Peter Graves-Morris

The epsilon algorithm and related topics

The epsilon algorithm is recommended as the best all-purpose acceleration method for slowly converging sequences. It exploits the numerical precision of the data to extrapolate the sequence to its limit. We explain its connections with Pade approximation and continued fractions which underpin its theoretical base. Then we review the most recent extensions of these principles to treat application of the epsilon algorithm to vector-valued sequences, and some related topics. In this paper, we consider the class of methods based on using generalised inverses of vectors, and the formulation specifically includes the complex case wherever possible.

Row convergence theorems for generalised inverse vector-valued Pade´ approximants

Abstract The approximants mentioned in the title are related to vector-valued continued fractions and the vector ϵ-algorithm devised by Wynn in 1963. Here we establish a unitary invariance property of these approximants and describe how the classical (1-dimensional) Pade approximants can be obtained as a special case. The main results of the paper consist of De Montessus—De Ballore type convergence theorems for row sequences (having fixed denominator degree) of vector-valued approximants to meromorphic vector functions.

From matrix to vector Pade´ approximants

Abstract Reliable modified Euclidean and Kronecker algorithms for real and complex vector-valued power series are constructed by exploiting an isomorphism between vectors and matrices. This is done using McLeods (1971) Clifford algebra representation, which preserves the Moore—Penrose inverse. This approach required new proofs of Wynns and Cordelliers identities which do not involve the use of determinants.

Solution of integral equations using generalised inverse, function-valued Pade´ approximants, I

Abstract Neumann series solutions for the linear integral equation φ(x) = 1 + λ∫ 1 0 {1 +⨍x − y⨍}φ(y) d y converge for small values of λ. We use generalised inverse, function-valued Pade approximants to accelerate the convergence of the Neumann series and to estimate the singular value of λ. These estimates are found to compare very favourably with those from the Fredholm determinant method and the Pade approximant method.

Problems and progress in vector Pade´ approximation

Abstract Some open problems in vector Pade approximation are stated, and some recent de Montessus-type theorems governing convergence of rows of the vector Pade table are contrasted. Weshow how these results indicate when it is more appropriate to use generalised inverse vector Pade approximants or their hybridised form. We show how a straightforward analogue of the Berlekamp-Massey algorithm may be used to calculate generalised inverse vector Pade approximants. This algorithm is applied to the derivation of a new low-order hybrid vector approximant. Related results include the case of a row convergence theorem for a complex-valued power series using a Clifford inverse instead of the Moore-Penrose inverse.

Cayley's theorem and its application in the theory of vector Pade´ approximants

Abstract Le A be a matrix of even dimension which is anti-symmetric after deletion of its r th row and column and let R , C be the anti-symmetric matrices formed by modifying the r th row and column of A , respectively. In this case, Cayleys (1857) theorem states that det A = Pf R · Pf C , where Pf R denotes the Pfaffian of R . A consequence of this theorem is an explicit factorisation of the standard determinantal representation of the denominator polynomial of a vector Pade approximant. We give a succinct, modern proof of Cayleys theorem. Then we prove a novel vector inequality arising from investigation of one such Pfaffian, and conjecture that all such Pfaffians are nonnegative.

Definition and uniqueness of integral approximants

Abstract Definitions are given for the integral approximant polynomials which insure their existence and uniqueness. A specification of minimality is required in these definitions. Existence of infinite subsequences of integral polynomials without common factors of z is proven. An equivalence theorem between the property of agreement of all high-order integral polynomials and the property that the function being approximated belongs to a particular function class is proved. The accuracy-through-order property is found to hold for all the cases we have investigated for the integral approximant. An example is given which proves that the series coefficients which uniquely determine the integral polynomials may not uniquely determine the integral approximant.

Existence of certain sequences of Hermite-Pade´ approximants

Abstract We prove the existence of infinite sequences of unique, minimal Hermite–Pade polynomials of horizontal, near-partially-diagonal and near-diagonal character.

Determinants and the structure of the integral approximant table

Abstract The structure of the table of integral approximants based on our minimality definition is studied. The key elements are simplices containing only a single linearly independent solution for the integral polynomials. We prove some new identities among determinants of the coefficients of the determining equations for integral polynomials of different types. Relations between the simplices and these determinants are also derived. We give determinantal conditions which are both necessary and sufficient for a full simplex to contain just one approximant. A nullity lemma is proved limiting the change between contiguous types in the number of linearly independent solutions for integral polynomials. Virtually all of our results are equally valid for the more general case of the Hermite-Pade approximants of the Latin type.

Results on sequences of Hermite-Pade´, integral approximants

Abstract The convergence of “horizontal” sequences of Hermite-Pade, integral approximants is studied by both numerical and theoretical methods. For functions with their nearest singularity of the algebraic type at z = 1 we complement previously proven convergence inside the unit circle by proving divergence outside the unit circle of sequences of the [ L / M ;1] type where M remains fixed and L → ∞.