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math::special(n) 0.1 "Math"
math::special - Special mathematical functions
TABLE OF
CONTENTS
SYNOPSIS
DESCRIPTION
OVERVIEW
PROCEDURES
THE
ORTHOGONAL POLYNOMIALS
REMARKS ON THE IMPLEMENTATION
KEYWORDS
COPYRIGHT
package require Tcl ?8.3?
package require math::special ?0.1?
This package implements several so-called special functions,
like the Gamma function, the Bessel functions and such.
Each function is implemented by a procedure that bears its name
(well, in close approximation):
- J0 for the zeroth-order Bessel function of the first kind
- J1 for the first-order Bessel function of the first kind
- Jn for the nth-order Bessel function of the first kind
- J1/2 for the half-order Bessel function of the first kind
- J-1/2 for the minus-half-order Bessel function of the first
kind
- I_n for the modified Bessel function of the first kind of order
n
- Gamma for the Gamma function, erf and erfc for the error
function and the complementary error function
- fresnel_C and fresnel_S for the Fresnel integrals
- elliptic_K and elliptic_E (complete elliptic integrals)
- exponent_Ei and other functions related to the so-called
exponential integrals
- legendre, hermite: some of the classical orthogonal
polynomials.
In the following table several characteristics of the functions
in this package are summarized: the domain for the argument, the
values for the parameters and error bounds.
|
Family | Function | Domain x | Parameter | Error bound
-------------+-------------+-------------+-------------+--------------
Bessel | J0, J1, | all of R | n = integer | < 1.0e-8
| Jn | | | (|x|<20, n<20)
Bessel | J1/2, J-1/2,| x > 0 | n = integer | exact
Bessel | I_n | all of R | n = integer | < 1.0e-6
| | | |
Elliptic | K | 0 <= x < 1 | -- | < 1.0e-6
integrals | E | 0 <= x < 1 | -- | < 1.0e-6
| | | |
Error | erf | | -- |
functions | erfc | | |
| ierfc_n | | |
| | | |
Exponential | Ei | x != 0 | -- | < 1.0e-10 (relative)
integrals | En | x > 0 | -- | as Ei
| li | x > 0 | -- | as Ei
| Chi | x > 0 | -- | < 1.0e-8
| Shi | x > 0 | -- | < 1.0e-8
| Ci | x > 0 | -- | < 2.0e-4
| Si | x > 0 | -- | < 2.0e-4
| | | |
Fresnel | C | all of R | -- | < 2.0e-3
integrals | S | all of R | -- | < 2.0e-3
| | | |
general | Beta | (see Gamma) | -- | < 1.0e-9
| Gamma | x != 0,-1, | -- | < 1.0e-9
| | -2, ... | |
| sinc | all of R | -- | exact
| | | |
orthogonal | Legendre | all of R | n = 0,1,... | exact
polynomials | Chebyshev | all of R | n = 0,1,... | exact
| Laguerre | all of R | n = 0,1,... | exact
| | | alpha el. R |
| Hermite | all of R | n = 0,1,... | exact
|
Note: Some of the error bounds are estimated, as no
"formal" bounds were available with the implemented approximation
method, others hold for the auxiliary functions used for estimating
the primary functions.
The following well-known functions are currently missing from
the package:
- Bessel functions of the second kind (Y_n, K_n)
- Bessel functions of arbitrary order (and hence the Airy
functions)
- Chebyshev polynomials of the second kind (U_n)
- The digamma function (psi)
- The incomplete gamma and beta functions
The package defines the following public procedures:
- ::math::special::Beta x y
- Compute the Beta function for arguments "x" and "y"
- float x
- First argument for the Beta function
- float y
- Second argument for the Beta function
- ::math::special::Gamma x y
- Compute the Gamma function for argument "x"
- float x
- Argument for the Gamma function
- ::math::special::erf x
- Compute the error function for argument "x"
- float x
- Argument for the error function
- ::math::special::erfc x
- Compute the complementary error function for argument "x"
- float x
- Argument for the complementary error function
- ::math::special::J0 x
- Compute the zeroth-order Bessel function of the first kind for
the argument "x"
- float x
- Argument for the Bessel function
- ::math::special::J1 x
- Compute the first-order Bessel function of the first kind for
the argument "x"
- float x
- Argument for the Bessel function
- ::math::special::Jn n x
- Compute the nth-order Bessel function of the first kind for the
argument "x"
- integer n
- Order of the Bessel function
- float x
- Argument for the Bessel function
- ::math::special::J1/2 x
- Compute the half-order Bessel function of the first kind for
the argument "x"
- float x
- Argument for the Bessel function
- ::math::special::J-1/2 x
- Compute the minus-half-order Bessel function of the first kind
for the argument "x"
- float x
- Argument for the Bessel function
- ::math::special::I_n x
- Compute the modified Bessel function of the first kind of order
n for the argument "x"
- int x
- Positive integer order of the function
- float x
- Argument for the function
- ::math::special::elliptic_K k
- Compute the complete elliptic integral of the first kind for
the argument "k"
- float k
- Argument for the function
- ::math::special::elliptic_E k
- Compute the complete elliptic integral of the second kind for
the argument "k"
- float k
- Argument for the function
- ::math::special::exponential_Ei
x
- Compute the exponential integral of the second kind for the
argument "x"
- float x
- Argument for the function (x != 0)
- ::math::special::exponential_En
n x
- Compute the exponential integral of the first kind for the
argument "x" and order n
- int n
- Order of the integral (n >= 0)
- float x
- Argument for the function (x >= 0)
- ::math::special::exponential_li
x
- Compute the logarithmic integral for the argument "x"
- float x
- Argument for the function (x > 0)
- ::math::special::exponential_Ci
x
- Compute the cosine integral for the argument "x"
- float x
- Argument for the function (x > 0)
- ::math::special::exponential_Si
x
- Compute the sine integral for the argument "x"
- float x
- Argument for the function (x > 0)
- ::math::special::exponential_Chi x
- Compute the hyperbolic cosine integral for the argument "x"
- float x
- Argument for the function (x > 0)
- ::math::special::exponential_Shi x
- Compute the hyperbolic sine integral for the argument "x"
- float x
- Argument for the function (x > 0)
- ::math::special::fresnel_C x
- Compute the Fresnel cosine integral for real argument x
- float x
- Argument for the function
- ::math::special::fresnel_S x
- Compute the Fresnel sine integral for real argument x
- float x
- Argument for the function
- ::math::special::sinc x
- Compute the sinc function for real argument x
- float x
- Argument for the function
- ::math::special::legendre n
- Return the Legendre polynomial of degree n (see THE ORTHOGONAL POLYNOMIALS)
- int n
- Degree of the polynomial
- ::math::special::chebyshev n
- Return the Chebyshev polynomial of degree n (of the first
kind)
- int n
- Degree of the polynomial
- ::math::special::laguerre alpha n
- Return the Laguerre polynomial of degree n with parameter
alpha
- float alpha
- Parameter of the Laguerre polynomial
- int n
- Degree of the polynomial
- ::math::special::hermite n
- Return the Hermite polynomial of degree n
- int n
- Degree of the polynomial
For dealing with the classical families of orthogonal
polynomials, the package relies on the math::polynomials
package. To evaluate the polynomial at some coordinate, use the
evalPolyn command:
|
set leg2 [::math::special::legendre 2]
puts "Value at x=$x: [::math::polynomials::evalPolyn $leg2 $x]"
|
The return value from the legendre and other commands
is actually the definition of the corresponding polynomial as used
in that package.
It should be noted, that the actual implementation of J0 and J1
depends on straightforward Gaussian quadrature formulas. The
(absolute) accuracy of the results is of the order 1.0e-4 or
better. The main reason to implement them like that was that it was
fast to do (the formulas are simple) and the computations are fast
too.
The implementation of J1/2 does not suffer from this: this
function can be expressed exactly in terms of elementary
functions.
The functions J0 and J1 are the ones you will encounter most
frequently in practice.
The computation of I_n is based on Miller's algorithm for
computing the minimal function from recurrence relations.
The computation of the Gamma and Beta functions relies on the
combinatorics package, whereas that of the error functions relies
on the statistics package.
The computation of the complete elliptic integrals uses the AGM
algorithm.
Much information about these functions can be found in:
Abramowitz and Stegun: Handbook of Mathematical
Functions (Dover, ISBN 486-61272-4)
Bessel functions , error function , math , special functions
Copyright © 2004 Arjen Markus
<arjenmarkus@users.sourceforge.net>