| 393 | betas *=0.25; |
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| 394 | if (iflag == 0) { goto _restore_variables; } |
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| 395 | |
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| 396 | // If linear iteration signals an almost double real |
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| 397 | // zero attempt quadratic iteration. |
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| 398 | // |
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| 399 | ui = -(xs+xs); |
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| 400 | vi = xs*xs; |
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| 401 | } |
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| 402 | |
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| 403 | _quadratic_iteration: |
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| 404 | |
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| 405 | do |
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| 406 | { |
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| 407 | QuadraticPolynomialIteration(&ui,&vi,nz); |
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| 408 | if (*nz > 0) { return; } |
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| 409 | |
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| 410 | // Quadratic iteration has failed. Flag that it has |
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| 411 | // been tried and decrease the convergence criterion. |
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| 412 | // |
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| 413 | vtry = 1; |
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| 414 | betav *= 0.25; |
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| 415 | |
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| 416 | // Try linear iteration if it has not been tried and |
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| 417 | // the S sequence is converging. |
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| 418 | // |
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| 419 | if (stry || !spass) { break; } |
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| 420 | for (i=0;i<n;i++) |
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| 421 | { |
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| 422 | k[i] = svk[i]; |
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| 423 | } |
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| 424 | RealPolynomialIteration(&xs,nz,&iflag); |
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| 425 | if (*nz > 0) { return; } |
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| 426 | |
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| 427 | // Linear iteration has failed. Flag that it has been |
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| 428 | // tried and decrease the convergence criterion. |
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| 429 | // |
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| 430 | stry = 1; |
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| 431 | betas *=0.25; |
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| 432 | if (iflag == 0) { break; } |
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| 433 | |
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| 434 | // If linear iteration signals an almost double real |
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| 435 | // zero attempt quadratic iteration. |
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| 436 | // |
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| 437 | ui = -(xs+xs); |
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| 438 | vi = xs*xs; |
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| 439 | } |
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| 440 | while (iflag != 0); |
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| 441 | |
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| 442 | // Restore variables. |
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| 443 | |
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| 444 | _restore_variables: |
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| 445 | |
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| 446 | u = svu; |
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| 447 | v = svv; |
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| 448 | for (i=0;i<n;i++) |
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| 449 | { |
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| 450 | k[i] = svk[i]; |
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| 451 | } |
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| 452 | |
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| 453 | // Try quadratic iteration if it has not been tried |
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| 454 | // and the V sequence is converging. |
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| 455 | // |
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| 456 | if (vpass && !vtry) { goto _quadratic_iteration; } |
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| 457 | |
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| 458 | // Recompute QP and scalar values to continue the |
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| 459 | // second stage. |
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| 460 | // |
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| 461 | QuadraticSyntheticDivision(n,&u,&v,p,qp,&a,&b); |
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| 462 | ComputeScalarFactors(&type); |
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| 463 | |
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| 464 | ovv = vv; |
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| 465 | oss = ss; |
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| 466 | otv = tv; |
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| 467 | ots = ts; |
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| 468 | } |
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| 469 | } |
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| 470 | |
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| 471 | void G4JTPolynomialSolver:: |
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| 472 | QuadraticPolynomialIteration(G4double *uu, G4double *vv, G4int *nz) |
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| 473 | { |
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| 474 | // Variable-shift k-polynomial iteration for a |
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| 475 | // quadratic factor converges only if the zeros are |
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| 476 | // equimodular or nearly so. |
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| 477 | // uu, vv - coefficients of starting quadratic. |
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| 478 | // nz - number of zeros found. |
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| 479 | // |
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| 480 | G4double ui=0.0, vi=0.0; |
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| 481 | G4double omp=0.0; |
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| 482 | G4double relstp=0.0; |
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| 483 | G4double mp=0.0, ee=0.0, t=0.0, zm=0.0; |
---|
| 484 | G4int type=0, i=1, j=0, tried=0; |
---|
| 485 | |
---|
| 486 | *nz = 0; |
---|
| 487 | tried = 0; |
---|
| 488 | u = *uu; |
---|
| 489 | v = *vv; |
---|
| 490 | |
---|
| 491 | // Main loop. |
---|
| 492 | |
---|
| 493 | while (1) |
---|
| 494 | { |
---|
| 495 | Quadratic(1.0,u,v,&szr,&szi,&lzr,&lzi); |
---|
| 496 | |
---|
| 497 | // Return if roots of the quadratic are real and not |
---|
| 498 | // close to multiple or nearly equal and of opposite |
---|
| 499 | // sign. |
---|
| 500 | // |
---|
| 501 | if (std::fabs(std::fabs(szr)-std::fabs(lzr)) > 0.01 * std::fabs(lzr)) |
---|
| 502 | { return; } |
---|
| 503 | |
---|
| 504 | // Evaluate polynomial by quadratic synthetic division. |
---|
| 505 | // |
---|
| 506 | QuadraticSyntheticDivision(n,&u,&v,p,qp,&a,&b); |
---|
| 507 | mp = std::fabs(a-szr*b) + std::fabs(szi*b); |
---|
| 508 | |
---|
| 509 | // Compute a rigorous bound on the rounding error in evaluating p. |
---|
| 510 | // |
---|
| 511 | zm = std::sqrt(std::fabs(v)); |
---|
| 512 | ee = 2.0*std::fabs(qp[0]); |
---|
| 513 | t = -szr*b; |
---|
| 514 | for (i=1;i<n;i++) |
---|
| 515 | { |
---|
| 516 | ee = ee*zm + std::fabs(qp[i]); |
---|
| 517 | } |
---|
| 518 | ee = ee*zm + std::fabs(a+t); |
---|
| 519 | ee *= (5.0 *mre + 4.0*are); |
---|
| 520 | ee = ee - (5.0*mre+2.0*are)*(std::fabs(a+t)+std::fabs(b)*zm) |
---|
| 521 | + 2.0*are*std::fabs(t); |
---|
| 522 | |
---|
| 523 | // Iteration has converged sufficiently if the |
---|
| 524 | // polynomial value is less than 20 times this bound. |
---|
| 525 | // |
---|
| 526 | if (mp <= 20.0*ee) |
---|
| 527 | { |
---|
| 528 | *nz = 2; |
---|
| 529 | return; |
---|
| 530 | } |
---|
| 531 | j++; |
---|
| 532 | |
---|
| 533 | // Stop iteration after 20 steps. |
---|
| 534 | // |
---|
| 535 | if (j > 20) { return; } |
---|
| 536 | if (j >= 2) |
---|
| 537 | { |
---|
| 538 | if (!(relstp > 0.01 || mp < omp || tried)) |
---|
| 539 | { |
---|
| 540 | // A cluster appears to be stalling the convergence. |
---|
| 541 | // Five fixed shift steps are taken with a u,v close to the cluster. |
---|
| 542 | // |
---|
| 543 | if (relstp < eta) { relstp = eta; } |
---|
| 544 | relstp = std::sqrt(relstp); |
---|
| 545 | u = u - u*relstp; |
---|
| 546 | v = v + v*relstp; |
---|
| 547 | QuadraticSyntheticDivision(n,&u,&v,p,qp,&a,&b); |
---|
| 548 | for (i=0;i<5;i++) |
---|
| 549 | { |
---|
| 550 | ComputeScalarFactors(&type); |
---|
| 551 | ComputeNextPolynomial(&type); |
---|
| 552 | } |
---|
| 553 | tried = 1; |
---|
| 554 | j = 0; |
---|
| 555 | } |
---|
| 556 | } |
---|
| 557 | omp = mp; |
---|
| 558 | |
---|
| 559 | // Calculate next k polynomial and new u and v. |
---|
| 560 | // |
---|
| 561 | ComputeScalarFactors(&type); |
---|
| 562 | ComputeNextPolynomial(&type); |
---|
| 563 | ComputeScalarFactors(&type); |
---|
| 564 | ComputeNewEstimate(type,&ui,&vi); |
---|
| 565 | |
---|
| 566 | // If vi is zero the iteration is not converging. |
---|
| 567 | // |
---|
| 568 | if (!(vi != 0.0)) { return; } |
---|
| 569 | relstp = std::fabs((vi-v)/vi); |
---|
| 570 | u = ui; |
---|
| 571 | v = vi; |
---|
| 572 | } |
---|
| 573 | } |
---|
| 574 | |
---|
| 575 | void G4JTPolynomialSolver:: |
---|
| 576 | RealPolynomialIteration(G4double *sss, G4int *nz, G4int *iflag) |
---|
| 577 | { |
---|
| 578 | // Variable-shift H polynomial iteration for a real zero. |
---|
| 579 | // sss - starting iterate |
---|
| 580 | // nz - number of zeros found |
---|
| 581 | // iflag - flag to indicate a pair of zeros near real axis. |
---|
| 582 | |
---|
| 583 | G4double t=0.; |
---|
| 584 | G4double omp=0.; |
---|
| 585 | G4double pv=0.0, kv=0.0, xs= *sss; |
---|
| 586 | G4double mx=0.0, mp=0.0, ee=0.0; |
---|
| 587 | G4int i=1, j=0; |
---|
| 588 | |
---|
| 589 | *nz = 0; |
---|
| 590 | *iflag = 0; |
---|
| 591 | |
---|
| 592 | // Main loop |
---|
| 593 | // |
---|
| 594 | while (1) |
---|
| 595 | { |
---|
| 596 | pv = p[0]; |
---|
| 597 | |
---|
| 598 | // Evaluate p at xs. |
---|
| 599 | // |
---|
| 600 | qp[0] = pv; |
---|
| 601 | for (i=1;i<=n;i++) |
---|
| 602 | { |
---|
| 603 | pv = pv*xs + p[i]; |
---|
| 604 | qp[i] = pv; |
---|
| 605 | } |
---|
| 606 | mp = std::fabs(pv); |
---|
| 607 | |
---|
| 608 | // Compute a rigorous bound on the error in evaluating p. |
---|
| 609 | // |
---|
| 610 | mx = std::fabs(xs); |
---|
| 611 | ee = (mre/(are+mre))*std::fabs(qp[0]); |
---|
| 612 | for (i=1;i<=n;i++) |
---|
| 613 | { |
---|
| 614 | ee = ee*mx + std::fabs(qp[i]); |
---|
| 615 | } |
---|
| 616 | |
---|
| 617 | // Iteration has converged sufficiently if the polynomial |
---|
| 618 | // value is less than 20 times this bound. |
---|
| 619 | // |
---|
| 620 | if (mp <= 20.0*((are+mre)*ee-mre*mp)) |
---|
| 621 | { |
---|
| 622 | *nz = 1; |
---|
| 623 | szr = xs; |
---|
| 624 | szi = 0.0; |
---|
| 625 | return; |
---|
| 626 | } |
---|
| 627 | j++; |
---|
| 628 | |
---|
| 629 | // Stop iteration after 10 steps. |
---|
| 630 | // |
---|
| 631 | if (j > 10) { return; } |
---|
| 632 | if (j >= 2) |
---|
| 633 | { |
---|
| 634 | if (!(std::fabs(t) > 0.001*std::fabs(xs-t) || mp < omp)) |
---|
| 635 | { |
---|
| 636 | // A cluster of zeros near the real axis has been encountered. |
---|
| 637 | // Return with iflag set to initiate a quadratic iteration. |
---|
| 638 | // |
---|
| 639 | *iflag = 1; |
---|
| 640 | *sss = xs; |
---|
| 641 | return; |
---|
| 642 | } // Return if the polynomial value has increased significantly. |
---|
| 643 | } |
---|
| 644 | |
---|
| 645 | omp = mp; |
---|
| 646 | |
---|
| 647 | // Compute t, the next polynomial, and the new iterate. |
---|
| 648 | // |
---|
| 649 | kv = k[0]; |
---|
| 650 | qk[0] = kv; |
---|
| 651 | for (i=1;i<n;i++) |
---|
| 652 | { |
---|
| 653 | kv = kv*xs + k[i]; |
---|
| 654 | qk[i] = kv; |
---|
| 655 | } |
---|
| 656 | if (std::fabs(kv) <= std::fabs(k[n-1])*10.0*eta) // Use unscaled form. |
---|
| 657 | { |
---|
| 658 | k[0] = 0.0; |
---|
| 659 | for (i=1;i<n;i++) |
---|
| 660 | { |
---|
| 661 | k[i] = qk[i-1]; |
---|
| 662 | } |
---|
| 663 | } |
---|
| 664 | else // Use the scaled form of the recurrence if k at xs is nonzero. |
---|
| 665 | { |
---|
| 666 | t = -pv/kv; |
---|
| 667 | k[0] = qp[0]; |
---|
| 668 | for (i=1;i<n;i++) |
---|
| 669 | { |
---|
| 670 | k[i] = t*qk[i-1] + qp[i]; |
---|
| 671 | } |
---|
| 672 | } |
---|
| 673 | kv = k[0]; |
---|
| 674 | for (i=1;i<n;i++) |
---|
| 675 | { |
---|
| 676 | kv = kv*xs + k[i]; |
---|
| 677 | } |
---|
| 678 | t = 0.0; |
---|
| 679 | if (std::fabs(kv) > std::fabs(k[n-1]*10.0*eta)) { t = -pv/kv; } |
---|
| 680 | xs += t; |
---|
| 681 | } |
---|
| 682 | } |
---|
| 683 | |
---|
| 684 | void G4JTPolynomialSolver::ComputeScalarFactors(G4int *type) |
---|
| 685 | { |
---|
| 686 | // This function calculates scalar quantities used to |
---|
| 687 | // compute the next k polynomial and new estimates of |
---|
| 688 | // the quadratic coefficients. |
---|
| 689 | // type - integer variable set here indicating how the |
---|
| 690 | // calculations are normalized to avoid overflow. |
---|
| 691 | |
---|
| 692 | // Synthetic division of k by the quadratic 1,u,v |
---|
| 693 | // |
---|
| 694 | QuadraticSyntheticDivision(n-1,&u,&v,k,qk,&c,&d); |
---|
| 695 | if (std::fabs(c) <= std::fabs(k[n-1]*100.0*eta)) |
---|
| 696 | { |
---|
| 697 | if (std::fabs(d) <= std::fabs(k[n-2]*100.0*eta)) |
---|
| 698 | { |
---|
| 699 | *type = 3; // Type=3 indicates the quadratic is almost a factor of k. |
---|
| 700 | return; |
---|
| 701 | } |
---|
| 702 | } |
---|
| 703 | |
---|
| 704 | if (std::fabs(d) < std::fabs(c)) |
---|
| 705 | { |
---|
| 706 | *type = 1; // Type=1 indicates that all formulas are divided by c. |
---|
| 707 | e = a/c; |
---|
| 708 | f = d/c; |
---|
| 709 | g = u*e; |
---|
| 710 | h = v*b; |
---|
| 711 | a3 = a*e + (h/c+g)*b; |
---|
| 712 | a1 = b - a*(d/c); |
---|
| 713 | a7 = a + g*d + h*f; |
---|
| 714 | return; |
---|
| 715 | } |
---|
| 716 | *type = 2; // Type=2 indicates that all formulas are divided by d. |
---|
| 717 | e = a/d; |
---|
| 718 | f = c/d; |
---|
| 719 | g = u*b; |
---|
| 720 | h = v*b; |
---|
| 721 | a3 = (a+g)*e + h*(b/d); |
---|
| 722 | a1 = b*f-a; |
---|
| 723 | a7 = (f+u)*a + h; |
---|
| 724 | } |
---|
| 725 | |
---|
| 726 | void G4JTPolynomialSolver::ComputeNextPolynomial(G4int *type) |
---|
| 727 | { |
---|
| 728 | // Computes the next k polynomials using scalars |
---|
| 729 | // computed in ComputeScalarFactors. |
---|
| 730 | |
---|
| 731 | G4int i=2; |
---|
| 732 | |
---|
| 733 | if (*type == 3) // Use unscaled form of the recurrence if type is 3. |
---|
| 734 | { |
---|
| 735 | k[0] = 0.0; |
---|
| 736 | k[1] = 0.0; |
---|
| 737 | for (i=2;i<n;i++) |
---|
| 738 | { |
---|
| 739 | k[i] = qk[i-2]; |
---|
| 740 | } |
---|
| 741 | return; |
---|
| 742 | } |
---|
| 743 | G4double temp = a; |
---|
| 744 | if (*type == 1) { temp = b; } |
---|
| 745 | if (std::fabs(a1) <= std::fabs(temp)*eta*10.0) |
---|
| 746 | { |
---|
| 747 | // If a1 is nearly zero then use a special form of the recurrence. |
---|
| 748 | // |
---|
| 749 | k[0] = 0.0; |
---|
| 750 | k[1] = -a7*qp[0]; |
---|
| 751 | for(i=2;i<n;i++) |
---|
| 752 | { |
---|
| 753 | k[i] = a3*qk[i-2] - a7*qp[i-1]; |
---|
| 754 | } |
---|
| 755 | return; |
---|
| 756 | } |
---|
| 757 | |
---|
| 758 | // Use scaled form of the recurrence. |
---|
| 759 | // |
---|
| 760 | a7 /= a1; |
---|
| 761 | a3 /= a1; |
---|
| 762 | k[0] = qp[0]; |
---|
| 763 | k[1] = qp[1] - a7*qp[0]; |
---|
| 764 | for (i=2;i<n;i++) |
---|
| 765 | { |
---|
| 766 | k[i] = a3*qk[i-2] - a7*qp[i-1] + qp[i]; |
---|
| 767 | } |
---|
| 768 | } |
---|
| 769 | |
---|
| 770 | void G4JTPolynomialSolver:: |
---|
| 771 | ComputeNewEstimate(G4int type, G4double *uu, G4double *vv) |
---|
| 772 | { |
---|
| 773 | // Compute new estimates of the quadratic coefficients |
---|
| 774 | // using the scalars computed in calcsc. |
---|
| 775 | |
---|
| 776 | G4double a4=0.0, a5=0.0, b1=0.0, b2=0.0, |
---|
| 777 | c1=0.0, c2=0.0, c3=0.0, c4=0.0, temp=0.0; |
---|
| 778 | |
---|
| 779 | // Use formulas appropriate to setting of type. |
---|
| 780 | // |
---|
| 781 | if (type == 3) // If type=3 the quadratic is zeroed. |
---|
| 782 | { |
---|
| 783 | *uu = 0.0; |
---|
| 784 | *vv = 0.0; |
---|
| 785 | return; |
---|
| 786 | } |
---|
| 787 | if (type == 2) |
---|
| 788 | { |
---|
| 789 | a4 = (a+g)*f + h; |
---|
| 790 | a5 = (f+u)*c + v*d; |
---|
| 791 | } |
---|
| 792 | else |
---|
| 793 | { |
---|
| 794 | a4 = a + u*b +h*f; |
---|
| 795 | a5 = c + (u+v*f)*d; |
---|
| 796 | } |
---|
| 797 | |
---|
| 798 | // Evaluate new quadratic coefficients. |
---|
| 799 | // |
---|
| 800 | b1 = -k[n-1]/p[n]; |
---|
| 801 | b2 = -(k[n-2]+b1*p[n-1])/p[n]; |
---|
| 802 | c1 = v*b2*a1; |
---|
| 803 | c2 = b1*a7; |
---|
| 804 | c3 = b1*b1*a3; |
---|
| 805 | c4 = c1 - c2 - c3; |
---|
| 806 | temp = a5 + b1*a4 - c4; |
---|
| 807 | if (!(temp != 0.0)) |
---|
| 808 | { |
---|
| 809 | *uu = 0.0; |
---|
| 810 | *vv = 0.0; |
---|
| 811 | return; |
---|
| 812 | } |
---|
| 813 | *uu = u - (u*(c3+c2)+v*(b1*a1+b2*a7))/temp; |
---|
| 814 | *vv = v*(1.0+c4/temp); |
---|
| 815 | return; |
---|
| 816 | } |
---|
| 817 | |
---|
| 818 | void G4JTPolynomialSolver:: |
---|
| 819 | QuadraticSyntheticDivision(G4int nn, G4double *uu, G4double *vv, |
---|
| 820 | std::vector<G4double> &pp, std::vector<G4double> &qq, |
---|
| 821 | G4double *aa, G4double *bb) |
---|
| 822 | { |
---|
| 823 | // Divides pp by the quadratic 1,uu,vv placing the quotient |
---|
| 824 | // in qq and the remainder in aa,bb. |
---|
| 825 | |
---|
| 826 | G4double cc=0.0; |
---|
| 827 | *bb = pp[0]; |
---|
| 828 | qq[0] = *bb; |
---|
| 829 | *aa = pp[1] - (*bb)*(*uu); |
---|
| 830 | qq[1] = *aa; |
---|
| 831 | for (G4int i=2;i<=nn;i++) |
---|
| 832 | { |
---|
| 833 | cc = pp[i] - (*aa)*(*uu) - (*bb)*(*vv); |
---|
| 834 | qq[i] = cc; |
---|
| 835 | *bb = *aa; |
---|
| 836 | *aa = cc; |
---|
| 837 | } |
---|
| 838 | } |
---|
| 839 | |
---|
| 840 | void G4JTPolynomialSolver::Quadratic(G4double aa,G4double b1, |
---|
| 841 | G4double cc,G4double *ssr,G4double *ssi, |
---|
| 842 | G4double *lr,G4double *li) |
---|
| 843 | { |
---|
| 844 | |
---|
| 845 | // Calculate the zeros of the quadratic aa*z^2 + b1*z + cc. |
---|
| 846 | // The quadratic formula, modified to avoid overflow, is used |
---|
| 847 | // to find the larger zero if the zeros are real and both |
---|
| 848 | // are complex. The smaller real zero is found directly from |
---|
| 849 | // the product of the zeros c/a. |
---|
| 850 | |
---|
| 851 | G4double bb=0.0, dd=0.0, ee=0.0; |
---|
| 852 | |
---|
| 853 | if (!(aa != 0.0)) // less than two roots |
---|
| 854 | { |
---|
| 855 | if (b1 != 0.0) |
---|
| 856 | { *ssr = -cc/b1; } |
---|
| 857 | else |
---|
| 858 | { *ssr = 0.0; } |
---|
| 859 | *lr = 0.0; |
---|
| 860 | *ssi = 0.0; |
---|
| 861 | *li = 0.0; |
---|
| 862 | return; |
---|
| 863 | } |
---|
| 864 | if (!(cc != 0.0)) // one real root, one zero root |
---|
| 865 | { |
---|
| 866 | *ssr = 0.0; |
---|
| 867 | *lr = -b1/aa; |
---|
| 868 | *ssi = 0.0; |
---|
| 869 | *li = 0.0; |
---|
| 870 | return; |
---|
| 871 | } |
---|
| 872 | |
---|
| 873 | // Compute discriminant avoiding overflow. |
---|
| 874 | // |
---|
| 875 | bb = b1/2.0; |
---|
| 876 | if (std::fabs(bb) < std::fabs(cc)) |
---|
| 877 | { |
---|
| 878 | if (cc < 0.0) |
---|
| 879 | { ee = -aa; } |
---|
| 880 | else |
---|
| 881 | { ee = aa; } |
---|
| 882 | ee = bb*(bb/std::fabs(cc)) - ee; |
---|
| 883 | dd = std::sqrt(std::fabs(ee))*std::sqrt(std::fabs(cc)); |
---|
| 884 | } |
---|
| 885 | else |
---|
| 886 | { |
---|
| 887 | ee = 1.0 - (aa/bb)*(cc/bb); |
---|
| 888 | dd = std::sqrt(std::fabs(ee))*std::fabs(bb); |
---|
| 889 | } |
---|
| 890 | if (ee < 0.0) // complex conjugate zeros |
---|
| 891 | { |
---|
| 892 | *ssr = -bb/aa; |
---|
| 893 | *lr = *ssr; |
---|
| 894 | *ssi = std::fabs(dd/aa); |
---|
| 895 | *li = -(*ssi); |
---|
| 896 | } |
---|
| 897 | else |
---|
| 898 | { |
---|
| 899 | if (bb >= 0.0) // real zeros. |
---|
| 900 | { dd = -dd; } |
---|
| 901 | *lr = (-bb+dd)/aa; |
---|
| 902 | *ssr = 0.0; |
---|
| 903 | if (*lr != 0.0) |
---|
| 904 | { *ssr = (cc/ *lr)/aa; } |
---|
| 905 | *ssi = 0.0; |
---|
| 906 | *li = 0.0; |
---|
| 907 | } |
---|
| 908 | } |
---|