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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 \(\log^ 2n)\) 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 numerical methods for linear systems and matrix inversion
65T40 Numerical methods for trigonometric approximation and interpolation
68Q25 Analysis of algorithms and problem complexity

Citations:

Zbl 0661.00019