BIT Numerical Mathematics

, Volume 36, Issue 2, pp 264–286

Optimized look-ahead recurrences for adjacent rows in the Padé table

  • Martin H. Gutknecht
  • Marlis Hochbruck

DOI: 10.1007/BF01731983

Cite this article as:
Gutknecht, M.H. & Hochbruck, M. Bit Numer Math (1996) 36: 264. doi:10.1007/BF01731983


A new look-ahead algorithm for recursively computing Padé approximants is introduced. It generates a subsequence of the Padé approximants on two adjacent rows (defined by fixed numerator degree) of the Padé 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 Padé forms.

The algorithm can be used to solve Toeplitz systems of equationsTx = b. In this application it comes in several versions: anO(N2) Levinson-type form, anO(N2) Schur-type form, and a superfastO(N log2N) Schur-type version. As an option of the first two versions, the corresponding block LDU decompositions ofT−1 orT, respectively, can be found.

Key words

Padé approximation row recurrence fast algorithm sawtooth recurrence look-ahead Toeplitz matrix Levinson algorithm Schur algorithm biorthogonal polynomials 

Copyright information

© BIT Foundation 1996

Authors and Affiliations

  • Martin H. Gutknecht
    • 1
  • Marlis Hochbruck
    • 2
  1. 1.Swiss Center for Scientific Computing (SCSC)ETH ZurichZurichSwitzerland
  2. 2.Mathematisches InstitutUniversität TübingenTübingenGermany

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