|
Laguerre polynomials are defined as solutions of Laguerre's differential equation:
xy'' + (1-x)y' + ny = 0Solutions corresponding to the non-negative integer n can be expressed using Rodrigues' formula
or can be constructed using the three term recurrence relation:
L0 (x) = 1The recurrence relation given above is the most efficient way to calculate the Laguerre polynomial. The LaguerreCalculate subroutine uses this relation to calculate Ln (x) for any given x.
The LaguerreSum subroutine calculates the sum of Laguerre polynomials c0 L0 (x) + c1 L1 (x) + ... + cn Ln (x) using Clenshaw's recurrence formula.
The LaguerreCoefficients subroutine can represent Ln (x) as a sum of powers of x: c0 + c1 x + ... + cn x n.
This article is intended for personal use only.
C# source.
C++ source.
C++ source. MPFR/GMP is used.
GMP source is available from gmplib.org. MPFR source is available from www.mpfr.org.
FreePascal version.
Delphi version.
VB.NET version.
VBA version.
Python version (CPython and IronPython are supported).
|
ALGLIB® - numerical analysis library, 1999-2012. |