Recursive Algorithms for Nonnormal Padé Tables

Journal of Computational and Applied Mathematics, 1976

In this paper two methods to compute Pad6 approximants are given. These methods are based on the interpretation of the e-algorithm of Wynn as the solution of a system of linear equations with an Hankel matrix. Both methods recursively computed a sub-diagonal of the Pad6 table. The first one gives the values of the approximants (the e-array) while the second provides the coefficient of numerators and denominators. The connection of this method with some polynomials and with continued fractions is studied. This method provides a way to compute recursively a diagonal of the q-d scheme. A variant of the second method can be used to compute the e-array and to settle the topological e-algorithm. The computational aspects of these methods are discussed. Some numerical examples are given. 1. THE e-ALGORITHM AND ITS ALGEBRAIC INTERPRETATION The e-algorithm is a recursive device found by Wynn [46] to compute the Shanks' transformation of sequences [38]. Let (s n } be a sequence of numbers; it can be transformed in a set of sequences by the relationship : H(n) k+l (Sn) n,k = O, 1 .... ek(Sn) = Hb(n) (a2Sn) where H(k n) (Un) is the Hankel determinant defined bY H(on)(Un): =1 n=0,1 ....

Recently McCabe and Murphy have considered the two-point Pad6 approximants to a function for which (formal) power series expansions at the origin and at infinity are given. In this paper these approximations are slightly modified and determinant representations for them are given. The existence of various three-term recursion relations for the numerators and denominators of these approximants is shown. Based on these, a new continued fraction representation for these approximants is obtained and also an efficient recursive method is proposed for the determination of the coefficients of all the approximants that obtain from a given number of terms of the power series.

The problem of minimal Padé approximation was conceived by A. Bultheel and M. Van Barel in their study of the Euclidean algorithm in the matrix case. In the scalar case, the minimal Padé approximants are generically equal to the classical Padé approximants, but a minimal Padé approximant is not characterized by a numerator and a denominator degree, but by two other parameters. A table with respect to these parameters is considered here. We characterize the structure of this minimal Padé approximation table by establishing connections with the classical Padé table. A property of square block structure of the table is formulated.

Journal of Computational and Applied Mathematics, 1980

Recently McCabe and Murphy have considered the two-point Pad6 approximants to a function for which (formal) power series expansions at the origin and at infinity are given. In this paper these approximations are slightly modified and determinant representations for them are given. The existence of various three-term recursion relations for the numerators and denominators of these approximants is shown. Based on these, a new continued fraction representation for these approximants is obtained and also an efficient recursive method is proposed for the determination of the coefficients of all the approximants that obtain from a given number of terms of the power series.

Journal of Computational and Applied Mathematics, 1997

In a recent paper, we introduced a new look-ahead algorithm for recursively computing Pad6 approximants. This algorithm generates a subsequence of the Pad6 approximants on two adjacent rows (defined by fixed numerator degree) of the Pad6 table. Its two basic versions reduce to the classical Levinson and Schur algorithms if no look-ahead is required. In this paper, we show that the computational overhead of the look-ahead steps in the O(N 2) versions of the look-ahead Levinson-and the look-ahead Schur-type algorithm can be further reduced. If the algorithms are used to solve Toeplitz systems of equations Tx-b, then the corresponding block LDU decompositions of T-1 or T, respectively, can be found with less computational effort than with any other look-ahead algorithm available today.

Lecture Notes in Computer Science, 2004

In this paper we interpret the Berlekamp-Massey algorithm (BMA) for synthesis of linear feedback shift register (LFSR) as an algorithm computing Pade approximants for Laurent series over arbitrary field. This interpretation of the BMA is based on a iterated procedure for computing of the sequence of polynomials orthogonal to some sequence of polynomial spaces with scalar product depending on the given Laurent series. It is shown that the BMA is equivalent to the Euclidean algorithm of a conversion of Laurent series in continued fractions.

BIT Numerical Mathematics, 1996

A new look-ahead algorithm for recursively computing Pade approximants is introduced. It generates a subsequence of the Pade approximants on two adjacent rows (defined by fixed numerator degree) of the Pade table. Its two basic versions reduce to the classical Levinson and Schur algorithms if no look-ahead is required. The new algorithm can be viewed as a combination of the look-ahead sawtooth and the look-ahead Levinson and Schur algorithms that we proposed before, but now the look-ahead step size is minimal (as in the sawtooth version) and the computational costs are as low as in the least expensive competing algorithms (including our look-ahead Levinson and Schur algorithms). The underlying recurrences link well-conditioned basic pairs, i, e ., pairs of sufficiently different neighboring Fade forms. The algorithm can be used to solve Toeplitz systems of equations Tx = b. In this application it comes in several versions : an O(N 2) Levinson-type form, an O(N2) Schur-type form, and a superfast O(Nlog2 N) Schur-type version. As an option of the first two versions, the corresponding block LDU decompositions of T-1 or T, respectively, can be found .

Mathematical Notes, 2006

The Berlekamp-Massey algorithm (further, the BMA) is interpreted as an algorithm for constructing Padé approximations to the Laurent series over an arbitrary field with singularity at infinity. It is shown that the BMA is an iterative procedure for constructing the sequence of polynomials orthogonal to the corresponding space of polynomials with respect to the inner product determined by the given series. The BMA is used to expand the exponential in continued fractions and calculate its Padé approximations.

Padé approximants are rational functions whose series expansion match a given series as far as possible. These approximants are usually written under a rational form. In this paper, we will show how to write them also under two different barycentric forms, and under a partial fraction form, depending on free parameters. According to the choice of these parameters, Padé-type approximants can be obtained under a barycentric or a partial fraction form.