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1.1 root 1: /*
2: floating point Bessel's function
3: of the first and second kinds
4: of order one
5:
6: j1(x) returns the value of J1(x)
7: for all real values of x.
8:
9: There are no error returns.
10: Calls sin, cos, sqrt.
11:
12: There is a niggling bug in J1 which
13: causes errors up to 2e-16 for x in the
14: interval [-8,8].
15: The bug is caused by an inappropriate order
16: of summation of the series. rhm will fix it
17: someday.
18:
19: Coefficients are from Hart & Cheney.
20: #6050 (20.98D)
21: #6750 (19.19D)
22: #7150 (19.35D)
23:
24: y1(x) returns the value of Y1(x)
25: for positive real values of x.
26: For x<=0, error number EDOM is set and a
27: large negative value is returned.
28:
29: Calls sin, cos, sqrt, log, j1.
30:
31: The values of Y1 have not been checked
32: to more than ten places.
33:
34: Coefficients are from Hart & Cheney.
35: #6447 (22.18D)
36: #6750 (19.19D)
37: #7150 (19.35D)
38: */
39:
40: #include <math.h>
41: #include <errno.h>
42:
43: int errno;
44: static double pzero, qzero;
45: static double tpi = .6366197723675813430755350535e0;
46: static double pio4 = .7853981633974483096156608458e0;
47: static double p1[] = {
48: 0.581199354001606143928050809e21,
49: -.6672106568924916298020941484e20,
50: 0.2316433580634002297931815435e19,
51: -.3588817569910106050743641413e17,
52: 0.2908795263834775409737601689e15,
53: -.1322983480332126453125473247e13,
54: 0.3413234182301700539091292655e10,
55: -.4695753530642995859767162166e7,
56: 0.2701122710892323414856790990e4,
57: };
58: static double q1[] = {
59: 0.1162398708003212287858529400e22,
60: 0.1185770712190320999837113348e20,
61: 0.6092061398917521746105196863e17,
62: 0.2081661221307607351240184229e15,
63: 0.5243710262167649715406728642e12,
64: 0.1013863514358673989967045588e10,
65: 0.1501793594998585505921097578e7,
66: 0.1606931573481487801970916749e4,
67: 1.0,
68: };
69: static double p2[] = {
70: -.4435757816794127857114720794e7,
71: -.9942246505077641195658377899e7,
72: -.6603373248364939109255245434e7,
73: -.1523529351181137383255105722e7,
74: -.1098240554345934672737413139e6,
75: -.1611616644324610116477412898e4,
76: 0.0,
77: };
78: static double q2[] = {
79: -.4435757816794127856828016962e7,
80: -.9934124389934585658967556309e7,
81: -.6585339479723087072826915069e7,
82: -.1511809506634160881644546358e7,
83: -.1072638599110382011903063867e6,
84: -.1455009440190496182453565068e4,
85: 1.0,
86: };
87: static double p3[] = {
88: 0.3322091340985722351859704442e5,
89: 0.8514516067533570196555001171e5,
90: 0.6617883658127083517939992166e5,
91: 0.1849426287322386679652009819e5,
92: 0.1706375429020768002061283546e4,
93: 0.3526513384663603218592175580e2,
94: 0.0,
95: };
96: static double q3[] = {
97: 0.7087128194102874357377502472e6,
98: 0.1819458042243997298924553839e7,
99: 0.1419460669603720892855755253e7,
100: 0.4002944358226697511708610813e6,
101: 0.3789022974577220264142952256e5,
102: 0.8638367769604990967475517183e3,
103: 1.0,
104: };
105: static double p4[] = {
106: -.9963753424306922225996744354e23,
107: 0.2655473831434854326894248968e23,
108: -.1212297555414509577913561535e22,
109: 0.2193107339917797592111427556e20,
110: -.1965887462722140658820322248e18,
111: 0.9569930239921683481121552788e15,
112: -.2580681702194450950541426399e13,
113: 0.3639488548124002058278999428e10,
114: -.2108847540133123652824139923e7,
115: 0.0,
116: };
117: static double q4[] = {
118: 0.5082067366941243245314424152e24,
119: 0.5435310377188854170800653097e22,
120: 0.2954987935897148674290758119e20,
121: 0.1082258259408819552553850180e18,
122: 0.2976632125647276729292742282e15,
123: 0.6465340881265275571961681500e12,
124: 0.1128686837169442121732366891e10,
125: 0.1563282754899580604737366452e7,
126: 0.1612361029677000859332072312e4,
127: 1.0,
128: };
129:
130: double
131: besj1(arg) double arg;{
132: double xsq, n, d, x;
133: double sin(), cos(), sqrt();
134: int i;
135:
136: x = arg;
137: if(x < 0.) x = -x;
138: if(x > 8.){
139: asympt(x);
140: n = x - 3.*pio4;
141: n = sqrt(tpi/x)*(pzero*cos(n) - qzero*sin(n));
142: if(arg <0.) n = -n;
143: return(n);
144: }
145: xsq = x*x;
146: for(n=0,d=0,i=8;i>=0;i--){
147: n = n*xsq + p1[i];
148: d = d*xsq + q1[i];
149: }
150: return(arg*n/d);
151: }
152:
153: double
154: besy1(arg) double arg;{
155: double xsq, n, d, x;
156: double sin(), cos(), sqrt(), log(), j1();
157: int i;
158:
159: errno = 0;
160: x = arg;
161: if(x <= 0.){
162: errno = EDOM;
163: return(-HUGE);
164: }
165: if(x > 8.){
166: asympt(x);
167: n = x - 3*pio4;
168: return(sqrt(tpi/x)*(pzero*sin(n) + qzero*cos(n)));
169: }
170: xsq = x*x;
171: for(n=0,d=0,i=9;i>=0;i--){
172: n = n*xsq + p4[i];
173: d = d*xsq + q4[i];
174: }
175: return(x*n/d + tpi*(j1(x)*log(x)-1./x));
176: }
177:
178: static
179: asympt(arg) double arg;{
180: double zsq, n, d;
181: int i;
182: zsq = 64./(arg*arg);
183: for(n=0,d=0,i=6;i>=0;i--){
184: n = n*zsq + p2[i];
185: d = d*zsq + q2[i];
186: }
187: pzero = n/d;
188: for(n=0,d=0,i=6;i>=0;i--){
189: n = n*zsq + p3[i];
190: d = d*zsq + q3[i];
191: }
192: qzero = (8./arg)*(n/d);
193: }
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