[3936ad3] | 1 | /* jn.c |
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| 2 | * |
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| 3 | * Bessel function of integer order |
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| 4 | * |
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| 5 | * |
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| 6 | * |
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| 7 | * SYNOPSIS: |
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| 8 | * |
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| 9 | * int n; |
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| 10 | * double x, y, jn(); |
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| 11 | * |
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| 12 | * y = jn( n, x ); |
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| 13 | * |
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| 14 | * |
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| 15 | * |
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| 16 | * DESCRIPTION: |
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| 17 | * |
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| 18 | * Returns Bessel function of order n, where n is a |
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| 19 | * (possibly negative) integer. |
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| 20 | * |
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| 21 | * The ratio of jn(x) to j0(x) is computed by backward |
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| 22 | * recurrence. First the ratio jn/jn-1 is found by a |
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| 23 | * continued fraction expansion. Then the recurrence |
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| 24 | * relating successive orders is applied until j0 or j1 is |
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| 25 | * reached. |
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| 26 | * |
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| 27 | * If n = 0 or 1 the routine for j0 or j1 is called |
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| 28 | * directly. |
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| 29 | * |
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| 30 | * |
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| 31 | * |
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| 32 | * ACCURACY: |
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| 33 | * |
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| 34 | * Absolute error: |
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| 35 | * arithmetic range # trials peak rms |
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| 36 | * DEC 0, 30 5500 6.9e-17 9.3e-18 |
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| 37 | * IEEE 0, 30 5000 4.4e-16 7.9e-17 |
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| 38 | * |
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| 39 | * |
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| 40 | * Not suitable for large n or x. Use jv() instead. |
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| 41 | * |
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| 42 | */ |
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| 43 | |
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| 44 | /* jn.c |
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| 45 | Cephes Math Library Release 2.8: June, 2000 |
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| 46 | Copyright 1984, 1987, 2000 by Stephen L. Moshier |
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| 47 | */ |
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| 48 | |
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[3f8584a2] | 49 | #if FLOAT_SIZE > 4 |
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[0a9d219] | 50 | |
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[3f8584a2] | 51 | double jn( int n, double x ); |
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| 52 | double jn( int n, double x ) { |
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[0a9d219] | 53 | |
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[3f8584a2] | 54 | // PAK: seems to be machine epsilon/2 |
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[0a9d219] | 55 | const double MACHEP = 1.11022302462515654042E-16; |
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[e6408d0] | 56 | double pkm2, pkm1, pk, xk, r, ans; |
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[0a9d219] | 57 | int k, sign; |
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| 58 | |
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| 59 | if( n < 0 ) { |
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[3f8584a2] | 60 | n = -n; |
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| 61 | if( (n & 1) == 0 ) /* -1**n */ |
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| 62 | sign = 1; |
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| 63 | else |
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| 64 | sign = -1; |
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| 65 | } else { |
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| 66 | sign = 1; |
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| 67 | } |
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[0a9d219] | 68 | |
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| 69 | if( x < 0.0 ) { |
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[3f8584a2] | 70 | if( n & 1 ) |
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| 71 | sign = -sign; |
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| 72 | x = -x; |
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| 73 | } |
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[3936ad3] | 74 | |
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[0a9d219] | 75 | if( n == 0 ) |
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[3f8584a2] | 76 | return( sign * j0(x) ); |
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[0a9d219] | 77 | if( n == 1 ) |
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[3f8584a2] | 78 | return( sign * j1(x) ); |
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[0a9d219] | 79 | if( n == 2 ) |
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[3f8584a2] | 80 | return( sign * (2.0 * j1(x) / x - j0(x)) ); |
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[3936ad3] | 81 | |
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[0a9d219] | 82 | if( x < MACHEP ) |
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[3f8584a2] | 83 | return( 0.0 ); |
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[3936ad3] | 84 | |
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[3f8584a2] | 85 | k = 53; |
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[0a9d219] | 86 | pk = 2 * (n + k); |
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| 87 | ans = pk; |
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| 88 | xk = x * x; |
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[3936ad3] | 89 | |
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[0a9d219] | 90 | do { |
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[3f8584a2] | 91 | pk -= 2.0; |
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| 92 | ans = pk - (xk/ans); |
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| 93 | } while( --k > 0 ); |
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[0a9d219] | 94 | |
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| 95 | /* backward recurrence */ |
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| 96 | |
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| 97 | pk = 1.0; |
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| 98 | |
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[3f8584a2] | 99 | ans = x/ans; |
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| 100 | pkm1 = 1.0/ans; |
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[0a9d219] | 101 | |
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[3f8584a2] | 102 | k = n-1; |
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| 103 | r = 2 * k; |
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[0a9d219] | 104 | |
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[3f8584a2] | 105 | do { |
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| 106 | pkm2 = (pkm1 * r - pk * x) / x; |
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| 107 | pk = pkm1; |
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| 108 | pkm1 = pkm2; |
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| 109 | r -= 2.0; |
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| 110 | } while( --k > 0 ); |
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| 111 | |
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| 112 | if( fabs(pk) > fabs(pkm1) ) |
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| 113 | ans = j1(x)/pk; |
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| 114 | else |
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| 115 | ans = j0(x)/pkm1; |
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[0a9d219] | 116 | |
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[3f8584a2] | 117 | return( sign * ans ); |
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| 118 | } |
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| 119 | |
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| 120 | #else |
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| 121 | |
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[1596de3] | 122 | float jnf(int n, float x); |
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[3f8584a2] | 123 | float jnf(int n, float x) |
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| 124 | { |
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| 125 | // PAK: seems to be machine epsilon/2 |
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| 126 | const double MACHEP = 5.9604645e-08; |
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| 127 | float pkm2, pkm1, pk, xk, r, ans; |
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| 128 | int k, sign; |
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| 129 | |
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| 130 | if( n < 0 ) { |
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| 131 | n = -n; |
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| 132 | if( (n & 1) == 0 ) /* -1**n */ |
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| 133 | sign = 1; |
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[0a9d219] | 134 | else |
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[3f8584a2] | 135 | sign = -1; |
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| 136 | } else { |
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| 137 | sign = 1; |
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| 138 | } |
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| 139 | |
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| 140 | if( x < 0.0 ) { |
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| 141 | if( n & 1 ) |
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| 142 | sign = -sign; |
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| 143 | x = -x; |
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| 144 | } |
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| 145 | |
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| 146 | if( n == 0 ) |
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| 147 | return( sign * j0f(x) ); |
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| 148 | if( n == 1 ) |
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| 149 | return( sign * j1f(x) ); |
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| 150 | if( n == 2 ) |
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| 151 | return( sign * (2.0 * j1f(x) / x - j0f(x)) ); |
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| 152 | |
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| 153 | if( x < MACHEP ) |
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| 154 | return( 0.0 ); |
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| 155 | |
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| 156 | k = 24; |
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| 157 | pk = 2 * (n + k); |
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| 158 | ans = pk; |
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| 159 | xk = x * x; |
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| 160 | |
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| 161 | do { |
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| 162 | pk -= 2.0; |
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| 163 | ans = pk - (xk/ans); |
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| 164 | } while( --k > 0 ); |
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| 165 | |
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| 166 | /* backward recurrence */ |
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| 167 | |
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| 168 | pk = 1.0; |
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| 169 | |
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| 170 | const float xinv = 1.0/x; |
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| 171 | pkm1 = ans * xinv; |
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| 172 | k = n-1; |
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| 173 | r = (float )(2 * k); |
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| 174 | |
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| 175 | do { |
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| 176 | pkm2 = (pkm1 * r - pk * x) * xinv; |
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| 177 | pk = pkm1; |
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| 178 | pkm1 = pkm2; |
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| 179 | r -= 2.0; |
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| 180 | } while( --k > 0 ); |
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| 181 | |
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| 182 | r = pk; |
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| 183 | if( r < 0 ) |
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| 184 | r = -r; |
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| 185 | ans = pkm1; |
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| 186 | if( ans < 0 ) |
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| 187 | ans = -ans; |
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| 188 | |
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| 189 | if( r > ans ) /* if( fabs(pk) > fabs(pkm1) ) */ |
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| 190 | ans = sign * j1f(x)/pk; |
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| 191 | else |
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| 192 | ans = sign * j0f(x)/pkm1; |
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| 193 | return( ans ); |
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[3936ad3] | 194 | } |
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[3f8584a2] | 195 | #endif |
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[3936ad3] | 196 | |
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[3f8584a2] | 197 | #if FLOAT_SIZE>4 |
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| 198 | #define sas_JN jn |
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| 199 | #else |
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| 200 | #define sas_JN jnf |
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| 201 | #endif |
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