NAME
Math::Polynomial::Chebyshev2 - Chebyshev polynomials of the second kind
SYNOPSIS
use Math::Polynomial::Chebyshev2;
# create a Chebyshev polynomial of the second kind of order 7
my $p = Math::Polynomial::Chebyshev2 -> chebyshev2(7);
# get the location of all roots
my $x = $p -> roots();
# use higher accuracy
use Math::BigFloat;
Math::BigFloat -> accuracy(60);
my $n = Math::BigFloat -> new(7);
$x = Math::Polynomial::Chebyshev2 -> chebyshev2($n);
DESCRIPTION
This package extends Math::Polynomial, so each instance polynomial created by this modules is a subclass of Math::Polynomial.
The Chebyshev polynomials of the second kind are orthogonal with respect to the weight function sqrt(1-x^2).
The first Chebyshev polynomials of the second kind are
U₀(x) = 1
U₁(x) = 2 x
U₂(x) = 4 x^2 - 1
U₃(x) = 8 x^3 - 4 x
U₄(x) = 16 x^4 - 12 x^2 + 1
U₅(x) = 32 x^5 - 32 x^3 + 6 x
U₆(x) = 64 x^6 - 80 x^4 + 24 x^2 - 1
U₇(x) = 128 x^7 - 192 x^5 + 80 x^3 - 8 x
U₈(x) = 256 x^8 - 448 x^6 + 240 x^4 - 40 x^2 + 1
U₉(x) = 512 x^9 - 1024 x^7 + 672 x^5 - 160 x^3 + 10 x
CLASS METHODS
Constructors
- chebyshev2($n)
-
Math::Polynomial::Chebyshev2->chebyshev2($n)
creates a new polynomial of order$n
, where$n
is a non-negative integer.# create a Chebyshev polynomial of the second kind of order 7 $p = Math::Polynomial::Chebyshev2 -> chebyshev2(7); # do the same, but with Math::BigFloat coefficients use Math::BigFloat; $n = Math::BigFloat -> new(7); $p = Math::Polynomial::Chebyshev2 -> chebyshev2($n); # do the same, but with Math::Complex coefficients use Math::Complex; $n = Math::Complex -> new(7); $p = Math::Polynomial::Chebyshev2 -> chebyshev2($n);
Property Accessors
- roots()
-
$p->roots
returns the location of all root of$p
. All roots are located in the open interval (-1,1).# get the roots of a polynomial @x = $p -> roots();
BUGS
Please report any bugs through the web interface at https://rt.cpan.org/Ticket/Create.html?Queue=Math-Polynomial-Chebyshev2 (requires login). We will be notified, and then you'll automatically be notified of progress on your bug as I make changes.
SUPPORT
You can find documentation for this module with the perldoc command.
perldoc Math::Polynomial::Chebyshev2
You can also look for information at:
GitHub Source Repository
RT: CPAN's request tracker
https://rt.cpan.org/Public/Dist/Display.html?Name=Math-Polynomial-Chebyshev2
CPAN Ratings
https://cpanratings.perl.org/dist/Math-Polynomial-Chebyshev2
MetaCPAN
CPAN Testers Matrix
http://matrix.cpantesters.org/?dist=Math-Polynomial-Chebyshev2
SEE ALSO
The Perl module Math::Polynomial.
The Wikipedia page https://en.wikipedia.org/wiki/Chebyshev_polynomials.
LICENSE AND COPYRIGHT
Copyright (c) 2020 Peter John Acklam.
This program is free software; you may redistribute it and/or modify it under the same terms as Perl itself.
AUTHOR
Peter John Acklam <pjacklam (at) gmail.com>.