Maxima Manual
16.1 Introduction to Orthogonal Polynomials
The specfun package, located in the share directory, contains Maxima
code for the evaluation of all orthogonal polynomials listed in
Chapter 22 of Abramowitz and Stegun. These include Chebyshev,
Laguerre, Hermite, Jacobi, Legendre, and ultraspherical (Gegenbauer)
polynomials. Additionally, specfun contains code for
spherical Bessel, spherical Hankel, and spherical harmonic functions.
The following table lists each function in specfun,
its Maxima name, restrictions on its arguments
( m and n must be integers), and a
reference to the algorithm specfun uses to evaluate it.
With few exceptions, specfun follows the conventions of
Abramowitz and Stegun. Before you use specfun, check
that specfun's conventions match your expectations.
A&S refers to Abramowitz and Stegun, Handbook of
Mathematical Functions (10th printing, December 1972),
G&R to Gradshteyn and Ryzhik,
Table of Integrals, Series, and Products
(1980 corrected and enlarged edition), and Merzbacher
to Quantum Mechanics (2ed, 1970).
| Function | Maxima Name | Restrictions | Reference(s) |
| Chebyshev T | chebyshev_t(n, x) | n > -1 | A&S 22.5.31 |
| Chebyshev U | chebyshev_u(n, x) | n > -1 | A&S 22.5.32 |
| generalized Laguerre | gen_laguerre(n,a,x) | n > -1 | A&S page 789 |
| Laguerre | laguerre(n,x) | n > -1 | A&S 22.5.67 |
| Hermite | hermite(n,x) | n > -1 | A&S 22.4.40, 22.5.41 |
| Jacobi | jacobi_p(n,a,b,x) | n > -1, a, b > -1 | A&S page 789 |
| associated Legendre P | assoc_legendre_p(n,m,x) | n > -1 | A&S 22.5.37, 8.6.6, 8.2.5 |
| associated Legendre Q | assoc_legendre_q(n,m,x) | n > -1, m > -1 | G & R 8.706 |
| Legendre P | legendre_p(n,m,x) | n > -1 | A&S 22.5.35 |
| Legendre Q | legendre_q(n,m,x) | n > -1 | A&S 8.6.19 |
| spherical Hankel 1st | spherical_hankel1(n, x) | n > -1 | A&S 10.1.36 |
| spherical Hankel 2nd | spherical_hankel2(n, x) | n > -1 | A&S 10.1.17 |
| spherical Bessel J | spherical_bessel_j(n,x) | n > -1 | A&S 10.1.8, 10.1.15 |
| spherical Bessel Y | spherical_bessel_y(n,x) | n > -1 | A&S 10.1.9, 10.1.15 |
| spherical harmonic | spherical_harmonic(n,m,x,y) | n > -1, |m| <= n | Merzbacher 9.64 |
| ultraspherical (Gegenbauer) | ultraspherical(n,a,x) | n > -1 | A&S 22.5.27 |
The specfun package is primarily intended for symbolic
computation. It is hoped that it gives accurate floating point
results as well; however, no claims are made that the algorithms
are well suited for numerical evaluation. Some effort, however,
has been made to provide good numerical performance.
When all arguments, except for the order, are floats (but not bfloats),
many functions in specfun call a float modedeclared version of the
Jacobi function. This greatly speeds floating point evaluation
of the orthogonal polynomials.
specfun handles most domain errors by returning an unevaluated
function. No attempt has been made to define simplification
rules (based on recursion relations) for unevaluated functions.
Users should be aware that it is possible for an expression
involving sums of unevaluated special functions to vanish, yet
Maxima is unable to reduce it to zero. Be careful.
To access functions in specfun, you must first load specfun.o.
Alternatively, you may append autoload statements to your
init.lsp file (located in your working directory). To autoload the
hermite function, for example, append
| | (defprop |$hermite| #"specfun.o" autoload)
(add2lnc '|$hermite| $props)
|
to your init.lsp file. An example use of specfun is
| | (c1) load("specfun.o")$
(c2) [hermite(0,x),hermite(1,x),hermite(2,x)];
(d2) [1,2*x,-2*(1-2*x^2)]
(c3) diff(hermite(n,x),x);
(d3) 2*n*hermite(n-1,x)
|
When using the compiled version of specfun, be especially
careful to use the correct number of function arguments; calling
them with too few arguments may generate a fatal error messages.
For example
| | (c1) load("specfun")$
/* chebyshev_t requires two arguments. */
(c2) chebyshev_t(8);
Error: Caught fatal error [memory may be damaged]
Fast links are on: do (si::use-fast-links nil) for debugging
Error signalled by MMAPCAR.
Broken at SIMPLIFY. Type :H for Help.
|
Maxima code translated into Lisp handles such errors more gracefully.
If specfun.LISP is installed on your machine, the same computation
results in a clear error message. For example
| | (c1) load("specfun.LISP")$
(c2) chebyshev_t(8);
Error: Expected 2 args but received 1 args
Fast links are on: do (si::use-fast-links nil) for debugging
Error signalled by MACSYMA-TOP-LEVEL.
Broken at |$CHEBYSHEV_T|. Type :H for Help.
|
Generally, compiled code runs faster than translated code; however,
translated code may be better for program development.
For some functions, when the order is symbolic but has been
declared to be an integer, specfun will return a series
representation. (The series representation is not
used by specfun for any computations.) You may use this feature to
find symbolic values for special values orthogonal polynomials.
An example:
| | (c1) load("specfun")$
(c2) legendre_p(n,1);
(d2) legendre_p(n, 1)
/* Declare n to be an integer; now legendre_p(n,1) evaluates to 1. */
(c3) declare(n,integer)$
(c4) legendre_p(n,1);
(d4) 1
(c5) ultraspherical(n,3/2,1);
(d4) (n+1)*gamma (n+3) / (2*gamma (n+2))
|
Although the preceding example doesn't show it, two terms
of the sum are added outside the summation. Removing these
two terms avoids errors associated with 0^0 terms
in a sum that should evaluate to 1, but evaluate to 0 in a Maxima
summation. Because the sum index runs from 1 to
n - 1, the lower sum
index will exceed the upper sum index when n = 0;
setting sumhack to true provides a fix. For example:
| | (c1) load("specfun.o")$
(c2) declare(n,integer)$
(c3) e : legendre_p(n,x)$
(c4) ev(e,sum,n=0);
Lower bound to SUM: 1
is greater than the upper bound: - 1
-- an error. Quitting. To debug this try DEBUGMODE(TRUE);)
(c5) ev(e,sum,n=0),sumhack : true;
(d5) 1
|
Most functions in specfun have a gradef property;
derivatives with respect to the order or other function parameters
aren't unevaluated.
The specfun package and its documentation were written by
Barton Willis of the University of Nebraska at Kearney. It is
released under the terms of the General Public License (GPL).
Send bug reports and comments on this package to
willisb@unk.edu. In your report, please include
Maxima and specfun version information. The specfun
version may be found using get:
| | (c2) get('specfun,'version);
(d2) 110
|
