Pade-type Approximants Research Articles

A new application of the extended Euclidean algorithm for matrix padé approximants

Work on Padé or Padé-type approximants ultimately involves the explicitdetermination of the polynomials forming the numerator and denominator of rational functions. These exist in the literature useful algorithms for constructing the polynomials when the starting coefficients of the given power series are of the ordinary kind. However, when one has the matrix coefficients forthe series, it becomes necessary to extend the procedure taking into account the special nature of the various operations. In this paper we present a new application of the extended Euclidean algorithm in order to obtain the sets of complete matrix Padé approximants. Also, an efficient Pascal procedure for implementation of the algorithm is given. In fact, the resulting algorithm generates the elements in the anti-diagonal of the Padé table. The concepts and constructive steps involved in the procedure are explained in detail and illustrated by a few suitable examples to show (i) how the algorithm works in general and (ii) the usefulness of the entire approach.

Convergence of a class of Borel-Padé-type approximants

In this paper we propose the use of a recently introduced set of rational functions, the so-called Pade-type approximants, to perform the analytic continuation of Borel transforms into a cut complex plane. It is shown that, by choosing a suitable set of orthogonal generating polynomials, the convergence of the Borel-Pade-type approximation method can be rigorously proven.

Quadratic Hermite-Padé approximation to the exponential function

Approximation to exp of the form wherepm,qm, andrm are polynomials of degree at mostm andpm has lead coefficient 1 is considered. Exact asymptotics and explicit formulas are obtained for the sequences {ℰm}, {pm}, {qm}, and {rm}. It is observed that the above sequences all satisfy the simple four-term recursion: $$\begin{array}{*{20}c} {T_{m + 3} = \frac{1}{{3m + 4}}[( - 6m - 14)z^3 T_m } \\ { + (9m + 15)(z^2 + (3m + 4)(3m + 7))T_{m + 1} + 3zT_{m + 2} ].} \\ \end{array} $$ It is also observed that these generalized Pade-type approximations can be used to asymptotically minimize expressions of the above form on the unit disk.

On interpolatory multivariate Padé-type approximants

It is proved that higher order interpolatory Pade-type approximants in two variables do not exist.

On the simplification of large linear systems using Padé -type approximations and Cauer continued fractions

Two algorithms ‘ PS.Routh’ and ‘ CF.Routh’ are derived to prove the equivalence of two approaches to model reduction, namely the matching of Markov parameters of the system and the Cauer first-form expansion method ( Shieh and Goldman 1974) The equivalence of the reduced-order models obtained by matching time-moments ( Lai and Mitra 1974) and by the Cauer second-form ( Shieh and Goldman 1974) is also proved using the same algorithms by replacing the subscripts of the coefficients in the expansions. Then the property of a ‘ singularity ’ incidental to the model reduction, is investigated where the ‘ singularity’ means the situation in which the system cannot be simplified by the Routh algorithm ( Shieh and Goldman 1974) or by matching time-moments ( or Markov parameters) Using the algorithm PS.Routh, it is proved that the condition of singularity for the continued fractions is equivalent to that for the Pade-type approximations. Under this condition, the order of the reduced model is limited. It is further...

Padé-type approximants of exp (−z) whose denominators are (1+z/n) n

In this paper, we show that the sequences of Pade-type approximants (k?1/k) and (k/k) converge to exp (?z), uniformly and geometrically on every compact subset of the plane. A numerical study has been done, which discriminates these sequences from the point of view ofA-acceptability.

Pade-Type Approximation and General Orthogonal Polynomials.

Pade-Type Approximation and General Orthogonal Polynomials.

The data on the electromagnetic pion form factor and P-wave isovector ππ phase-shift are definitely consistent

The data for the absolute value of the pion form factor (free of the omega-meson contribution) are combined with the P-wave isovector pi pi phase-shift. The resultant real and imaginary parts of the pion form factor are described by means of a Pade-type approximation. All the data which involve pion form factor experimental points from the range of momenta -0.8432 GeV2<t<1 GeV2, the pion charge radius and the P-wave isovector pi pi phase-shift in the elastic region (including also the generally accepted value of the scattering length a11) are mutually consistent. The data themselves reveal through a Pade-type approximation that the consistency can be achieved only if the pion form factor left-hand cut from the second Riemann sheet is taken into account. In almost all the Pade-type approximations considered one stable pion form factor zero is found in the space-like region; this might indicate the existence of a diffraction minimum in the differential cross section for elastic e- pi scattering as a consequence of the constituent structure of the pion, as in the case of electron elastic scattering on nuclei.

Time moments and Markov parameters of composite systems

A theorem is given to show that a reduced-order composite system, formed from the interconnection of a number of reduced-order models that are Pade-type approximants of some subsystems, matches the initial time moments and Markov parameters of the composite system formed from the interconnection of these subsystems. Thus the reduced-order composite system has the same tracking properties as the original composite system.

Canterbury approximants in potential scattering

The two variable rational approximant schemes devised in Canterbury are shown, by examples, to be quite satisfactory for obtaining amplitudes from a two variable Neumann-Born series. The (N+1/N) and (N/N) are markedly the best approximants in this context, and attention is drawn to the numerical accuracy in the Neumann-Born series coefficients which is essential in conjunction with Pade-type approximants.