David E Roberts

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.

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.

On the convergence of rows of vector Pade´ approximants

We state and prove a de Montessus like theorem for vector-valued meromorphic functions using vector Pade approximants which are defined in the context of Clifford algebras. The proof is based on that used by Saff for the scalar case. The theorem may be regarded as an answer to a general problem posed by Graves-Morris and Saff.

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.

On a representation of vector continued fractions

Abstract Vector Pade approximants to power series with vector coefficients may be calculated using the three-term recurrence relations of vector continued fractions if formulated in the framework of Clifford algebras. We show that the numerator and denominator polynomials of these fractions take particularly simple forms which require just a few degrees of freedom in their representation. The new description also allows the calculation of “hybrid” approximants.

On a vector q-d algorithm.

Using the framework provided by Clifford algebras, we consider a non‐commutative quotient‐difference algorithm for obtaining the elements of a continued fraction corresponding to a given vector‐valued power series. We demonstrate that these elements are ratios of vectors, which may be calculated with the aid of a cross rule using only vector operations. For vector‐valued meromorphic functions we derive the asymptotic behaviour of these vectors, and hence of the continued fraction elements themselves. The behaviour of these elements is similar to that in the scalar case, while the vectors are linked with the residues of the given function. In the particular case of vector power series arising from matrix iteration the new algorithm amounts to a generalisation of the power method to sub‐dominant eigenvalues, and their eigenvectors.

Iterative control in automotive testing.

Abstract Road simulator test rigs investigate both the durability and the squeak and rattle characteristics of vehicle structures. The test specimens are the whole vehicle or a sub-structure taken from the vehicle. In laboratory simulation tests, random time series of strain, displacement, or acceleration, which represent service conditions, are reproduced at specified locations within the structure. These responses are formulated from measurements made on the road or at a specialized proving ground. The premise is that reproducing these responses recreates service strain throughout the structure. The test rigs are normally driven by hydraulics and form multiple channel non-linear dynamic systems. Time series drive files that reproduce the strain are generated to excite the test rig. These are determined using an iterative procedure that overcomes the deficiencies of the hydraulics, channel interaction, and non-linearity in the system. The current paper outlines a new iterative algorithm with better convergence than the commonly used iterative algorithm which does not always converge. This new iterative algorithm is demonstrated in numerical simulations that represent typical automotive engineering systems.

Vector continued fraction algorithms

We consider the construction of rational approximations to given power series whose coefficients are vectors. The approximants are in the form of vector-valued continued fractions which may be used to obtain vector Pade approximants using recurrence relations. Algorithms for the determination of the vector elements of these fractions have been established using Clifford algebras. We devise new algorithms based on these which involve operations on vectors and scalars only — a desirable characteristic for computations involving vectors of large dimension. As a consequence, we are able to form new expressions for the numerator and denominator polynomials of these approximants as products of vectors, thus retaining their Clifford nature.

Road Simulators: The Iterative Algorithm for Drive File Creation

Road simulators reproduce measured service environments in laboratory based test rigs and have contributed significantly to improving the structural integrity and quality of modern vehicles. These rigs are driven by data that are derived from a specified response and the frequency response function of the test rig in an iterative process. This paper introduces an alternative iterative procedure that converges to a valid drive file in fewer iteration steps than the current algorithm.