Immittance-Domain Levinson Algorithms

Several computationally efficient versions of the Levinson algorithm for solving linear equations with Toeplitz and quasi-Toeplitz matrices are presented, motivated by a new stability test due to Bistritz (1983). The new versions require half the number of multiplications and the same number of additions as the conventional form of the Levinson algorithm. The saving is achieved by using three-term (rather than two-term) recursions and propagating them in an impedance /admittance (or immittance) domain rather than the conventional scattering domain. One of our recursions coincides with the recent results of Delsarte and Genin on “split-Levinson” algorithms for symmetric Toeplitz matrices, where the efficiency is gained by using the symmetric and skew-symmetric versions of the usual polynomials. This special structure is lost in the quasi-Toeplitz case, but one still can obtain similar computational reductions by suitably using three-term recursions in the immittance domain.