Math 2.0

Numerical analysis Nonlinear equations Theorem

Aberth method

Builds on Newton's methodRoot of a polynomialRate of convergenceComplex numbers

Statement

Let pp be a polynomial of degree n≥2n \ge 2 with complex coefficients, whose roots r1,…,rnr_1, \dots, r_n are pairwise distinct. The Aberth method, also called the Aberth–Ehrlich method, updates nn approximations z1,…,znz_1, \dots, z_n simultaneously by zi↦zi−wiz_i \mapsto z_i - w_i, where wi=p(zi)p′(zi)1−p(zi)p′(zi)∑j≠i1zi−zj,i=1,…,n.w_i = \frac{\dfrac{p(z_i)}{p'(z_i)}}{1 - \dfrac{p(z_i)}{p'(z_i)} \displaystyle\sum_{j \ne i} \frac{1}{z_i - z_j}}, \qquad i = 1, \dots, n. Then there is δ>0\delta > 0 such that, if each starting value satisfies ∣zi−ri∣<δ\abs{z_i - r_i} < \delta, the iterates are defined and converge to r1,…,rnr_1, \dots, r_n, with cubic convergence.

Remarks

The step is Newton's method applied to the rational function Fi(x)=p(x)∏j≠i(x−zj),F_i(x) = \frac{p(x)}{\prod_{j \ne i} (x - z_j)}, which has the same root rir_i as pp but whose poles push ziz_i away from the other approximations. Indeed Fi′/Fi=p′/p−∑j≠i1/(x−zj)F_i'/F_i = p'/p - \sum_{j \ne i} 1/(x - z_j), and x−Fi(x)/Fi′(x)x - F_i(x)/F_i'(x) at x=zix = z_i is exactly zi−wiz_i - w_i. This repulsion tends to keep two approximations from converging to the same root. At multiple roots the convergence is only linear.

Going deeper

For a student

It improves the Durand–Kerner method, which converges only quadratically near simple roots. Each step costs O(n2)O(n^2) operations, and the steps can be done in parallel or in Gauss–Seidel fashion, using new values as soon as they are known. It is the core of the MPSolve software for computing polynomial roots to high precision.

Reading path

To understand this page from the ground up, read these 9 pages in order. The foundations come first.

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Complex numbers, object, no evidence levelComplex numbersobjectLimit point of a set, definition, no evidence levelLimit point of a setdefinitionRate of convergence, concept, no evidence levelRate of convergenceconceptRoot of a polynomial, concept, no evidence levelRoot of a polynomialconceptLimit of a function, definition, no evidence levelLimit of a functiondefinitionDerivative, definition, no evidence levelDerivativedefinitionTangent line, concept, no evidence levelTangent lineconceptNewton's method, concept, no evidence levelNewton's methodconceptAberth method, statement, no evidence levelAberth methodstatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Complex numbers object
  2. Limit point of a set definition
  3. Rate of convergence concept
  4. Root of a polynomial concept
  5. Limit of a function definition
  6. Derivative definition
  7. Tangent line concept
  8. Newton's method concept
  9. Aberth method statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Aberth_method
About this page
TagBWQ8PVA9Q3
URIm2://BWQ8PVA9Q3
Kindstatement, theorem
WritingW0 stub
MSC65H04
StatusThis copy of the site has no build data. Run m2 status BWQ8PVA9Q3.
AboutRoot of a polynomial
UsesNewton's method, Root of a polynomial, Rate of convergence, Complex numbers
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