[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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| 49 | double jn( int n, double x ); |
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[0a9d219] | 50 | |
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| 51 | double jn( int n, double x ) { |
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| 52 | |
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| 53 | const double MACHEP = 1.11022302462515654042E-16; |
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| 54 | double pkm2, pkm1, pk, xk, r, ans, xinv; |
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| 55 | int k, sign; |
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| 56 | |
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| 57 | if( n < 0 ) { |
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| 58 | n = -n; |
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| 59 | if( (n & 1) == 0 ) /* -1**n */ |
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| 60 | sign = 1; |
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| 61 | else |
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| 62 | sign = -1; |
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[3936ad3] | 63 | } |
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[0a9d219] | 64 | else |
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| 65 | sign = 1; |
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| 66 | |
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| 67 | if( x < 0.0 ) { |
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| 68 | if( n & 1 ) |
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| 69 | sign = -sign; |
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| 70 | x = -x; |
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[3936ad3] | 71 | } |
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| 72 | |
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[0a9d219] | 73 | if( n == 0 ) |
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| 74 | return( sign * j0(x) ); |
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| 75 | if( n == 1 ) |
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| 76 | return( sign * j1(x) ); |
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| 77 | if( n == 2 ) |
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| 78 | return( sign * (2.0 * j1(x) / x - j0(x)) ); |
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[3936ad3] | 79 | |
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[0a9d219] | 80 | if( x < MACHEP ) |
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| 81 | return( 0.0 ); |
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[3936ad3] | 82 | |
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[07142f3] | 83 | #if FLOAT_SIZE > 4 |
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[0a9d219] | 84 | k = 53; |
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[07142f3] | 85 | #else |
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[0a9d219] | 86 | k = 24; |
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[07142f3] | 87 | #endif |
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[3936ad3] | 88 | |
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[0a9d219] | 89 | pk = 2 * (n + k); |
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| 90 | ans = pk; |
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| 91 | xk = x * x; |
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[3936ad3] | 92 | |
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[0a9d219] | 93 | do { |
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| 94 | pk -= 2.0; |
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| 95 | ans = pk - (xk/ans); |
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| 96 | } while( --k > 0 ); |
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| 97 | |
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| 98 | /* backward recurrence */ |
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| 99 | |
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| 100 | pk = 1.0; |
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| 101 | |
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[07142f3] | 102 | #if FLOAT_SIZE > 4 |
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[0a9d219] | 103 | ans = x/ans; |
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| 104 | pkm1 = 1.0/ans; |
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| 105 | |
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| 106 | k = n-1; |
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| 107 | r = 2 * k; |
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| 108 | |
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| 109 | do { |
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| 110 | pkm2 = (pkm1 * r - pk * x) / x; |
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| 111 | pk = pkm1; |
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| 112 | pkm1 = pkm2; |
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| 113 | r -= 2.0; |
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| 114 | } while( --k > 0 ); |
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| 115 | |
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| 116 | if( fabs(pk) > fabs(pkm1) ) |
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| 117 | ans = j1(x)/pk; |
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| 118 | else |
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| 119 | ans = j0(x)/pkm1; |
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| 120 | |
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| 121 | return( sign * ans ); |
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[07142f3] | 122 | |
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| 123 | #else |
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[0a9d219] | 124 | xinv = 1.0/x; |
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| 125 | pkm1 = ans * xinv; |
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| 126 | k = n-1; |
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| 127 | r = (float )(2 * k); |
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| 128 | |
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| 129 | do { |
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| 130 | pkm2 = (pkm1 * r - pk * x) * xinv; |
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| 131 | pk = pkm1; |
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| 132 | pkm1 = pkm2; |
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| 133 | r -= 2.0; |
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| 134 | } |
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| 135 | while( --k > 0 ); |
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| 136 | |
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| 137 | r = pk; |
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| 138 | if( r < 0 ) |
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| 139 | r = -r; |
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| 140 | ans = pkm1; |
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| 141 | if( ans < 0 ) |
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| 142 | ans = -ans; |
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[3936ad3] | 143 | |
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[0a9d219] | 144 | if( r > ans ) /* if( fabs(pk) > fabs(pkm1) ) */ |
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| 145 | ans = sign * j1(x)/pk; |
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| 146 | else |
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| 147 | ans = sign * j0(x)/pkm1; |
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| 148 | return( ans ); |
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[07142f3] | 149 | #endif |
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[3936ad3] | 150 | } |
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| 151 | |
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