1math::special(n) Tcl Math Library math::special(n)
2______________________________________________________________________________
6
8 math::special - Special mathematical functions
9
11 package require Tcl ?8.5?
12 package require math::special ?0.4?
14 ::math::special::Beta x y
16 ::math::special::incBeta a b x
18 ::math::special::regIncBeta a b x
20 ::math::special::Gamma x
22 ::math::special::digamma x
24 ::math::special::erf x
26 ::math::special::erfc x
28 ::math::special::invnorm p
30 ::math::special::J0 x
32 ::math::special::J1 x
34 ::math::special::Jn n x
36 ::math::special::J1/2 x
38 ::math::special::J-1/2 x
40 ::math::special::I_n x
42 ::math::special::cn u k
44 ::math::special::dn u k
46 ::math::special::sn u k
48 ::math::special::elliptic_K k
50 ::math::special::elliptic_E k
52 ::math::special::exponential_Ei x
54 ::math::special::exponential_En n x
56 ::math::special::exponential_li x
58 ::math::special::exponential_Ci x
60 ::math::special::exponential_Si x
62 ::math::special::exponential_Chi x
64 ::math::special::exponential_Shi x
66 ::math::special::fresnel_C x
68 ::math::special::fresnel_S x
70 ::math::special::sinc x
72 ::math::special::legendre n
74 ::math::special::chebyshev n
76 ::math::special::laguerre alpha n
78 ::math::special::hermite n
80______________________________________________________________________________
82
84 This package implements several so-called special functions, like the
85 Gamma function, the Bessel functions and such.
86 Each function is implemented by a procedure that bears its name (well,
88 in close approximation):
89 • J0 for the zeroth-order Bessel function of the first kind
91 • J1 for the first-order Bessel function of the first kind
93 • Jn for the nth-order Bessel function of the first kind
95 • J1/2 for the half-order Bessel function of the first kind
97 • J-1/2 for the minus-half-order Bessel function of the first kind
99 • I_n for the modified Bessel function of the first kind of order
101 n
102 • Gamma for the Gamma function, erf and erfc for the error func‐
104 tion and the complementary error function
105 • fresnel_C and fresnel_S for the Fresnel integrals
107 • elliptic_K and elliptic_E (complete elliptic integrals)
109 • exponent_Ei and other functions related to the so-called expo‐
111 nential integrals
112 • legendre, hermite: some of the classical orthogonal polynomials.
114
116 In the following table several characteristics of the functions in this
117 package are summarized: the domain for the argument, the values for the
118 parameters and error bounds.
119 Family | Function | Domain x | Parameter | Error bound
122 -------------+-------------+-------------+-------------+--------------
123 Bessel | J0, J1, | all of R | n = integer | < 1.0e-8
124 | Jn | | | (|x|<20, n<20)
125 Bessel | J1/2, J-1/2,| x > 0 | n = integer | exact
126 Bessel | I_n | all of R | n = integer | < 1.0e-6
127 | | | |
128 Elliptic | cn | 0 <= x <= 1 | -- | < 1.0e-10
129 functions | dn | 0 <= x <= 1 | -- | < 1.0e-10
130 | sn | 0 <= x <= 1 | -- | < 1.0e-10
131 Elliptic | K | 0 <= x < 1 | -- | < 1.0e-6
132 integrals | E | 0 <= x < 1 | -- | < 1.0e-6
133 | | | |
134 Error | erf | | -- |
135 functions | erfc | | |
136 | | | |
137 Inverse | invnorm | 0 < x < 1 | -- | < 1.2e-9
138 normal | | | |
139 distribution | | | |
140 | | | |
141 Exponential | Ei | x != 0 | -- | < 1.0e-10 (relative)
142 integrals | En | x > 0 | -- | as Ei
143 | li | x > 0 | -- | as Ei
144 | Chi | x > 0 | -- | < 1.0e-8
145 | Shi | x > 0 | -- | < 1.0e-8
146 | Ci | x > 0 | -- | < 2.0e-4
147 | Si | x > 0 | -- | < 2.0e-4
148 | | | |
149 Fresnel | C | all of R | -- | < 2.0e-3
150 integrals | S | all of R | -- | < 2.0e-3
151 | | | |
152 general | Beta | (see Gamma) | -- | < 1.0e-9
153 | Gamma | x != 0,-1, | -- | < 1.0e-9
154 | | -2, ... | |
155 | incBeta | | a, b > 0 | < 1.0e-9
156 | regIncBeta | | a, b > 0 | < 1.0e-9
157 | digamma | x != 0,-1 | | < 1.0e-9
158 | | -2, ... | |
159 | | | |
160 | sinc | all of R | -- | exact
161 | | | |
162 orthogonal | Legendre | all of R | n = 0,1,... | exact
163 polynomials | Chebyshev | all of R | n = 0,1,... | exact
164 | Laguerre | all of R | n = 0,1,... | exact
165 | | | alpha el. R |
166 | Hermite | all of R | n = 0,1,... | exact
167 Note: Some of the error bounds are estimated, as no "formal" bounds
169 were available with the implemented approximation method, others hold
170 for the auxiliary functions used for estimating the primary functions.
171 The following well-known functions are currently missing from the pack‐
173 age:
174 • Bessel functions of the second kind (Y_n, K_n)
176 • Bessel functions of arbitrary order (and hence the Airy func‐
178 tions)
179 • Chebyshev polynomials of the second kind (U_n)
181 • The incomplete gamma function
183
185 The package defines the following public procedures:
186 ::math::special::Beta x y
188 Compute the Beta function for arguments "x" and "y"
189 float x
191 First argument for the Beta function
192 float y
194 Second argument for the Beta function
195 ::math::special::incBeta a b x
198 Compute the incomplete Beta function for argument "x" with pa‐
199 rameters "a" and "b"
200 float a
202 First parameter for the incomplete Beta function, a > 0
203 float b
205 Second parameter for the incomplete Beta function, b > 0
206 float x
208 Argument for the incomplete Beta function
209 ::math::special::regIncBeta a b x
212 Compute the regularized incomplete Beta function for argument
213 "x" with parameters "a" and "b"
214 float a
216 First parameter for the incomplete Beta function, a > 0
217 float b
219 Second parameter for the incomplete Beta function, b > 0
220 float x
222 Argument for the regularized incomplete Beta function
223 ::math::special::Gamma x
226 Compute the Gamma function for argument "x"
227 float x
229 Argument for the Gamma function
230 ::math::special::digamma x
233 Compute the digamma function (psi) for argument "x"
234 float x
236 Argument for the digamma function
237 ::math::special::erf x
240 Compute the error function for argument "x"
241 float x
243 Argument for the error function
244 ::math::special::erfc x
247 Compute the complementary error function for argument "x"
248 float x
250 Argument for the complementary error function
251 ::math::special::invnorm p
254 Compute the inverse of the normal distribution function for ar‐
255 gument "p"
256 float p
258 Argument for the inverse normal distribution function (p
259 must be greater than 0 and lower than 1)
260 ::math::special::J0 x
263 Compute the zeroth-order Bessel function of the first kind for
264 the argument "x"
265 float x
267 Argument for the Bessel function
268 ::math::special::J1 x
270 Compute the first-order Bessel function of the first kind for
271 the argument "x"
272 float x
274 Argument for the Bessel function
275 ::math::special::Jn n x
277 Compute the nth-order Bessel function of the first kind for the
278 argument "x"
279 integer n
281 Order of the Bessel function
282 float x
284 Argument for the Bessel function
285 ::math::special::J1/2 x
287 Compute the half-order Bessel function of the first kind for the
288 argument "x"
289 float x
291 Argument for the Bessel function
292 ::math::special::J-1/2 x
294 Compute the minus-half-order Bessel function of the first kind
295 for the argument "x"
296 float x
298 Argument for the Bessel function
299 ::math::special::I_n x
301 Compute the modified Bessel function of the first kind of order
302 n for the argument "x"
303 int x Positive integer order of the function
305 float x
307 Argument for the function
308 ::math::special::cn u k
310 Compute the elliptic function cn for the argument "u" and param‐
311 eter "k".
312 float u
314 Argument for the function
315 float k
317 Parameter
318 ::math::special::dn u k
320 Compute the elliptic function dn for the argument "u" and param‐
321 eter "k".
322 float u
324 Argument for the function
325 float k
327 Parameter
328 ::math::special::sn u k
330 Compute the elliptic function sn for the argument "u" and param‐
331 eter "k".
332 float u
334 Argument for the function
335 float k
337 Parameter
338 ::math::special::elliptic_K k
340 Compute the complete elliptic integral of the first kind for the
341 argument "k"
342 float k
344 Argument for the function
345 ::math::special::elliptic_E k
347 Compute the complete elliptic integral of the second kind for
348 the argument "k"
349 float k
351 Argument for the function
352 ::math::special::exponential_Ei x
354 Compute the exponential integral of the second kind for the ar‐
355 gument "x"
356 float x
358 Argument for the function (x != 0)
359 ::math::special::exponential_En n x
361 Compute the exponential integral of the first kind for the argu‐
362 ment "x" and order n
363 int n Order of the integral (n >= 0)
365 float x
367 Argument for the function (x >= 0)
368 ::math::special::exponential_li x
370 Compute the logarithmic integral for the argument "x"
371 float x
373 Argument for the function (x > 0)
374 ::math::special::exponential_Ci x
376 Compute the cosine integral for the argument "x"
377 float x
379 Argument for the function (x > 0)
380 ::math::special::exponential_Si x
382 Compute the sine integral for the argument "x"
383 float x
385 Argument for the function (x > 0)
386 ::math::special::exponential_Chi x
388 Compute the hyperbolic cosine integral for the argument "x"
389 float x
391 Argument for the function (x > 0)
392 ::math::special::exponential_Shi x
394 Compute the hyperbolic sine integral for the argument "x"
395 float x
397 Argument for the function (x > 0)
398 ::math::special::fresnel_C x
400 Compute the Fresnel cosine integral for real argument x
401 float x
403 Argument for the function
404 ::math::special::fresnel_S x
406 Compute the Fresnel sine integral for real argument x
407 float x
409 Argument for the function
410 ::math::special::sinc x
412 Compute the sinc function for real argument x
413 float x
415 Argument for the function
416 ::math::special::legendre n
418 Return the Legendre polynomial of degree n (see THE ORTHOGONAL
419 POLYNOMIALS)
420 int n Degree of the polynomial
422 ::math::special::chebyshev n
425 Return the Chebyshev polynomial of degree n (of the first kind)
426 int n Degree of the polynomial
428 ::math::special::laguerre alpha n
431 Return the Laguerre polynomial of degree n with parameter alpha
432 float alpha
434 Parameter of the Laguerre polynomial
435 int n Degree of the polynomial
437 ::math::special::hermite n
440 Return the Hermite polynomial of degree n
441 int n Degree of the polynomial
443
446 For dealing with the classical families of orthogonal polynomials, the
447 package relies on the math::polynomials package. To evaluate the poly‐
448 nomial at some coordinate, use the evalPolyn command:
449 set leg2 [::math::special::legendre 2]
452 puts "Value at x=$x: [::math::polynomials::evalPolyn $leg2 $x]"
453 The return value from the legendre and other commands is actually the
456 definition of the corresponding polynomial as used in that package.
457
459 It should be noted, that the actual implementation of J0 and J1 depends
460 on straightforward Gaussian quadrature formulas. The (absolute) accu‐
461 racy of the results is of the order 1.0e-4 or better. The main reason
462 to implement them like that was that it was fast to do (the formulas
463 are simple) and the computations are fast too.
464 The implementation of J1/2 does not suffer from this: this function can
466 be expressed exactly in terms of elementary functions.
467 The functions J0 and J1 are the ones you will encounter most frequently
469 in practice.
470 The computation of I_n is based on Miller's algorithm for computing the
472 minimal function from recurrence relations.
473 The computation of the Gamma and Beta functions relies on the combina‐
475 torics package, whereas that of the error functions relies on the sta‐
476 tistics package.
477 The computation of the complete elliptic integrals uses the AGM algo‐
479 rithm.
480 Much information about these functions can be found in:
482 Abramowitz and Stegun: Handbook of Mathematical Functions (Dover, ISBN
484 486-61272-4)
485
487 This document, and the package it describes, will undoubtedly contain
488 bugs and other problems. Please report such in the category math ::
489 special of the Tcllib Trackers [http://core.tcl.tk/tcllib/reportlist].
490 Please also report any ideas for enhancements you may have for either
491 package and/or documentation.
492 When proposing code changes, please provide unified diffs, i.e the out‐
494 put of diff -u.
495 Note further that attachments are strongly preferred over inlined
497 patches. Attachments can be made by going to the Edit form of the
498 ticket immediately after its creation, and then using the left-most
499 button in the secondary navigation bar.
500
502 Bessel functions, error function, math, special functions
503
505 Mathematics
506
508 Copyright (c) 2004 Arjen Markus <arjenmarkus@users.sourceforge.net>
509tcllib 0.4 math::special(n)