GNU Octave  4.2.1 A high-level interpreted language, primarily intended for numerical computations, mostly compatible with Matlab
psi.cc
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1 /*
2
3 Copyright (C) 2016-2017 CarnĂ« Draug
4
5 This file is part of Octave.
6
7 Octave is free software; you can redistribute it and/or modify it
9 Free Software Foundation; either version 3 of the License, or (at your
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12 Octave is distributed in the hope that it will be useful, but WITHOUT
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14 FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
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17 You should have received a copy of the GNU General Public License
18 along with Octave; see the file COPYING. If not, see
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21 */
22
23 #if defined (HAVE_CONFIG_H)
24 # include "config.h"
25 #endif
26
27 #include "ov.h"
28 #include "defun.h"
29 #include "error.h"
30 #include "dNDArray.h"
31 #include "fNDArray.h"
32
33 #include "lo-specfun.h"
34
35 DEFUN (psi, args, ,
36  doc: /* -*- texinfo -*-
37 @deftypefn {} {} psi (@var{z})
38 @deftypefnx {} {} psi (@var{k}, @var{z})
39 Compute the psi (polygamma) function.
40
41 The polygamma functions are the @var{k}th derivative of the logarithm
42 of the gamma function. If unspecified, @var{k} defaults to zero. A value
43 of zero computes the digamma function, a value of 1, the trigamma function,
44 and so on.
45
46 The digamma function is defined:
47
48 @tex
49 $$50 \Psi (z) = {d (log (\Gamma (z))) \over dx} 51$$
52 @end tex
53 @ifnottex
54
55 @example
56 @group
57 psi (z) = d (log (gamma (z))) / dx
58 @end group
59 @end example
60
61 @end ifnottex
62
63 When computing the digamma function (when @var{k} equals zero), @var{z}
64 can have any value real or complex value. However, for polygamma functions
65 (@var{k} higher than 0), @var{z} must be real and non-negative.
66
67 @seealso{gamma, gammainc, gammaln}
68 @end deftypefn */)
69 {
70  int nargin = args.length ();
71
72  if (nargin < 1 || nargin > 2)
73  print_usage ();
74
75  const octave_value oct_z = (nargin == 1) ? args(0) : args(1);
76  const octave_idx_type k = (nargin == 1) ? 0 : args(0).idx_type_value ("psi: K must be an integer");
77  if (k < 0)
78  error ("psi: K must be non-negative");
79
81
82  if (k == 0)
83  {
84 #define FLOAT_BRANCH(T, A, M, E) \
85  if (oct_z.is_ ## T ##_type ()) \
86  { \
87  const A ## NDArray z = oct_z.M ## array_value (); \
88  A ## NDArray psi_z (z.dims ()); \
89  \
90  const E* zv = z.data (); \
91  E* psi_zv = psi_z.fortran_vec (); \
92  const octave_idx_type n = z.numel (); \
93  for (octave_idx_type i = 0; i < n; i++) \
94  *psi_zv++ = octave::math::psi (*zv++); \
95  \
96  retval = psi_z; \
97  }
98
99  if (oct_z.is_complex_type ())
100  {
101  FLOAT_BRANCH(double, Complex, complex_, Complex)
102  else FLOAT_BRANCH(single, FloatComplex, float_complex_, FloatComplex)
103  else
104  error ("psi: Z must be a floating point");
105  }
106  else
107  {
108  FLOAT_BRANCH(double, , , double)
109  else FLOAT_BRANCH(single, Float, float_, float)
110  else
111  error ("psi: Z must be a floating point");
112  }
113
114 #undef FLOAT_BRANCH
115  }
116  else
117  {
118  if (! oct_z.is_real_type ())
119  error ("psi: Z must be real value for polygamma (K > 0)");
120
121 #define FLOAT_BRANCH(T, A, M, E) \
122  if (oct_z.is_ ## T ##_type ()) \
123  { \
124  const A ## NDArray z = oct_z.M ## array_value (); \
125  A ## NDArray psi_z (z.dims ()); \
126  \
127  const E* zv = z.data (); \
128  E* psi_zv = psi_z.fortran_vec (); \
129  const octave_idx_type n = z.numel (); \
130  for (octave_idx_type i = 0; i < n; i++) \
131  { \
132  if (*zv < 0) \
133  error ("psi: Z must be non-negative for polygamma (K > 0)"); \
134  \
135  *psi_zv++ = octave::math::psi (k, *zv++); \
136  } \
137  retval = psi_z; \
138  }
139
140  FLOAT_BRANCH(double, , , double)
141  else FLOAT_BRANCH(single, Float, float_, float)
142  else
143  error ("psi: Z must be a floating point for polygamma (K > 0)");
144
145 #undef FLOAT_BRANCH
146  }
147
148  return retval;
149 }
150
151 /*
152 %!shared em
153 %! em = 0.577215664901532860606512090082402431042; # Euler-Mascheroni Constant
154
155 %!assert (psi (ones (7, 3, 5)), repmat (-em, [7 3 5]))
156 %!assert (psi ([0 1]), [-Inf -em])
157 %!assert (psi ([-20:1]), [repmat(-Inf, [1 21]) -em])
158 %!assert (psi (single ([0 1])), single ([-Inf -em]))
159
160 ## Abramowitz and Stegun, page 258, eq 6.3.5
161 %!test
162 %! z = [-100:-1 1:200] ./ 10; # drop the 0
163 %! assert (psi (z + 1), psi (z) + 1 ./ z, eps*1000);
164
165 ## Abramowitz and Stegun, page 258, eq 6.3.2
166 %!assert (psi (1), -em)
167
168 ## Abramowitz and Stegun, page 258, eq 6.3.3
169 %!assert (psi (1/2), -em - 2 * log (2))
170
171 ## The following tests are from Pascal Sebah and Xavier Gourdon (2002)
172 ## "Introduction to the Gamma Function"
173
174 ## Interesting identities of the digamma function, in section of 5.1.3
175 %!assert (psi (1/3), - em - (3/2) * log(3) - ((sqrt (3) / 6) * pi), eps*10)
176 %!assert (psi (1/4), - em -3 * log (2) - pi/2, eps*10)
177 %!assert (psi (1/6), - em -2 * log (2) - (3/2) * log (3) - ((sqrt (3) / 2) * pi), eps*10)
178
179 ## First 6 zeros of the digamma function, in section of 5.1.5 (and also on
180 ## Abramowitz and Stegun, page 258, eq 6.3.19)
181 %!assert (psi ( 1.46163214496836234126265954232572132846819620400644), 0, eps)
182 %!assert (psi (-0.504083008264455409258269304533302498955385182368579), 0, eps*2)
183 %!assert (psi (-1.573498473162390458778286043690434612655040859116846), 0, eps*2)
184 %!assert (psi (-2.610720868444144650001537715718724207951074010873480), 0, eps*10)
185 %!assert (psi (-3.635293366436901097839181566946017713948423861193530), 0, eps*10)
186 %!assert (psi (-4.653237761743142441714598151148207363719069416133868), 0, eps*100)
187
188 ## Tests for complex values
189 %!shared z
190 %! z = [-100:-1 1:200] ./ 10; # drop the 0
191
192 ## Abramowitz and Stegun, page 259 eq 6.3.10
193 %!assert (real (psi (i*z)), real (psi (1 - i*z)))
194
195 ## Abramowitz and Stegun, page 259 eq 6.3.11
196 %!assert (imag (psi (i*z)), 1/2 .* 1./z + 1/2 * pi * coth (pi * z), eps *10)
197
198 ## Abramowitz and Stegun, page 259 eq 6.3.12
199 %!assert (imag (psi (1/2 + i*z)), 1/2 * pi * tanh (pi * z), eps*10)
200
201 ## Abramowitz and Stegun, page 259 eq 6.3.13
202 %!assert (imag (psi (1 + i*z)), - 1./(2*z) + 1/2 * pi * coth (pi * z), eps*10)
203
204 ## Abramowitz and Stegun, page 260 eq 6.4.5
205 %!test
206 %! for z = 0:20
207 %! assert (psi (1, z + 0.5),
208 %! 0.5 * (pi^2) - 4 * sum ((2*(1:z) -1) .^(-2)),
209 %! eps*10);
210 %! endfor
211
212 ## Abramowitz and Stegun, page 260 eq 6.4.6
213 %!test
214 %! z = 0.1:0.1:20;
215 %! for n = 0:8
216 %! ## our precision goes down really quick when computing n is too high.
217 %! assert (psi (n, z+1),
218 %! psi (n, z) + ((-1)^n) * factorial (n) * (z.^(-n-1)), 0.1);
219 %! endfor
220
221 ## Test input validation
222 %!error psi ()
223 %!error psi (1, 2, 3)
224 %!error <Z must be> psi ("non numeric")
225 %!error <conversion of 5.3 to int.* value failed> psi (5.3, 1)
226 %!error <K must be non-negative> psi (-5, 1)
227 %!error <Z must be non-negative for polygamma> psi (5, -1)
228 %!error <Z must be a floating point> psi (5, uint8 (-1))
229 %!error <Z must be real value for polygamma> psi (5, 5i)
230
231 */
double psi(double z)
Definition: lo-specfun.cc:3714
bool is_real_type(void) const
Definition: ov.h:667
OCTINTERP_API void print_usage(void)
Definition: defun.cc:52
for large enough k
Definition: lu.cc:606
#define DEFUN(name, args_name, nargout_name, doc)
Definition: defun.h:46
void error(const char *fmt,...)
Definition: error.cc:570
#define FLOAT_BRANCH(T, A, M, E)
issues an error eealso single
Definition: ov-bool-mat.cc:594
JNIEnv void * args
Definition: ov-java.cc:67
int nargin
Definition: graphics.cc:10115
bool is_complex_type(void) const
Definition: ov.h:670
octave_value retval
Definition: data.cc:6294
std::complex< float > FloatComplex
Definition: oct-cmplx.h:32
std::complex< double > Complex
Definition: oct-cmplx.h:31