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Summary: SUPERFAST SOLUTION OF REAL POSITIVE DEFINITE TOEPLITZ
SYSTEMS
GREGORY S. AMMAR AND WILLIAM B. GRAGG
Abstract. We describe an implementation of the generalized Schur algorithm for the superfast
solution of real positive definite Toeplitz systems of order n + 1, where n = 2 . Our implementation
uses the split-radix fast Fourier transform algorithms for real data of Duhamel. We are able to obtain
the nth Szego polynomial using fewer than 8n log2
2 n real arithmetic operations without explicit use of
the bit-reversal permutation. Since Levinson's algorithm requires slightly more than 2n2 operations
to obtain this polynomial, we achieve crossover with Levinson's algorithm at n = 256.
Key words. Toeplitz matrix, Schur's algorithm, split-radix Fast Fourier Transform
AMS subject classifications. 65F05, 65E05
1. Introduction. Consider the linear system of equations Mx = b, where
M = Mn+1 =
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