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Math-Polynomial-Chebyshev-0.02
River stage zero No dependents
/ Math::Polynomial::Chebyshev

NAME

Math::Polynomial::Chebyshev - Chebyshev polynomials of the first kind

SYNOPSIS

use Math::Polynomial::Chebyshev;

# create a Chebyshev polynomial of the first kind of order 7
my $p = Math::Polynomial::Chebyshev -> chebyshev(7);

# get the location of all extremas
my @xe = $p -> extremas();

# get the location of all roots
my @xn = $p -> roots();

# use higher accuracy
use Math::BigFloat;
Math::BigFloat -> accuracy(60);
my $n_mbf = Math::BigFloat -> new(7);
my $p_mbf = Math::Polynomial::Chebyshev -> chebyshev($n_mbf);

DESCRIPTION

This package extends Math::Polynomial, so each instance polynomial created by this module is a subclass of Math::Polynomial.

The Chebyshev polynomials of the first kind are orthogonal with respect to the weight function 1/sqrt(1-x^2).

The first Chebyshev polynomials of the first kind are

T₀(x) = 1
T₁(x) = x
T₂(x) = 2 x^2 - 1
T₃(x) = 4 x^3 - 3 x
T₄(x) = 8 x^4 - 8 x^2 + 1
T₅(x) = 16 x^5 - 20 x^3 + 5 x
T₆(x) = 32 x^6 - 48 x^4 + 18 x^2 - 1
T₇(x) = 64 x^7 - 112 x^5 + 56 x^3 - 7 x
T₈(x) = 128 x^8 - 256 x^6 + 160 x^4 - 32 x^2 + 1
T₉(x) = 256 x^9 - 576 x^7 + 432 x^5 - 120 x^3 + 9 x

CLASS METHODS

Constructors

chebyshev($n)

Math::Polynomial::Chebyshev->chebyshev($n) creates a new polynomial of order $n, where $n is a non-negative integer.

# create a Chebyshev polynomial of the first kind of order 7
$p = Math::Polynomial::Chebyshev -> chebyshev(7);

# do the same, but with Math::BigFloat coefficients
use Math::BigFloat;
$n = Math::BigFloat -> new(7);
$p = Math::Polynomial::Chebyshev -> chebyshev($n);

# do the same, but with Math::Complex coefficients
use Math::Complex;
$n = Math::Complex -> new(7);
$p = Math::Polynomial::Chebyshev -> chebyshev($n);
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();
extremas()

$p->extremas() returns the location of all extremas of $p located in the closed interval [-1,1]. There are no extremas outside of this interval. Only the extremas in the closed interval (-1,1) are local extremas. All extremas have values +/-1.

# get the extremas of a polynomial
@x = $p -> extremas();

BUGS

Please report any bugs through the web interface at https://rt.cpan.org/Ticket/Create.html?Queue=Math-Polynomial-Chebyshev (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::Chebyshev

You can also look for information at:

SEE ALSO

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>.