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 = &#915;
32 *      THETA = &theta;
33 *      INTEGRAL = &#8747;
34 *      INTEGRATE = $(BIG &#8747;<sub>$(SMALL $1)</sub><sup>$2</sup>)
35 *      POWER = $1<sup>$2</sup>
36 *      SUB = $1<sub>$2</sub>
37 *      BIGSUM = $(BIG &Sigma; <sup>$2</sup><sub>$(SMALL $1)</sub>)
38 *      CHOOSE = $(BIG &#40;) <sup>$(SMALL $1)</sup><sub>$(SMALL $2)</sub> $(BIG &#41;)
39 *      PLUSMN = &plusmn;
40 *      INFIN = &infin;
41 *      PLUSMNINF = &plusmn;&infin;
42 *      PI = &pi;
43 *      LT = &lt;
44 *      GT = &gt;
45 *      SQRT = &radic;
46 *      HALF = &frac12;
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 */
55module std.mathspecial;
56import std.internal.math.errorfunction;
57import std.internal.math.gammafunction;
58public import std.math;
59
60/* ***********************************************
61 *            GAMMA AND RELATED FUNCTIONS        *
62 * ***********************************************/
63
64pure:
65nothrow:
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 */
88real 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 */
107real 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 * $(NAN) if sign is indeterminate.
116 *
117 * Note that this function can be used in conjunction with logGamma(x) to
118 * evaluate gamma for very large values of x.
119 */
120real sgnGamma(real x)
121{
122    /* Author: Don Clugston. */
123    if (isNaN(x)) return x;
124    if (x > 0) return 1.0;
125    if (x < -1/real.epsilon)
126    {
127        // Large negatives lose all precision
128        return real.nan;
129    }
130    long n = rndtol(x);
131    if (x == n)
132    {
133        return x == 0 ?  copysign(1, x) : real.nan;
134    }
135    return n & 1 ? 1.0 : -1.0;
136}
137
138@safe unittest
139{
140    assert(sgnGamma(5.0) == 1.0);
141    assert(isNaN(sgnGamma(-3.0)));
142    assert(sgnGamma(-0.1) == -1.0);
143    assert(sgnGamma(-55.1) == 1.0);
144    assert(isNaN(sgnGamma(-real.infinity)));
145    assert(isIdentical(sgnGamma(NaN(0xABC)), NaN(0xABC)));
146}
147
148/** Beta function
149 *
150 * The beta function is defined as
151 *
152 * beta(x, y) = ($(GAMMA)(x) * $(GAMMA)(y)) / $(GAMMA)(x + y)
153 */
154real 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 */
176real 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 */
187real 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 */
198real 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 */
223real 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 */
236real 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 */
255real gammaIncomplete(real a, real x )
256in {
257   assert(x >= 0);
258   assert(a > 0);
259}
260body {
261    return std.internal.math.gammafunction.gammaIncomplete(a, x);
262}
263
264/** ditto */
265real gammaIncompleteCompl(real a, real x )
266in {
267   assert(x >= 0);
268   assert(a > 0);
269}
270body {
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 */
280real gammaIncompleteComplInverse(real a, real p)
281in {
282  assert(p >= 0 && p <= 1);
283  assert(a > 0);
284}
285body {
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 */
304real 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( - $(POWER t, 2)) dt
314 *
315 * This function has high relative accuracy for
316 * values of x far from zero. (For values near zero, use erf(x)).
317 */
318real erfc(real x)
319{
320    return std.internal.math.errorfunction.erfc(x);
321}
322
323
324/** Normal distribution function.
325 *
326 * The normal (or Gaussian, or bell-shaped) distribution is
327 * defined as:
328 *
329 * normalDist(x) = 1/$(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 */
341real 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 */
354real normalDistributionInverse(real p)
355in {
356  assert(p >= 0.0L && p <= 1.0L, "Domain error");
357}
358body
359{
360    return std.internal.math.errorfunction.normalDistributionInvImpl(p);
361}
362