1math::special(n) Tcl Math Library math::special(n)
2______________________________________________________________________________
6
8 math::special - Special mathematical functions
9
11 package require Tcl ?8.3?
12 package require math::special ?0.3?
14 ::math::special::Beta x y
16 ::math::special::Gamma x
18 ::math::special::erf x
20 ::math::special::erfc x
22 ::math::special::invnorm p
24 ::math::special::J0 x
26 ::math::special::J1 x
28 ::math::special::Jn n x
30 ::math::special::J1/2 x
32 ::math::special::J-1/2 x
34 ::math::special::I_n x
36 ::math::special::cn u k
38 ::math::special::dn u k
40 ::math::special::sn u k
42 ::math::special::elliptic_K k
44 ::math::special::elliptic_E k
46 ::math::special::exponential_Ei x
48 ::math::special::exponential_En n x
50 ::math::special::exponential_li x
52 ::math::special::exponential_Ci x
54 ::math::special::exponential_Si x
56 ::math::special::exponential_Chi x
58 ::math::special::exponential_Shi x
60 ::math::special::fresnel_C x
62 ::math::special::fresnel_S x
64 ::math::special::sinc x
66 ::math::special::legendre n
68 ::math::special::chebyshev n
70 ::math::special::laguerre alpha n
72 ::math::special::hermite n
74______________________________________________________________________________
76
78 This package implements several so-called special functions, like the
79 Gamma function, the Bessel functions and such.
80 Each function is implemented by a procedure that bears its name (well,
82 in close approximation):
83 · J0 for the zeroth-order Bessel function of the first kind
85 · J1 for the first-order Bessel function of the first kind
87 · Jn for the nth-order Bessel function of the first kind
89 · J1/2 for the half-order Bessel function of the first kind
91 · J-1/2 for the minus-half-order Bessel function of the first kind
93 · I_n for the modified Bessel function of the first kind of order
95 n
96 · Gamma for the Gamma function, erf and erfc for the error func‐
98 tion and the complementary error function
99 · fresnel_C and fresnel_S for the Fresnel integrals
101 · elliptic_K and elliptic_E (complete elliptic integrals)
103 · exponent_Ei and other functions related to the so-called expo‐
105 nential integrals
106 · legendre, hermite: some of the classical orthogonal polynomials.
108
110 In the following table several characteristics of the functions in this
111 package are summarized: the domain for the argument, the values for the
112 parameters and error bounds.
113 Family | Function | Domain x | Parameter | Error bound
116 -------------+-------------+-------------+-------------+--------------
117 Bessel | J0, J1, | all of R | n = integer | < 1.0e-8
118 | Jn | | | (|x|<20, n<20)
119 Bessel | J1/2, J-1/2,| x > 0 | n = integer | exact
120 Bessel | I_n | all of R | n = integer | < 1.0e-6
121 | | | |
122 Elliptic | cn | 0 <= x <= 1 | -- | < 1.0e-10
123 functions | dn | 0 <= x <= 1 | -- | < 1.0e-10
124 | sn | 0 <= x <= 1 | -- | < 1.0e-10
125 Elliptic | K | 0 <= x < 1 | -- | < 1.0e-6
126 integrals | E | 0 <= x < 1 | -- | < 1.0e-6
127 | | | |
128 Error | erf | | -- |
129 functions | erfc | | |
130 | | | |
131 Inverse | invnorm | 0 < x < 1 | -- | < 1.2e-9
132 normal | | | |
133 distribution | | | |
134 | | | |
135 Exponential | Ei | x != 0 | -- | < 1.0e-10 (relative)
136 integrals | En | x > 0 | -- | as Ei
137 | li | x > 0 | -- | as Ei
138 | Chi | x > 0 | -- | < 1.0e-8
139 | Shi | x > 0 | -- | < 1.0e-8
140 | Ci | x > 0 | -- | < 2.0e-4
141 | Si | x > 0 | -- | < 2.0e-4
142 | | | |
143 Fresnel | C | all of R | -- | < 2.0e-3
144 integrals | S | all of R | -- | < 2.0e-3
145 | | | |
146 general | Beta | (see Gamma) | -- | < 1.0e-9
147 | Gamma | x != 0,-1, | -- | < 1.0e-9
148 | | -2, ... | |
149 | sinc | all of R | -- | exact
150 | | | |
151 orthogonal | Legendre | all of R | n = 0,1,... | exact
152 polynomials | Chebyshev | all of R | n = 0,1,... | exact
153 | Laguerre | all of R | n = 0,1,... | exact
154 | | | alpha el. R |
155 | Hermite | all of R | n = 0,1,... | exact
156 Note: Some of the error bounds are estimated, as no "formal" bounds
158 were available with the implemented approximation method, others hold
159 for the auxiliary functions used for estimating the primary functions.
160 The following well-known functions are currently missing from the pack‐
162 age:
163 · Bessel functions of the second kind (Y_n, K_n)
165 · Bessel functions of arbitrary order (and hence the Airy func‐
167 tions)
168 · Chebyshev polynomials of the second kind (U_n)
170 · The digamma function (psi)
172 · The incomplete gamma and beta functions
174
176 The package defines the following public procedures:
177 ::math::special::Beta x y
179 Compute the Beta function for arguments "x" and "y"
180 float x
182 First argument for the Beta function
183 float y
185 Second argument for the Beta function
186 ::math::special::Gamma x
189 Compute the Gamma function for argument "x"
190 float x
192 Argument for the Gamma function
193 ::math::special::erf x
196 Compute the error function for argument "x"
197 float x
199 Argument for the error function
200 ::math::special::erfc x
203 Compute the complementary error function for argument "x"
204 float x
206 Argument for the complementary error function
207 ::math::special::invnorm p
210 Compute the inverse of the normal distribution function for
211 argument "p"
212 float p
214 Argument for the inverse normal distribution function (p
215 must be greater than 0 and lower than 1)
216 ::math::special::J0 x
219 Compute the zeroth-order Bessel function of the first kind for
220 the argument "x"
221 float x
223 Argument for the Bessel function
224 ::math::special::J1 x
226 Compute the first-order Bessel function of the first kind for
227 the argument "x"
228 float x
230 Argument for the Bessel function
231 ::math::special::Jn n x
233 Compute the nth-order Bessel function of the first kind for the
234 argument "x"
235 integer n
237 Order of the Bessel function
238 float x
240 Argument for the Bessel function
241 ::math::special::J1/2 x
243 Compute the half-order Bessel function of the first kind for the
244 argument "x"
245 float x
247 Argument for the Bessel function
248 ::math::special::J-1/2 x
250 Compute the minus-half-order Bessel function of the first kind
251 for the argument "x"
252 float x
254 Argument for the Bessel function
255 ::math::special::I_n x
257 Compute the modified Bessel function of the first kind of order
258 n for the argument "x"
259 int x Positive integer order of the function
261 float x
263 Argument for the function
264 ::math::special::cn u k
266 Compute the elliptic function cn for the argument "u" and param‐
267 eter "k".
268 float u
270 Argument for the function
271 float k
273 Parameter
274 ::math::special::dn u k
276 Compute the elliptic function dn for the argument "u" and param‐
277 eter "k".
278 float u
280 Argument for the function
281 float k
283 Parameter
284 ::math::special::sn u k
286 Compute the elliptic function sn for the argument "u" and param‐
287 eter "k".
288 float u
290 Argument for the function
291 float k
293 Parameter
294 ::math::special::elliptic_K k
296 Compute the complete elliptic integral of the first kind for the
297 argument "k"
298 float k
300 Argument for the function
301 ::math::special::elliptic_E k
303 Compute the complete elliptic integral of the second kind for
304 the argument "k"
305 float k
307 Argument for the function
308 ::math::special::exponential_Ei x
310 Compute the exponential integral of the second kind for the
311 argument "x"
312 float x
314 Argument for the function (x != 0)
315 ::math::special::exponential_En n x
317 Compute the exponential integral of the first kind for the argu‐
318 ment "x" and order n
319 int n Order of the integral (n >= 0)
321 float x
323 Argument for the function (x >= 0)
324 ::math::special::exponential_li x
326 Compute the logarithmic integral for the argument "x"
327 float x
329 Argument for the function (x > 0)
330 ::math::special::exponential_Ci x
332 Compute the cosine integral for the argument "x"
333 float x
335 Argument for the function (x > 0)
336 ::math::special::exponential_Si x
338 Compute the sine integral for the argument "x"
339 float x
341 Argument for the function (x > 0)
342 ::math::special::exponential_Chi x
344 Compute the hyperbolic cosine integral for the argument "x"
345 float x
347 Argument for the function (x > 0)
348 ::math::special::exponential_Shi x
350 Compute the hyperbolic sine integral for the argument "x"
351 float x
353 Argument for the function (x > 0)
354 ::math::special::fresnel_C x
356 Compute the Fresnel cosine integral for real argument x
357 float x
359 Argument for the function
360 ::math::special::fresnel_S x
362 Compute the Fresnel sine integral for real argument x
363 float x
365 Argument for the function
366 ::math::special::sinc x
368 Compute the sinc function for real argument x
369 float x
371 Argument for the function
372 ::math::special::legendre n
374 Return the Legendre polynomial of degree n (see THE ORTHOGONAL
375 POLYNOMIALS)
376 int n Degree of the polynomial
378 ::math::special::chebyshev n
381 Return the Chebyshev polynomial of degree n (of the first kind)
382 int n Degree of the polynomial
384 ::math::special::laguerre alpha n
387 Return the Laguerre polynomial of degree n with parameter alpha
388 float alpha
390 Parameter of the Laguerre polynomial
391 int n Degree of the polynomial
393 ::math::special::hermite n
396 Return the Hermite polynomial of degree n
397 int n Degree of the polynomial
399
402 For dealing with the classical families of orthogonal polynomials, the
403 package relies on the math::polynomials package. To evaluate the poly‐
404 nomial at some coordinate, use the evalPolyn command:
405 set leg2 [::math::special::legendre 2]
408 puts "Value at x=$x: [::math::polynomials::evalPolyn $leg2 $x]"
409 The return value from the legendre and other commands is actually the
412 definition of the corresponding polynomial as used in that package.
413
415 It should be noted, that the actual implementation of J0 and J1 depends
416 on straightforward Gaussian quadrature formulas. The (absolute) accu‐
417 racy of the results is of the order 1.0e-4 or better. The main reason
418 to implement them like that was that it was fast to do (the formulas
419 are simple) and the computations are fast too.
420 The implementation of J1/2 does not suffer from this: this function can
422 be expressed exactly in terms of elementary functions.
423 The functions J0 and J1 are the ones you will encounter most frequently
425 in practice.
426 The computation of I_n is based on Miller's algorithm for computing the
428 minimal function from recurrence relations.
429 The computation of the Gamma and Beta functions relies on the combina‐
431 torics package, whereas that of the error functions relies on the sta‐
432 tistics package.
433 The computation of the complete elliptic integrals uses the AGM algo‐
435 rithm.
436 Much information about these functions can be found in:
438 Abramowitz and Stegun: Handbook of Mathematical Functions (Dover, ISBN
440 486-61272-4)
441
443 This document, and the package it describes, will undoubtedly contain
444 bugs and other problems. Please report such in the category math ::
445 special of the Tcllib Trackers [http://core.tcl.tk/tcllib/reportlist].
446 Please also report any ideas for enhancements you may have for either
447 package and/or documentation.
448 When proposing code changes, please provide unified diffs, i.e the out‐
450 put of diff -u.
451 Note further that attachments are strongly preferred over inlined
453 patches. Attachments can be made by going to the Edit form of the
454 ticket immediately after its creation, and then using the left-most
455 button in the secondary navigation bar.
456
458 Bessel functions, error function, math, special functions
459
461 Mathematics
462
464 Copyright (c) 2004 Arjen Markus <arjenmarkus@users.sourceforge.net>
465tcllib 0.3 math::special(n)