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 ν . Our implementation uses the split-radix fast Fourier transform algorithms for real data of Duhamel. We are able to obtain the nth Szegő polynomial using fewer than 8n log 2 2 n real arithmetic operations without explicit use of the bit-reversal permutation. Since Levinson's algorithm requires slightly more than 2n 2 operations to obtain this polynomial, we achieve crossover with Levinson's algorithm at n = 256.

Numerically stable algorithms are given for updating the Gram-Schmidt QR factorization of an m x n matrix A (m > n) when A is modified by a matrix of rank one, or when a row or column is inserted or deleted. The algorithms require 0(mn) operations per update, and are based on the use of elementary two-by-two reflection matrices and the Gram-Schmidt process with reorthogonalization. An error analysis of the reorthogonalization process provides rigorous justification for the corresponding ALGOL procedures.

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Summary. We present an expos6 of the elementary theory of Jacobi matrices and, in particular, their reconstruction from the Gaussian weights and abscissas. Many recent works propose use of the diagonal Hermitian Lanczos process for this purpose. We show that this process is numerically unstable. We recall Rutishauser's elegant and stable algorithm of 1963, based on plane rotations, implement it efficiently, and discuss our numerical experience. We also apply Rutishauser's algorithm to reconstruct a persymmetric Jacobi matrix from its spectrum in an efficient and stable manner.

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