1 // Written in the D programming language.
2
3 /**
4 * Mathematical Special Functions
5 *
6 * The technical term 'Special Functions' includes several families of
7 * transcendental functions, which have important applications in particular
8 * branches of mathematics and physics.
9 *
10 * The gamma and related functions, and the error function are crucial for
11 * mathematical statistics.
12 * The Bessel and related functions arise in problems involving wave propagation
13 * (especially in optics).
14 * Other major categories of special functions include the elliptic integrals
15 * (related to the arc length of an ellipse), and the hypergeometric functions.
16 *
17 * Status:
18 * Many more functions will be added to this module.
19 * The naming convention for the distribution functions (gammaIncomplete, etc)
20 * is not yet finalized and will probably change.
21 *
22 * Macros:
23 * TABLE_SV = <table border="1" cellpadding="4" cellspacing="0">
24 * <caption>Special Values</caption>
25 * $0</table>
26 * SVH = $(TR $(TH $1) $(TH $2))
27 * SV = $(TR $(TD $1) $(TD $2))
28 *
29 * NAN = $(RED NAN)
30 * SUP = <span style="vertical-align:super;font-size:smaller">$0</span>
31 * GAMMA = Γ
32 * THETA = θ
33 * INTEGRAL = ∫
34 * INTEGRATE = $(BIG ∫<sub>$(SMALL $1)</sub><sup>$2</sup>)
35 * POWER = $1<sup>$2</sup>
36 * SUB = $1<sub>$2</sub>
37 * BIGSUM = $(BIG Σ <sup>$2</sup><sub>$(SMALL $1)</sub>)
38 * CHOOSE = $(BIG () <sup>$(SMALL $1)</sup><sub>$(SMALL $2)</sub> $(BIG ))
39 * PLUSMN = ±
40 * INFIN = ∞
41 * PLUSMNINF = ±∞
42 * PI = π
43 * LT = <
44 * GT = >
45 * SQRT = √
46 * HALF = ½
47 *
48 *
49 * Copyright: Based on the CEPHES math library, which is
50 * Copyright (C) 1994 Stephen L. Moshier (moshier@world.std.com).
51 * License: $(HTTP www.boost.org/LICENSE_1_0.txt, Boost License 1.0).
52 * Authors: Stephen L. Moshier (original C code). Conversion to D by Don Clugston
53 * Source: $(PHOBOSSRC std/_mathspecial.d)
54 */
55 module std.mathspecial;
56 import std.internal.math.errorfunction;
57 import std.internal.math.gammafunction;
58 public import std.math;
59
60 /* ***********************************************
61 * GAMMA AND RELATED FUNCTIONS *
62 * ***********************************************/
63
64 pure:
65 nothrow:
66 @safe:
67 @nogc:
68
69 /** The Gamma function, $(GAMMA)(x)
70 *
71 * $(GAMMA)(x) is a generalisation of the factorial function
72 * to real and complex numbers.
73 * Like x!, $(GAMMA)(x+1) = x * $(GAMMA)(x).
74 *
75 * Mathematically, if z.re > 0 then
76 * $(GAMMA)(z) = $(INTEGRATE 0, $(INFIN)) $(POWER t, z-1)$(POWER e, -t) dt
77 *
78 * $(TABLE_SV
79 * $(SVH x, $(GAMMA)(x) )
80 * $(SV $(NAN), $(NAN) )
81 * $(SV $(PLUSMN)0.0, $(PLUSMNINF))
82 * $(SV integer > 0, (x-1)! )
83 * $(SV integer < 0, $(NAN) )
84 * $(SV +$(INFIN), +$(INFIN) )
85 * $(SV -$(INFIN), $(NAN) )
86 * )
87 */
gamma(real x)88 real gamma(real x)
89 {
90 return std.internal.math.gammafunction.gamma(x);
91 }
92
93 /** Natural logarithm of the gamma function, $(GAMMA)(x)
94 *
95 * Returns the base e (2.718...) logarithm of the absolute
96 * value of the gamma function of the argument.
97 *
98 * For reals, logGamma is equivalent to log(fabs(gamma(x))).
99 *
100 * $(TABLE_SV
101 * $(SVH x, logGamma(x) )
102 * $(SV $(NAN), $(NAN) )
103 * $(SV integer <= 0, +$(INFIN) )
104 * $(SV $(PLUSMNINF), +$(INFIN) )
105 * )
106 */
logGamma(real x)107 real logGamma(real x)
108 {
109 return std.internal.math.gammafunction.logGamma(x);
110 }
111
112 /** The sign of $(GAMMA)(x).
113 *
114 * Returns -1 if $(GAMMA)(x) < 0, +1 if $(GAMMA)(x) > 0,
115 * $(NAN) if sign is indeterminate.
116 *
117 * Note that this function can be used in conjunction with logGamma(x) to
118 * evaluate gamma for very large values of x.
119 */
sgnGamma(real x)120 real sgnGamma(real x)
121 {
122 /* Author: Don Clugston. */
123 if (isNaN(x)) return x;
124 if (x > 0) return 1.0;
125 if (x < -1/real.epsilon)
126 {
127 // Large negatives lose all precision
128 return real.nan;
129 }
130 long n = rndtol(x);
131 if (x == n)
132 {
133 return x == 0 ? copysign(1, x) : real.nan;
134 }
135 return n & 1 ? 1.0 : -1.0;
136 }
137
138 @safe unittest
139 {
140 assert(sgnGamma(5.0) == 1.0);
141 assert(isNaN(sgnGamma(-3.0)));
142 assert(sgnGamma(-0.1) == -1.0);
143 assert(sgnGamma(-55.1) == 1.0);
144 assert(isNaN(sgnGamma(-real.infinity)));
145 assert(isIdentical(sgnGamma(NaN(0xABC)), NaN(0xABC)));
146 }
147
148 /** Beta function
149 *
150 * The beta function is defined as
151 *
152 * beta(x, y) = ($(GAMMA)(x) * $(GAMMA)(y)) / $(GAMMA)(x + y)
153 */
beta(real x,real y)154 real beta(real x, real y)
155 {
156 if ((x+y)> MAXGAMMA)
157 {
158 return exp(logGamma(x) + logGamma(y) - logGamma(x+y));
159 } else return gamma(x) * gamma(y) / gamma(x+y);
160 }
161
162 @safe unittest
163 {
164 assert(isIdentical(beta(NaN(0xABC), 4), NaN(0xABC)));
165 assert(isIdentical(beta(2, NaN(0xABC)), NaN(0xABC)));
166 }
167
168 /** Digamma function
169 *
170 * The digamma function is the logarithmic derivative of the gamma function.
171 *
172 * digamma(x) = d/dx logGamma(x)
173 *
174 * See_Also: $(LREF logmdigamma), $(LREF logmdigammaInverse).
175 */
digamma(real x)176 real digamma(real x)
177 {
178 return std.internal.math.gammafunction.digamma(x);
179 }
180
181 /** Log Minus Digamma function
182 *
183 * logmdigamma(x) = log(x) - digamma(x)
184 *
185 * See_Also: $(LREF digamma), $(LREF logmdigammaInverse).
186 */
logmdigamma(real x)187 real logmdigamma(real x)
188 {
189 return std.internal.math.gammafunction.logmdigamma(x);
190 }
191
192 /** Inverse of the Log Minus Digamma function
193 *
194 * Given y, the function finds x such log(x) - digamma(x) = y.
195 *
196 * See_Also: $(LREF logmdigamma), $(LREF digamma).
197 */
logmdigammaInverse(real x)198 real logmdigammaInverse(real x)
199 {
200 return std.internal.math.gammafunction.logmdigammaInverse(x);
201 }
202
203 /** Incomplete beta integral
204 *
205 * Returns incomplete beta integral of the arguments, evaluated
206 * from zero to x. The regularized incomplete beta function is defined as
207 *
208 * betaIncomplete(a, b, x) = $(GAMMA)(a + b) / ( $(GAMMA)(a) $(GAMMA)(b) ) *
209 * $(INTEGRATE 0, x) $(POWER t, a-1)$(POWER (1-t), b-1) dt
210 *
211 * and is the same as the the cumulative distribution function.
212 *
213 * The domain of definition is 0 <= x <= 1. In this
214 * implementation a and b are restricted to positive values.
215 * The integral from x to 1 may be obtained by the symmetry
216 * relation
217 *
218 * betaIncompleteCompl(a, b, x ) = betaIncomplete( b, a, 1-x )
219 *
220 * The integral is evaluated by a continued fraction expansion
221 * or, when b * x is small, by a power series.
222 */
betaIncomplete(real a,real b,real x)223 real betaIncomplete(real a, real b, real x )
224 {
225 return std.internal.math.gammafunction.betaIncomplete(a, b, x);
226 }
227
228 /** Inverse of incomplete beta integral
229 *
230 * Given y, the function finds x such that
231 *
232 * betaIncomplete(a, b, x) == y
233 *
234 * Newton iterations or interval halving is used.
235 */
betaIncompleteInverse(real a,real b,real y)236 real betaIncompleteInverse(real a, real b, real y )
237 {
238 return std.internal.math.gammafunction.betaIncompleteInv(a, b, y);
239 }
240
241 /** Incomplete gamma integral and its complement
242 *
243 * These functions are defined by
244 *
245 * gammaIncomplete = ( $(INTEGRATE 0, x) $(POWER e, -t) $(POWER t, a-1) dt )/ $(GAMMA)(a)
246 *
247 * gammaIncompleteCompl(a,x) = 1 - gammaIncomplete(a,x)
248 * = ($(INTEGRATE x, $(INFIN)) $(POWER e, -t) $(POWER t, a-1) dt )/ $(GAMMA)(a)
249 *
250 * In this implementation both arguments must be positive.
251 * The integral is evaluated by either a power series or
252 * continued fraction expansion, depending on the relative
253 * values of a and x.
254 */
gammaIncomplete(real a,real x)255 real gammaIncomplete(real a, real x )
256 in {
257 assert(x >= 0);
258 assert(a > 0);
259 }
260 body {
261 return std.internal.math.gammafunction.gammaIncomplete(a, x);
262 }
263
264 /** ditto */
gammaIncompleteCompl(real a,real x)265 real gammaIncompleteCompl(real a, real x )
266 in {
267 assert(x >= 0);
268 assert(a > 0);
269 }
270 body {
271 return std.internal.math.gammafunction.gammaIncompleteCompl(a, x);
272 }
273
274 /** Inverse of complemented incomplete gamma integral
275 *
276 * Given a and p, the function finds x such that
277 *
278 * gammaIncompleteCompl( a, x ) = p.
279 */
gammaIncompleteComplInverse(real a,real p)280 real gammaIncompleteComplInverse(real a, real p)
281 in {
282 assert(p >= 0 && p <= 1);
283 assert(a > 0);
284 }
285 body {
286 return std.internal.math.gammafunction.gammaIncompleteComplInv(a, p);
287 }
288
289
290 /* ***********************************************
291 * ERROR FUNCTIONS & NORMAL DISTRIBUTION *
292 * ***********************************************/
293
294 /** Error function
295 *
296 * The integral is
297 *
298 * erf(x) = 2/ $(SQRT)($(PI))
299 * $(INTEGRATE 0, x) exp( - $(POWER t, 2)) dt
300 *
301 * The magnitude of x is limited to about 106.56 for IEEE 80-bit
302 * arithmetic; 1 or -1 is returned outside this range.
303 */
erf(real x)304 real erf(real x)
305 {
306 return std.internal.math.errorfunction.erf(x);
307 }
308
309 /** Complementary error function
310 *
311 * erfc(x) = 1 - erf(x)
312 * = 2/ $(SQRT)($(PI))
313 * $(INTEGRATE x, $(INFIN)) exp( - $(POWER t, 2)) dt
314 *
315 * This function has high relative accuracy for
316 * values of x far from zero. (For values near zero, use erf(x)).
317 */
erfc(real x)318 real erfc(real x)
319 {
320 return std.internal.math.errorfunction.erfc(x);
321 }
322
323
324 /** Normal distribution function.
325 *
326 * The normal (or Gaussian, or bell-shaped) distribution is
327 * defined as:
328 *
329 * normalDist(x) = 1/$(SQRT)(2$(PI)) $(INTEGRATE -$(INFIN), x) exp( - $(POWER t, 2)/2) dt
330 * = 0.5 + 0.5 * erf(x/sqrt(2))
331 * = 0.5 * erfc(- x/sqrt(2))
332 *
333 * To maintain accuracy at values of x near 1.0, use
334 * normalDistribution(x) = 1.0 - normalDistribution(-x).
335 *
336 * References:
337 * $(LINK http://www.netlib.org/cephes/ldoubdoc.html),
338 * G. Marsaglia, "Evaluating the Normal Distribution",
339 * Journal of Statistical Software <b>11</b>, (July 2004).
340 */
normalDistribution(real x)341 real normalDistribution(real x)
342 {
343 return std.internal.math.errorfunction.normalDistributionImpl(x);
344 }
345
346 /** Inverse of Normal distribution function
347 *
348 * Returns the argument, x, for which the area under the
349 * Normal probability density function (integrated from
350 * minus infinity to x) is equal to p.
351 *
352 * Note: This function is only implemented to 80 bit precision.
353 */
normalDistributionInverse(real p)354 real normalDistributionInverse(real p)
355 in {
356 assert(p >= 0.0L && p <= 1.0L, "Domain error");
357 }
358 body
359 {
360 return std.internal.math.errorfunction.normalDistributionInvImpl(p);
361 }
362