1 | ;;;; -*- Mode: lisp -*- |
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2 | ;;;; |
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3 | ;;;; Copyright (c) 2011 Raymond Toy |
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4 | ;;;; Permission is hereby granted, free of charge, to any person |
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5 | ;;;; obtaining a copy of this software and associated documentation |
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6 | ;;;; files (the "Software"), to deal in the Software without |
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7 | ;;;; restriction, including without limitation the rights to use, |
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8 | ;;;; copy, modify, merge, publish, distribute, sublicense, and/or sell |
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9 | ;;;; copies of the Software, and to permit persons to whom the |
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10 | ;;;; Software is furnished to do so, subject to the following |
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11 | ;;;; conditions: |
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12 | ;;;; |
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13 | ;;;; The above copyright notice and this permission notice shall be |
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14 | ;;;; included in all copies or substantial portions of the Software. |
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15 | ;;;; |
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16 | ;;;; THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, |
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17 | ;;;; EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES |
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18 | ;;;; OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND |
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19 | ;;;; NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT |
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20 | ;;;; HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, |
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21 | ;;;; WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING |
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22 | ;;;; FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR |
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23 | ;;;; OTHER DEALINGS IN THE SOFTWARE. |
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24 | |
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25 | (in-package #:oct) |
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26 | |
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27 | ;;; References: |
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28 | ;;; |
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29 | ;;; [1] Borwein, Borwein, Crandall, "Effective Laguerre Asymptotics", |
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30 | ;;; http://people.reed.edu/~crandall/papers/Laguerre-f.pdf |
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31 | ;;; |
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32 | ;;; [2] Borwein, Borwein, Chan, "The Evaluation of Bessel Functions |
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33 | ;;; via Exp-Arc Integrals", http://web.cs.dal.ca/~jborwein/bessel.pdf |
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34 | ;;; |
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35 | |
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36 | (defvar *debug-exparc* nil) |
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37 | |
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38 | ;; B[k](p) = 1/2^(k+3/2)*integrate(exp(-p*u)*u^(k-1/2),u,0,1) |
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39 | ;; = 1/2^(k+3/2)/p^(k+1/2)*integrate(t^(k-1/2)*exp(-t),t,0,p) |
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40 | ;; = 1/2^(k+3/2)/p^(k+1/2) * g(k+1/2, p) |
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41 | ;; |
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42 | ;; where g(a,z) is the lower incomplete gamma function. |
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43 | ;; |
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44 | ;; There is the continued fraction expansion for g(a,z) (see |
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45 | ;; cf-incomplete-gamma in qd-gamma.lisp): |
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46 | ;; |
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47 | ;; g(a,z) = z^a*exp(-z)/ CF |
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48 | ;; |
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49 | ;; So |
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50 | ;; |
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51 | ;; B[k](p) = 1/2^(k+3/2)/p^(k+1/2)*p^(k+1/2)*exp(-p)/CF |
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52 | ;; = exp(-p)/2^(k+3/2)/CF |
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53 | ;; |
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54 | ;; |
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55 | ;; Note also that [2] gives a recurrence relationship for B[k](p) in |
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56 | ;; eq (2.6), but there is an error there. The correct relationship is |
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57 | ;; |
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58 | ;; B[k](p) = -exp(-p)/(p*sqrt(2)*2^(k+1)) + (k-1/2)*B[k-1](p)/(2*p) |
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59 | ;; |
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60 | ;; The paper is missing the division by p in the term containing |
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61 | ;; B[k-1](p). This is easily derived from the recurrence relationship |
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62 | ;; for the (lower) incomplete gamma function. |
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63 | ;; |
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64 | ;; Note too that as k increases, the recurrence appears to be unstable |
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65 | ;; and B[k](p) begins to increase even though it is strictly bounded. |
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66 | ;; (This is also easy to see from the integral.) Hence, we do not use |
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67 | ;; the recursion. However, it might be stable for use with |
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68 | ;; double-float precision; this has not been tested. |
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69 | ;; |
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70 | (defun bk (k p) |
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71 | (/ (exp (- p)) |
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72 | (* (sqrt (float 2 (realpart p))) (ash 1 (+ k 1))) |
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73 | (let ((a (float (+ k 1/2) (realpart p)))) |
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74 | (lentz #'(lambda (n) |
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75 | (+ n a)) |
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76 | #'(lambda (n) |
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77 | (if (evenp n) |
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78 | (* (ash n -1) p) |
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79 | (- (* (+ a (ash n -1)) p)))))))) |
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80 | |
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81 | ;; exp-arc I function, as given in the Laguerre paper |
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82 | ;; |
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83 | ;; I(p, q) = 4*exp(p) * sum(g[k](-2*%i*q)/(2*k)!*B[k](p), k, 0, inf) |
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84 | ;; |
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85 | ;; where g[k](p) = product(p^2+(2*j-1)^2, j, 1, k) and B[k](p) as above. |
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86 | ;; |
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87 | ;; For computation, note that g[k](p) = g[k-1](p) * (p^2 + (2*k-1)^2) |
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88 | ;; and (2*k)! = (2*k-2)! * (2*k-1) * (2*k). Then, let |
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89 | ;; |
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90 | ;; R[k](p) = g[k](p)/(2*k)! |
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91 | ;; |
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92 | ;; Then |
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93 | ;; |
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94 | ;; R[k](p) = g[k](p)/(2*k)! |
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95 | ;; = g[k-1](p)/(2*k-2)! * (p^2 + (2*k-1)^2)/((2*k-1)*(2*k) |
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96 | ;; = R[k-1](p) * (p^2 + (2*k-1)^2)/((2*k-1)*(2*k) |
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97 | ;; |
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98 | ;; In the exp-arc paper, the function is defined (equivalently) as |
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99 | ;; |
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100 | ;; I(p, q) = 2*%i*exp(p)/q * sum(r[2*k+1](-2*%i*q)/(2*k)!*B[k](p), k, 0, inf) |
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101 | ;; |
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102 | ;; where r[2*k+1](p) = p*product(p^2 + (2*j-1)^2, j, 1, k) |
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103 | ;; |
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104 | ;; Let's note some properties of I(p, q). |
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105 | ;; |
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106 | ;; I(-%i*z, v) = 2*%i*exp(-%i*z)/q * sum(r[2*k+1](-2*%i*v)/(2*k)!*B[k](-%i*z)) |
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107 | ;; |
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108 | ;; Note thate B[k](-%i*z) = 1/2^(k+3/2)*integrate(exp(%i*z*u)*u^(k-1/2),u,0,1) |
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109 | ;; = conj(B[k](%i*z). |
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110 | ;; |
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111 | ;; Hence I(-%i*z, v) = conj(I(%i*z, v)) when both z and v are real. |
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112 | (defun exp-arc-i (p q) |
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113 | (let* ((sqrt2 (sqrt (float 2 (realpart p)))) |
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114 | (exp/p/sqrt2 (/ (exp (- p)) p sqrt2)) |
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115 | (v (* #c(0 -2) q)) |
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116 | (v2 (expt v 2)) |
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117 | (eps (epsilon (realpart p)))) |
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118 | (when *debug-exparc* |
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119 | (format t "sqrt2 = ~S~%" sqrt2) |
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120 | (format t "exp/p/sqrt2 = ~S~%" exp/p/sqrt2)) |
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121 | (do* ((k 0 (1+ k)) |
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122 | (bk (/ (incomplete-gamma 1/2 p) |
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123 | 2 sqrt2 (sqrt p)) |
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124 | (- (/ (* bk (- k 1/2)) 2 p) |
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125 | (/ exp/p/sqrt2 (ash 1 (+ k 1))))) |
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126 | ;; ratio[k] = r[2*k+1](v)/(2*k)!. |
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127 | ;; r[1] = v and r[2*k+1](v) = r[2*k-1](v)*(v^2 + (2*k-1)^2) |
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128 | ;; ratio[0] = v |
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129 | ;; and ratio[k] = r[2*k-1](v)*(v^2+(2*k-1)^2) / ((2*k-2)! * (2*k-1) * 2*k) |
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130 | ;; = ratio[k]*(v^2+(2*k-1)^2)/((2*k-1) * 2 * k) |
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131 | (ratio v |
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132 | (* ratio (/ (+ v2 (expt (1- (* 2 k)) 2)) |
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133 | (* 2 k (1- (* 2 k)))))) |
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134 | (term (* ratio bk) |
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135 | (* ratio bk)) |
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136 | (sum term (+ sum term))) |
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137 | ((< (abs term) (* (abs sum) eps)) |
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138 | (* sum #c(0 2) (/ (exp p) q))) |
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139 | (when *debug-exparc* |
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140 | (format t "k = ~D~%" k) |
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141 | (format t " bk = ~S~%" bk) |
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142 | (format t " ratio = ~S~%" ratio) |
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143 | (format t " term = ~S~%" term) |
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144 | (format t " sum - ~S~%" sum))))) |
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145 | |
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146 | (defun exp-arc-i-2 (p q) |
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147 | (let* ((sqrt2 (sqrt (float 2 (realpart p)))) |
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148 | (exp/p/sqrt2 (/ (exp (- p)) p sqrt2)) |
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149 | (v (* #c(0 -2) q)) |
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150 | (v2 (expt v 2)) |
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151 | (eps (epsilon (realpart p)))) |
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152 | (when *debug-exparc* |
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153 | (format t "sqrt2 = ~S~%" sqrt2) |
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154 | (format t "exp/p/sqrt2 = ~S~%" exp/p/sqrt2)) |
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155 | (do* ((k 0 (1+ k)) |
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156 | (bk (bk 0 p) |
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157 | (bk k p)) |
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158 | (ratio v |
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159 | (* ratio (/ (+ v2 (expt (1- (* 2 k)) 2)) |
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160 | (* 2 k (1- (* 2 k)))))) |
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161 | (term (* ratio bk) |
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162 | (* ratio bk)) |
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163 | (sum term (+ sum term))) |
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164 | ((< (abs term) (* (abs sum) eps)) |
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165 | (* sum #c(0 2) (/ (exp p) q))) |
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166 | (when *debug-exparc* |
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167 | (format t "k = ~D~%" k) |
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168 | (format t " bk = ~S~%" bk) |
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169 | (format t " ratio = ~S~%" ratio) |
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170 | (format t " term = ~S~%" term) |
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171 | (format t " sum - ~S~%" sum))))) |
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172 | |
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173 | |
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174 | ;; This currently only works for v an integer. |
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175 | ;; |
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176 | (defun integer-bessel-j-exp-arc (v z) |
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177 | (let* ((iz (* #c(0 1) z)) |
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178 | (i+ (exp-arc-i-2 iz v)) |
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179 | (i- (exp-arc-i-2 (- iz ) v))) |
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180 | (/ (+ (* (cis (* v (float-pi i+) -1/2)) |
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181 | i+) |
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182 | (* (cis (* v (float-pi i+) 1/2)) |
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183 | i-)) |
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184 | (float-pi i+) |
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185 | 2))) |
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186 | |
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187 | (defun paris-series (v z n) |
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188 | (labels ((pochhammer (a k) |
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189 | (/ (gamma (+ a k)) |
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190 | (gamma a))) |
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191 | (a (v k) |
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192 | (* (/ (pochhammer (+ 1/2 v) k) |
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193 | (gamma (float (1+ k) z))) |
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194 | (pochhammer (- 1/2 v) k)))) |
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195 | (* (loop for k from 0 below n |
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196 | sum (* (/ (a v k) |
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197 | (expt (* 2 z) k)) |
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198 | (/ (cf-incomplete-gamma (+ k v 1/2) (* 2 z)) |
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199 | (gamma (+ k v 1/2))))) |
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200 | (/ (exp z) |
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201 | (sqrt (* 2 (float-pi z) z)))))) |
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202 | |
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