# 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 \$x = \$p -> extremas();

# get the location of all roots
\$x = \$p -> roots();

# use higher accuracy
use Math::BigFloat;
Math::BigFloat -> accuracy(60);
my \$n = Math::BigFloat -> new(7);
\$x = Math::Polynomial::Chebyshev -> chebyshev(\$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 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` return 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: