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.
Abstract. We present an efficient and reliable algorithm for discrete least squares approximation of a real-valued function given at arbitrary distinct nodes in [0, 2tt) by trigonometric polynomials. The algorithm is based on a scheme for the solution of an inverse eigenproblem for unitary Hessenberg matrices, and requires only O(mn) arithmetic operations as compared with 0(mn ) operations needed for algorithms that ignore the structure of the problem. Moreover, the proposed algorithm produces consistently accurate results that are often better than those obtained by general QR decomposition methods for the least squares problem. Our algorithm can also be used for discrete least squares approximation on the unit circle by algebraic polynomials.
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