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Superfast solution of real positive definite Toeplitz systems. (English) Zbl 0671.65021
Linear algebra in signals, systems, and control, Proc. SIAM Conf., Boston/Mass. 1986, 107-125 (1988).

[For the entire collection see Zbl 0661.00019.]

The authors derive a generalized Schur algorithm for the “superfast” solution of real positive definite Toeplitz systems $Mx=b$ of order $n+1$, where $n={2}^{\nu }$. Fast Toeplitz solvers are based on ideas from the classical theory of Szegö polynomials. In particular, the Szegö polynomials can be identified with the columns of the reverse Cholesky factorization of ${M}^{-1}$. Schur’s algorithm generates a continued fraction representation of the holomorphic function mapping the unit disk in the complex plane into its closure.

The generalized Schur algorithm is a doubling procedure for calculating the linear fractional transformation that results from n steps of Schur’s algorithm. By using standard fast Fourier transform, the authors construct the linear fractional transformation that results from n steps of Schur’s algorithm in O(n ${log}^{2}n\right)$ complex multiplications. The implementation uses the split-radix fast Fourier transform algorithms for real data of Duhamel that exploit the inherent symmetries of the real data.

Reviewer: Th.Beth
##### MSC:
 65F05 Direct methods for linear systems and matrix inversion (numerical linear algebra) 65T40 Trigonometric approximation and interpolation (numerical methods) 68Q25 Analysis of algorithms and problem complexity