Roots of equations by functional iteration

On Schröder’s families of root-finding methods

“…Actually, D m (x) in (4) is the functional determinant given by (1), that is, D m (x) ≡ ∆ m (x), D 0 (x) = 1. Then the Basic Family (in the terminology of the authors of [3]) is defined by…”

“…In Concluding remark of the paper [3] In 1946 Hamilton [1] introduced functional iterations using functional determinants. Let α be the zero of an analytic function f and let x be its approximation.…”

“…where ∆ m (x) is just the determinant given by (1). ∆ m,r (x) is the determinant which is obtained from ∆ m by replacing the rth column by the column (f (x) 0 0 · · · 0) T .…”

“…Our attention is restricted to the rediscovered families of rational iteration methods for solving nonlinear equations. It is initiated by the results that appeared after the World War II (e.g., [1,2]) and recent papers (e.g., [3][4][5][6]). …”

Comments on a root finding method using Padé approximation

“…The family of iteration formulae derived in [8] are well known, and can be found in [2,4]. Additionally, the fifth order iteration formula in [8, p. 294] is in Kiss [6] also.…”