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If such a matrix is transformed by a step of the $QR$ algorithm, there is a corresponding transformation in the measure. The tridiagonal matrices are also exploited for the construction of Gaussian quadrature formulas for measures on the line. The developments on the real line are replicated with suitable modifications on the unit circle via Lanczos-like procedures for unitary operators. The best-known procedure of this type is the recursion of Szego for computing orthogonal polynomials on the unit circle. The approach taken in this paper is to develop recursions that compute orthogonal Laurent polynomials (rational functions) rather than polynomials. 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