Polynomial manipulations in Julia

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

Polynomials Polynomials Polynomials

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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)

+, -, *, /, div, ==

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)

julia> degree(p^2)

julia> degree(p-p)

polyval(p::Poly, x::Number)

Evaluate the polynomial p at x.

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

A call method is also available:

julia> p(0.1)

polyint(p::Poly, k::Number=0)

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}:

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

julia> roots(Poly([0, 0, 1]))
2-element Array{Float64,1}:


  • polyfit: fits a polynomial of minimal degree fitting the points specified by x and y using the least-squares fit. julia> xs = 1:4; ys = exp(xs); polyfit(xs, ys) Poly(-7.717211620141281 + 17.9146616149694x - 9.77757245502143x^2 + 2.298404288652356x^3)

Other methods

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.

See also

  • 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