Pade table

Pade table

[′päd·ə ‚tā·bəl]
(mathematics)
A table associated to a power series having in its p th row and q th column the ratio of a polynomial of degree q by one of degree p so that this fraction expanded into a power series agrees with the original up to the p + q term.
References in periodicals archive ?
The so-called valleys in the c-table are the lines of minimal absolute values of entries, which indicate the lines of best Pade approximants (BPA) in the Pade table [9].
The Pade table (p-table) of the formal power series (1) is a doubly infinite array of irreducible rational functions {m/n} called reduced Padeforms, determined in such a manner that the Maclaurin expansion of {m/n} agrees with C(z) as far as possible.
Interest in the c-table is due to its direct relation to the Pade table [8, 11] and to the particular ease of its computation.
They correspond to the positions of the best Pade approximants on the corresponding antidiagonals in the Pade table. Joining these minima we obtain the valley structure in the c-table mentioned in Section 6 (see Table 5).
Gragg, "The Pade table and its relation to certain algorithms of numerical analysis," SIAM Review, vol.
Simpler than the Nuttall--Pommerenke theorem is an earlier theorem of de Montessus de Ballore, concerning rows of the Pade table rather than diagonals, which asserts that as m [right arrow] [infinity] with fixed n, the poles of approximants [r.sub.mn] must approach those of a meromorphic function like tan(4z) [1, 17].
See in particular the so-called Gragg example on page 13 and its Pade table on page 23.
The investigations in [29] - [35] deal with Padd approximants in the whole Pade table. Thus, also non-diagonal ray sequences have been considered, and among many other interesting things, it is quite instructive to see how Szego's earlier results about Taylor polynomials reappear as a limiting case in [35].
In [35] not only diagonal sequences, but the whole range of non-diagonal ray sequences of Pade polynomials has been studied, and among other very interesting results it has been shown how the asymptotic cluster sets of the zeros of non-diagonal Pade polynomials continuously change with the angle of the ray sequences in the Pade table. As a consequence one can see how the typical situation of diagonal Pade approximants transforms step by step into that of Taylor polynomials.
JAGER, A multidimensional generalisation of the Pade table, proc.
VARGA, The behavior of the Pade table for the exponential, in Approximation Theory II, Proc.
1992] for computing all the nonsingular Pade systems along a particular path of the corresponding Pade tables. The success of the algorithm depends on the ability to recognize nonsingular systems and is provided by the nonsingularity condition above.