1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
|
/* fdtrl.c
*
* F distribution, long double precision
*
*
*
* SYNOPSIS:
*
* int df1, df2;
* long double x, y, fdtrl();
*
* y = fdtrl( df1, df2, x );
*
*
*
* DESCRIPTION:
*
* Returns the area from zero to x under the F density
* function (also known as Snedcor's density or the
* variance ratio density). This is the density
* of x = (u1/df1)/(u2/df2), where u1 and u2 are random
* variables having Chi square distributions with df1
* and df2 degrees of freedom, respectively.
*
* The incomplete beta integral is used, according to the
* formula
*
* P(x) = incbetl( df1/2, df2/2, (df1*x/(df2 + df1*x) ).
*
*
* The arguments a and b are greater than zero, and x
* x is nonnegative.
*
* ACCURACY:
*
* Tested at random points (a,b,x) in the indicated intervals.
* x a,b Relative error:
* arithmetic domain domain # trials peak rms
* IEEE 0,1 1,100 10000 9.3e-18 2.9e-19
* IEEE 0,1 1,10000 10000 1.9e-14 2.9e-15
* IEEE 1,5 1,10000 10000 5.8e-15 1.4e-16
*
* ERROR MESSAGES:
*
* message condition value returned
* fdtrl domain a<0, b<0, x<0 0.0
*
*/
/* fdtrcl()
*
* Complemented F distribution
*
*
*
* SYNOPSIS:
*
* int df1, df2;
* long double x, y, fdtrcl();
*
* y = fdtrcl( df1, df2, x );
*
*
*
* DESCRIPTION:
*
* Returns the area from x to infinity under the F density
* function (also known as Snedcor's density or the
* variance ratio density).
*
*
* inf.
* -
* 1 | | a-1 b-1
* 1-P(x) = ------ | t (1-t) dt
* B(a,b) | |
* -
* x
*
* (See fdtr.c.)
*
* The incomplete beta integral is used, according to the
* formula
*
* P(x) = incbet( df2/2, df1/2, (df2/(df2 + df1*x) ).
*
*
* ACCURACY:
*
* See incbet.c.
* Tested at random points (a,b,x).
*
* x a,b Relative error:
* arithmetic domain domain # trials peak rms
* IEEE 0,1 0,100 10000 4.2e-18 3.3e-19
* IEEE 0,1 1,10000 10000 7.2e-15 2.6e-16
* IEEE 1,5 1,10000 10000 1.7e-14 3.0e-15
*
* ERROR MESSAGES:
*
* message condition value returned
* fdtrcl domain a<0, b<0, x<0 0.0
*
*/
/* fdtril()
*
* Inverse of complemented F distribution
*
*
*
* SYNOPSIS:
*
* int df1, df2;
* long double x, p, fdtril();
*
* x = fdtril( df1, df2, p );
*
* DESCRIPTION:
*
* Finds the F density argument x such that the integral
* from x to infinity of the F density is equal to the
* given probability p.
*
* This is accomplished using the inverse beta integral
* function and the relations
*
* z = incbi( df2/2, df1/2, p )
* x = df2 (1-z) / (df1 z).
*
* Note: the following relations hold for the inverse of
* the uncomplemented F distribution:
*
* z = incbi( df1/2, df2/2, p )
* x = df2 z / (df1 (1-z)).
*
* ACCURACY:
*
* See incbi.c.
* Tested at random points (a,b,p).
*
* a,b Relative error:
* arithmetic domain # trials peak rms
* For p between .001 and 1:
* IEEE 1,100 40000 4.6e-18 2.7e-19
* IEEE 1,10000 30000 1.7e-14 1.4e-16
* For p between 10^-6 and .001:
* IEEE 1,100 20000 1.9e-15 3.9e-17
* IEEE 1,10000 30000 2.7e-15 4.0e-17
*
* ERROR MESSAGES:
*
* message condition value returned
* fdtril domain p <= 0 or p > 1 0.0
* v < 1
*/
/*
Cephes Math Library Release 2.3: March, 1995
Copyright 1984, 1995 by Stephen L. Moshier
*/
#include <math.h>
#ifdef ANSIPROT
extern long double incbetl ( long double, long double, long double );
extern long double incbil ( long double, long double, long double );
#else
long double incbetl(), incbil();
#endif
long double fdtrcl( ia, ib, x )
int ia, ib;
long double x;
{
long double a, b, w;
if( (ia < 1) || (ib < 1) || (x < 0.0L) )
{
mtherr( "fdtrcl", DOMAIN );
return( 0.0L );
}
a = ia;
b = ib;
w = b / (b + a * x);
return( incbetl( 0.5L*b, 0.5L*a, w ) );
}
long double fdtrl( ia, ib, x )
int ia, ib;
long double x;
{
long double a, b, w;
if( (ia < 1) || (ib < 1) || (x < 0.0L) )
{
mtherr( "fdtrl", DOMAIN );
return( 0.0L );
}
a = ia;
b = ib;
w = a * x;
w = w / (b + w);
return( incbetl(0.5L*a, 0.5L*b, w) );
}
long double fdtril( ia, ib, y )
int ia, ib;
long double y;
{
long double a, b, w, x;
if( (ia < 1) || (ib < 1) || (y <= 0.0L) || (y > 1.0L) )
{
mtherr( "fdtril", DOMAIN );
return( 0.0L );
}
a = ia;
b = ib;
/* Compute probability for x = 0.5. */
w = incbetl( 0.5L*b, 0.5L*a, 0.5L );
/* If that is greater than y, then the solution w < .5.
Otherwise, solve at 1-y to remove cancellation in (b - b*w). */
if( w > y || y < 0.001L)
{
w = incbil( 0.5L*b, 0.5L*a, y );
x = (b - b*w)/(a*w);
}
else
{
w = incbil( 0.5L*a, 0.5L*b, 1.0L - y );
x = b*w/(a*(1.0L-w));
}
return(x);
}
|
