Methods to find zeros (roots) of polynomials over given domains



Methods to find zeros (roots) of polynomials over given domains

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This package provides the method poly_roots to find roots of univariate polynomial functions over the complex numbers, the real numbers, the rationals, the integers, or Z_p. (A "root" is the name for a "zero" of a polynomial function.) The package takes advantage of other root-finding packages for polynomials within Julia (e.g., PolynomialRoots for numeric solutions over the complex numbers and PolynomialFactors for exact solutions over the rationals and integers).

The basic interface is

poly_roots(f, domain)

The polynomial, f, is specified through a function, a vector of coefficients ([p0, p1, ..., pn]), or as a Poly{T} object, from the the Polynomials.jl package. The domain is specified by Over.C (the default), Over.R, Over.Q, Over.Z, or over.Zp{p}, with variants for specifying an underlying type.


julia> poly_roots(x -> x^4 - 1, Over.C)  # uses `roots` from `PolynomialRoots.jl`
4-element Array{Complex{Float64},1}:

julia> poly_roots(x -> x^4 - 1, Over.R)
2-element Array{Float64,1}:

julia> poly_roots(x -> x^4 - 1, Over.Q) # uses `PolynomialFactors.jl`
2-element Array{Rational{Int64},1}:

julia> poly_roots(x -> x^4 - 1, Over.Z) # uses `PolynomialFactors.jl`
2-element Array{Int64,1}:

julia> poly_roots(x -> x^4 - 1, Over.Zp{5}) # uses `PolynomialFactors.jl`
4-element Array{Int64,1}:

Domains can also have their underlying types specified. For example, to solve over the BigFloat type, we have:

poly_roots(x -> x^4 - 1, Over.CC{BigFloat})  # `CC{BigFloat}` not just `C`
using DoubleFloats  # significantly faster than BigFloat
poly_roots(x -> x^4 - 1, Over.CC{Double64})
4-element Array{Complex{DoubleFloat{Float64}},1}:
                    1.0 + 0.0im
                   -1.0 + 0.0im
 -7.851872429582108e-35 - 1.0im
 -7.851872429582108e-35 + 1.0im

There are other methods for Over.C. This will use the AMVW method:

julia> poly_roots(x -> x^4 - 1, Over.C, method=:amvw) # might differ slightly
4-element Array{Complex{Float64},1}:
 -2.1603591655723396e-16 - 0.9999999999999999im
 -2.1603591655723396e-16 + 0.9999999999999999im
                     1.0 + 0.0im
                    -1.0 + 0.0im

This method is useful for high-degree polynomials (cf. FastPolynomialRoots):

n = 5000
rs = poly_roots(randn(n+1))
sum(isreal, rs) # 0
sum(!isnan, rs) # 1 (should be n)
rs = poly_roots(randn(n+1), method=:amvw)
sum(isreal, rs)  # 6 ~  6.04... = 2/π*log(n) + 0.6257358072 + 2/(n*π)
sum(!isnan, rs) # 5000 = n


This package uses:

  • The PolynomialRoots package to find roots over the complex numbers. The Roots package can also be used. As well, an implementation of the AMVW algorithm can be used. The default seems to be faster and as accurate as the others, but for very high degree polynomials, the :amvw method should be used, as it will be faster and more reliable.

  • The PolynomialFactors package to return roots over the rationals, integers, and integers modulo a prime.

  • As well, it provides an algorithm to find the real roots of polynomials.

The main motivation for this package was to move the polynomial specific code out of the Roots package. This makes the Roots package have fewer dependencies and a more focused task. In addition, the polynomial specific code could use some better implementations of the underlying algorithms.

In the process of doing this, making a common interface to the other root-finding packages seemed to make sense.

Other possibly useful methods

The package also provides

  • PolynomialZeros.AGCD.agcd for computing an approximate GCD of polynomials p and q over Poly{Float64}. (This is used to reduce a polynomial over the reals to a square-free polynomial. Square-free polynomials are needed for the algorithm used. This algorithm can become unreliable for degree 15 or more polynomials.)

  • PolynomialZeros.MultRoot.multroot for finding roots of p in Poly{Float64} over Complex{Float64} which has some advantage if p has high multiplicities. The roots function from the Polynomials package will find all the roots of a polynomial. Its performance degrades when the polynomial has high multiplicities. The multroot function is provided to handle this case a bit better. The function follows algorithms due to Zeng, "Computing multiple roots of inexact polynomials", Math. Comp. 74 (2005), 869-903.

    x = variable(Float64)
    p = (x-1)^4 * (x-2)^3 * (x-3)^2 * (x-4)
    q = polyder(p)
    gcd(p,q) # should be (x-1)^3 * (x-2)^2 * (x-3), but is a constant
    u,v,w,resid = PolynomialZeros.AGCD.agcd(p,q)
    LinearAlgebra.norm(u - (x-1)^3*(x-2)^2*(x-3), Inf)  ~ 3.8e-6
poly_roots(p) # 2 real, 8 complex
PolynomialZeros.MultRoot.multroot(p) # ([4.0,3.0,2.0,1.0], [1, 2, 3, 4])

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