## Fast solution of Toeplitz systems of equations and computation of
Padé approximants

59. R. P. Brent, F. G. Gustavson and D. Y. Y. Yun,
Fast solution of Toeplitz systems of equations and computation of
Padé approximants,
* J. of Algorithms* 1 (1980), 259-295.
MR 82d:65033.
Also appeared as Report TR RC 8173, IBM Research (January 1980).

Abstract:
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Paper:
pdf (2063K).

## Abstract

We present two new algorithms, ADT and MDT, for solving order-*n*
Toeplitz
systems of linear equations *Tz = b*
in time O(*n* log^{2}*n*)
and space O(*n*).
The fastest algorithms previously known, such as Trench's algorithm,
require time of order *n*^{2}
and require that all principal submatrices of *T*
be nonsingular. Our algorithm ADT requires only that *T* be nonsingular.
Both our algorithms for Toeplitz systems are derived from algorithms for
computing entries in the Padé table for a given power series.
We prove that entries in the Padé table can be computed by the Extended
Euclidean Algorithm. We describe an algorithm EMGCD (Extended Middle Greatest
Common Divisor) which is faster than the algorithm HGCD of Aho, Hopcroft and
Ullman, although both require time
O(*n* log^{2}*n*),
and we generalize
EMGCD to produce PRSDC (Polynomial Remainder Sequence Divide and Conquer)
which produces any iterate in the PRS, not just the middle term,
in time O(*n* log^{2}*n*).
Applying PRSDC to the polynomials
*U*_{0}(*x*) =
*x*^{2n+1} and
*U*_{1}(*x*) =
*a*_{0} + *a*_{1}*x*
+ ... +
*a*_{2n}*x*^{2n} gives
algorithm AD (Anti-Diagonal), which computes any (*m,p*)
entry along the
antidiagonal *m* + *p* = 2*n*
of the Padé
table for *U*_{1} in time
O(*n* log^{2}*n*).

Our other algorithm, MD (Main-Diagonal), computes any diagonal
entry (*n,n*) in the Padé table for a normal power series,
also in time
O(*n* log^{2}*n*).
MD is related to Schönhage's fast continued fraction
algorithm. A Toeplitz matrix *T* is naturally associated with
*U*_{1},
and the (*n,n*) Padé approximation to
*U*_{1} gives the first column of
*T*^{-1}.
Thus, the Padé table algorithms AD and MD give
O(*n* log^{2}*n*)
Toeplitz algorithms ADT and MDT. Trench's formula breaks down in certain
degenerate cases, but in such cases a companion formula, the discrete analog
of the Christoffel-Darboux formula, is valid and may be used to compute
*z*
in time O(*n* log^{2}*n*)
via the fast computation (by algorithm AD) of at most
four Padé approximants.

We also apply our results to obtain new complexity
bounds for the solution of banded Toeplitz systems and for BCH decoding via
Berlekamp's algorithm.

## Erratum

In the last line on page 268, "EMGCD(G_{0}, G_{0})"
should be "EMGCD(G_{0}, G_{1})".
## Comments

This paper gave the first "superfast" algorithm for the solution of
Toeplitz systems.
The term "superfast" was introduced by Ammar and Gragg to distinguish
algorithms which are faster than order *n*^{2}.
For related work and references
see
[143,
144,
177].
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