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
|
////////////////////////////////////////////////////////////////////////
//
// Copyright (C) 2016-2021 The Octave Project Developers
//
// See the file COPYRIGHT.md in the top-level directory of this
// distribution or <https://octave.org/copyright/>.
//
// This file is part of Octave.
//
// Octave is free software: you can redistribute it and/or modify it
// under the terms of the GNU General Public License as published by
// the Free Software Foundation, either version 3 of the License, or
// (at your option) any later version.
//
// Octave is distributed in the hope that it will be useful, but
// WITHOUT ANY WARRANTY; without even the implied warranty of
// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
// GNU General Public License for more details.
//
// You should have received a copy of the GNU General Public License
// along with Octave; see the file COPYING. If not, see
// <https://www.gnu.org/licenses/>.
//
////////////////////////////////////////////////////////////////////////
#if defined (HAVE_CONFIG_H)
# include "config.h"
#endif
#include "ov.h"
#include "defun.h"
#include "error.h"
#include "dNDArray.h"
#include "fNDArray.h"
#include "lo-specfun.h"
DEFUN (psi, args, ,
doc: /* -*- texinfo -*-
@deftypefn {} {} psi (@var{z})
@deftypefnx {} {} psi (@var{k}, @var{z})
Compute the psi (polygamma) function.
The polygamma functions are the @var{k}th derivative of the logarithm
of the gamma function. If unspecified, @var{k} defaults to zero. A value
of zero computes the digamma function, a value of 1, the trigamma function,
and so on.
The digamma function is defined:
@tex
$$
\Psi (z) = {d (log (\Gamma (z))) \over dx}
$$
@end tex
@ifnottex
@example
@group
psi (z) = d (log (gamma (z))) / dx
@end group
@end example
@end ifnottex
When computing the digamma function (when @var{k} equals zero), @var{z}
can have any value real or complex value. However, for polygamma functions
(@var{k} higher than 0), @var{z} must be real and non-negative.
@seealso{gamma, gammainc, gammaln}
@end deftypefn */)
{
int nargin = args.length ();
if (nargin < 1 || nargin > 2)
print_usage ();
const octave_value oct_z = (nargin == 1) ? args(0) : args(1);
const octave_idx_type k = (nargin == 1) ? 0 : args(0).xidx_type_value ("psi: K must be an integer");
if (k < 0)
error ("psi: K must be non-negative");
octave_value retval;
if (k == 0)
{
#define FLOAT_BRANCH(T, A, M, E) \
if (oct_z.is_ ## T ##_type ()) \
{ \
const A ## NDArray z = oct_z.M ## array_value (); \
A ## NDArray psi_z (z.dims ()); \
\
const E *zv = z.data (); \
E *psi_zv = psi_z.fortran_vec (); \
const octave_idx_type n = z.numel (); \
for (octave_idx_type i = 0; i < n; i++) \
*psi_zv++ = octave::math::psi (*zv++); \
\
retval = psi_z; \
}
if (oct_z.iscomplex ())
{
FLOAT_BRANCH(double, Complex, complex_, Complex)
else FLOAT_BRANCH(single, FloatComplex, float_complex_, FloatComplex)
else
error ("psi: Z must be a floating point");
}
else
{
FLOAT_BRANCH(double, , , double)
else FLOAT_BRANCH(single, Float, float_, float)
else
error ("psi: Z must be a floating point");
}
#undef FLOAT_BRANCH
}
else
{
if (! oct_z.isreal ())
error ("psi: Z must be real value for polygamma (K > 0)");
#define FLOAT_BRANCH(T, A, M, E) \
if (oct_z.is_ ## T ##_type ()) \
{ \
const A ## NDArray z = oct_z.M ## array_value (); \
A ## NDArray psi_z (z.dims ()); \
\
const E *zv = z.data (); \
E *psi_zv = psi_z.fortran_vec (); \
const octave_idx_type n = z.numel (); \
for (octave_idx_type i = 0; i < n; i++) \
{ \
if (*zv < 0) \
error ("psi: Z must be non-negative for polygamma (K > 0)"); \
\
*psi_zv++ = octave::math::psi (k, *zv++); \
} \
retval = psi_z; \
}
FLOAT_BRANCH(double, , , double)
else FLOAT_BRANCH(single, Float, float_, float)
else
error ("psi: Z must be a floating point for polygamma (K > 0)");
#undef FLOAT_BRANCH
}
return retval;
}
/*
%!shared em
%! em = 0.577215664901532860606512090082402431042; # Euler-Mascheroni Constant
%!assert (psi (ones (7, 3, 5)), repmat (-em, [7 3 5]))
%!assert (psi ([0 1]), [-Inf -em])
%!assert (psi ([-20:1]), [repmat(-Inf, [1 21]) -em])
%!assert (psi (single ([0 1])), single ([-Inf -em]))
## Abramowitz and Stegun, page 258, eq 6.3.5
%!test
%! z = [-100:-1 1:200] ./ 10; # drop the 0
%! assert (psi (z + 1), psi (z) + 1 ./ z, eps*1000);
## Abramowitz and Stegun, page 258, eq 6.3.2
%!assert (psi (1), -em)
## Abramowitz and Stegun, page 258, eq 6.3.3
%!assert (psi (1/2), -em - 2 * log (2))
## The following tests are from Pascal Sebah and Xavier Gourdon (2002)
## "Introduction to the Gamma Function"
## Interesting identities of the digamma function, in section of 5.1.3
%!assert (psi (1/3), - em - (3/2) * log(3) - ((sqrt (3) / 6) * pi), eps*10)
%!assert (psi (1/4), - em -3 * log (2) - pi/2, eps*10)
%!assert (psi (1/6), - em -2 * log (2) - (3/2) * log (3) - ((sqrt (3) / 2) * pi), eps*10)
## First 6 zeros of the digamma function, in section of 5.1.5 (and also on
## Abramowitz and Stegun, page 258, eq 6.3.19)
%!assert (psi ( 1.46163214496836234126265954232572132846819620400644), 0, eps)
%!assert (psi (-0.504083008264455409258269304533302498955385182368579), 0, eps*2)
%!assert (psi (-1.573498473162390458778286043690434612655040859116846), 0, eps*2)
%!assert (psi (-2.610720868444144650001537715718724207951074010873480), 0, eps*10)
%!assert (psi (-3.635293366436901097839181566946017713948423861193530), 0, eps*10)
%!assert (psi (-4.653237761743142441714598151148207363719069416133868), 0, eps*100)
## Tests for complex values
%!shared z
%! z = [-100:-1 1:200] ./ 10; # drop the 0
## Abramowitz and Stegun, page 259 eq 6.3.10
%!assert (real (psi (i*z)), real (psi (1 - i*z)))
## Abramowitz and Stegun, page 259 eq 6.3.11
%!assert (imag (psi (i*z)), 1/2 .* 1./z + 1/2 * pi * coth (pi * z), eps *10)
## Abramowitz and Stegun, page 259 eq 6.3.12
%!assert (imag (psi (1/2 + i*z)), 1/2 * pi * tanh (pi * z), eps*10)
## Abramowitz and Stegun, page 259 eq 6.3.13
%!assert (imag (psi (1 + i*z)), - 1./(2*z) + 1/2 * pi * coth (pi * z), eps*10)
## Abramowitz and Stegun, page 260 eq 6.4.5
%!test
%! for z = 0:20
%! assert (psi (1, z + 0.5),
%! 0.5 * (pi^2) - 4 * sum ((2*(1:z) -1) .^(-2)),
%! eps*10);
%! endfor
## Abramowitz and Stegun, page 260 eq 6.4.6
%!test
%! z = 0.1:0.1:20;
%! for n = 0:8
%! ## our precision goes down really quick when computing n is too high.
%! assert (psi (n, z+1),
%! psi (n, z) + ((-1)^n) * factorial (n) * (z.^(-n-1)), 0.1);
%! endfor
## Test input validation
%!error psi ()
%!error psi (1, 2, 3)
%!error <Z must be> psi ("non numeric")
%!error <K must be an integer> psi ({5.3}, 1)
%!error <K must be non-negative> psi (-5, 1)
%!error <Z must be non-negative for polygamma> psi (5, -1)
%!error <Z must be a floating point> psi (5, uint8 (-1))
%!error <Z must be real value for polygamma> psi (5, 5i)
*/
|
