Solving A Toeplitz System
A matrix equation of the form
is called a Toeplitz system if A is a Toeplitz matrix. If A is an Toeplitz matrix, then the system has only 2n−1 degrees of freedom, rather than n2. We might therefore expect that the solution of a Toeplitz system would be easier, and indeed that is the case.
Toeplitz systems can be solved by the Levinson algorithm in Θ(n2) time. Variants of this algorithm have been shown to be weakly stable (i.e. they exhibit numerical stability for well-conditioned linear systems). The algorithm can also be used to find the determinant of a Toeplitz matrix in O(n2) time.
A Toeplitz matrix can also be decomposed (i.e. factored) in O(n2) time. The Bareiss algorithm for an LU decomposition is stable. An LU decomposition gives a quick method for solving a Toeplitz system, and also for computing the determinant.
Algorithms that are asymptotically faster than those of Bareiss and Levinson have been described in the literature.
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