A new approach to acceleration of convergence of a sequence of vectors

The genesis and early developments of Aitken’s process, Shanks’ transformation, the ε–algorithm, and related fixed point methods

“…However, the vectors ε (n) 2k can be expressed, as in the scalar case, by ratios of determinants but of dimension 2k + 1 instead of k + 1, as proved by Peter R. Graves-Morris and Chris Jenkins [120,121]. The proof of this result was obtained via the relation of this algorithm to Pfaffians and vector Padé approximants [119]. They can also be expressed as ratios of designants, a generalization of determinants in the non-commutative case, a work due to Ahmed Salam [249][250][251].…”

Row Convergence Theorems for Vector-Valued Padé Approximants

“…[n&kÂk] (z) in the scalar (d=1) case, Graves-Morris [8] introduced a polynomial _(z)r-Q(z). His expressions for Q(z) and _(z) involve the umbral calculus; here we need the equivalent expression based on (2.4), which follows immediately from (2.24) and it is…”

“…Because vector Pade approximants which converge to a vector-valued meromorphic function f(z) having k poles in a disk have denominators of degree 2k and not k, hybrid vector Pade approximants of type [nÂk] were introduced by Graves-Morris [7]. These approximants also satisfy a McLeod-type theorem [8], and Roberts has shown that they have a number of attractive properties [22].…”