Symmetric functions and root-finding algorithms

We study a fundamental family of root-finding iteration functions in the context of symmetric functions. This family, which we refer to as the Basic Family, goes back to Schröder's 1870 paper, and admits numerous different representations. In one representation, it is known as König's family. A purely algebraic derivation by Kalantari et al. leads to the discovery of many minimality and uniqueness properties of this family. Our new perspective reveals a symmetric algebraic structure of the Basic Family, which gives rise to simple combinatorial proofs of many important properties of this family and two of its variants. The first variant maintains high order of convergence for multiple roots. The second variant, called the Truncated Basic Family, is an infinite family of mth order methods for every m ≥ 3, using only the first m - 1 derivatives. Our result extends Kalantari's analysis of Halley's family, the special case of the Truncated Basic Family where m = 3. Finally, we give a recipe for constructing new high order root-finding algorithms, and use it to derive an interesting family of iteration functions that have higher orders of convergence for multiple roots than for simple roots.