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The Aberth method, or Aberth–Ehrlich method, named after Oliver Aberth and Louis W. Ehrlich, is a root-finding algorithm for simultaneous approximation of all the roots of a univariate polynomial.

The fundamental theorem of algebra states that for each polynomial with complex coefficients there are as many roots as the degree of the polynomial. Numerical algorithms that approximate all roots at once include the Weierstrass–(Durand–Kerner) method and the Aberth–Ehrlich method.

Description

Let $p(x)=p_{n}x^{n}+p_{n-1}x^{n-1}+\cdots +p_{1}x+p_{0}$ be a univariate polynomial of degree n with real or complex coefficients. Then there exist complex numbers $z_{1}^{*},\,z_{2}^{*},\dots ,z_{n}^{*}$, the roots of p(x), that give the factorisation:

$p(x)=p_{n}\cdot (x-z_{1}^{*})\cdot (x-z_{2}^{*})\cdots (x-z_{n}^{*}).$

Although those numbers are unknown, upper and lower bounds for their absolute values are computable from the coefficients of the polynomial. Now one can pick n distinct numbers in the complex plane—randomly or evenly distributed—such such that their absolute values are within the same bounds. A set of such numbers is called an initial approximation of the set of roots of p(x). This approximation can be iteratively improved using the following procedure.

Let $z_{1},\dots ,z_{n}\in \mathbb {C}$ be the current approximations of the zeros of p(x). Then offset numbers $w_{1},\dots ,w_{n}\in \mathbb {C}$ are computed as

$w_{k}=-{\frac {\frac {p(z_{k})}{p'(z_{k})}}{1-{\frac {p(z_{k})}{p'(z_{k})}}\cdot \sum _{j\neq k}{\frac {1}{z_{k}-z_{j}}}}},$

where p'(z) is the polynomial derivative of p evaluated in the point z.

The next set of approximations of roots of p(x) is then $z_{1}+w_{1},\dots ,z_{n}+w_{n}$. One can measure the quality of the current approximation by the values of the polynomial or by the size of the offsets.

Inside the formula of the Aberth method one can find elements of Newton's method and the Weierstrass–(Durand–Kerner) method. Details for an efficient implementation, esp. on the choice of good initial approximations, can be found in Bini (1996).

Derivation from Newton's method

The iteration formula is the univariate Newton iteration for the function

$F(x)={\frac {p(x)}{\prod _{j=0;\,j\neq k}^{n}(x-z_{j})}}$

If the values $z_{1},\dots ,z_{n}$ are already close to the roots of p(x), then the rational function F(x) is almost linear with poles at $z_{1},\dots ,z_{k-1},z_{k+1},\dots ,z_{n}$ that direct the Newton iteration away from the roots of p(x) that are close to them. That is, the corresponding basins of attraction get rather small.

The Newton step in the univariate case is the reciprocal value to the logarithmic derivative

${\begin{aligned}{\frac {F'(x)}{F(x)}}&={\frac {d}{dx}}\ln |F(x)|\\&={\frac {d}{dx}}{\big (}\ln |p(x)|-\sum _{j=0;\,j\neq k}^{n}\ln |x-z_{j}|{\big )}\\&={\frac {p'(x)}{p(x)}}-\sum _{j=0;\,j\neq k}^{n}{\frac {1}{x-z_{j}}}\end{aligned}}$

Thus,

$z_{k}'=z_{k}-{\frac {F(z_{k})}{F'(z_{k})}}=z_{k}-{\frac {1}{{\frac {p'(z_{k})}{p(z_{k})}}-\sum _{j=0;\,j\neq k}^{n}{\frac {1}{z_{k}-z_{j}}}}}$

results in the Aberth–Ehrlich iteration.

The iteration may be executed in a simultaneous Jacobi-like iteration where first all new approximations are computed from the old approximations or in a sequential Gauss–Seidel-like iteration that uses each new approximation from the time it is computed.

See also

• MPSolve A package for numerical computation of polynomial roots. Free usage for scientific purpose.