Basic arithmetic, integration, differentiation, evaluation, and root finding over dense univariate polynomials.

Construct a polynomial from its coefficients, lowest order first.

```
julia> Poly([1,0,3,4])
Poly(1 + 3x^2 + 4x^3)
```

An optional variable parameter can be added.

```
julia> Poly([1,2,3], :s)
Poly(1 + 2s + 3s^2)
```

Construct a polynomial from its roots. This is in contrast to the
`Poly`

constructor, which constructs a polynomial from its
coefficients.

```
// Represents (x-1)*(x-2)*(x-3)
julia> poly([1,2,3])
Poly(-6 + 11x - 6x^2 + x^3)
```

The usual arithmetic operators are overloaded to work on polynomials, and combinations of polynomials and scalars.

```
julia> p = Poly([1,2])
Poly(1 + 2x)
julia> q = Poly([1, 0, -1])
Poly(1 - x^2)
julia> 2p
Poly(2 + 4x)
julia> 2+p
Poly(3 + 2x)
julia> p - q
Poly(2x + x^2)
julia> p * q
Poly(1 + 2x - x^2 - 2x^3)
julia> q / 2
Poly(0.5 - 0.5x^2)
julia> q ÷ p # `div`, also `rem` and `divrem`
Poly(0.25 - 0.5x)
```

Note that operations involving polynomials with different variables will error.

```
julia> p = Poly([1, 2, 3], :x)
julia> q = Poly([1, 2, 3], :s)
julia> p + q
ERROR: Polynomials must have same variable.
```

To get the degree of the polynomial use the `degree`

method

```
julia> degree(p)
2
julia> degree(p^2)
4
julia> degree(p-p)
-1
```

Evaluate the polynomial `p`

at `x`

.

```
julia> p = Poly([1, 0, -1])
julia> polyval(p, 0.1)
0.99
```

A call method is also available:

```
julia> p(0.1)
0.99
```

Integrate the polynomial `p`

term by term, optionally adding constant
term `k`

. The order of the resulting polynomial is one higher than the
order of `p`

.

```
julia> polyint(Poly([1, 0, -1]))
Poly(x - 0.3333333333333333x^3)
julia> polyint(Poly([1, 0, -1]), 2)
Poly(2.0 + x - 0.3333333333333333x^3)
```

Differentiate the polynomial `p`

term by term. The order of the
resulting polynomial is one lower than the order of `p`

.

```
julia> polyder(Poly([1, 3, -1]))
Poly(3 - 2x)
```

Return the roots (zeros) of `p`

, with multiplicity. The number of
roots returned is equal to the order of `p`

. By design, this is not type-stable,
the returned roots may be real or complex.

```
julia> roots(Poly([1, 0, -1]))
2-element Array{Float64,1}:
-1.0
1.0
julia> roots(Poly([1, 0, 1]))
2-element Array{Complex{Float64},1}:
0.0+1.0im
0.0-1.0im
julia> roots(Poly([0, 0, 1]))
2-element Array{Float64,1}:
0.0
0.0
```

`polyfit`

: fits a polynomial (of order`n`

) to`x`

and`y`

using a least-squares approximation.`julia> xs = 1:4; ys = exp.(xs); polyfit(xs, ys) Poly(-7.717211620141281 + 17.9146616149694x - 9.77757245502143x^2 + 2.298404288652356x^3)`

Visual example:

Polynomial objects also have other methods:

0-based indexing is used to extract the coefficients of $a_0 + a_1 x + a_2 x^2 + ...$, coefficients may be changed using indexing notation.

`coeffs`

: returns the entire coefficient vector`degree`

: returns the polynomial degree,`length`

is 1 plus the degree`variable`

: returns the polynomial symbol as a degree 1 polynomial`norm`

: find the`p`

-norm of a polynomial`conj`

: finds the conjugate of a polynomial over a complex fiel`truncate`

: set to 0 all small terms in a polynomial;`chop`

chops off any small leading values that may arise due to floating point operations.`gcd`

: greatest common divisor of two polynomials.`Pade`

: Return the Pade approximant of order`m/n`

for a polynomial as a`Pade`

object.

MultiPoly.jl for sparse multivariate polynomials

MultivariatePolynomials.jl for multivariate polynomials and moments of commutative or non-commutative variables

Nemo.jl for generic polynomial rings, matrix spaces, fraction fields, residue rings, power series

PolynomialRoots.jl for a fast complex polynomial root finder

02/24/2014

5 days ago

249 commits