This paper is concerned with the numerical stability of inversion algorithms for banded Toeplitz systems. We analyze the numerical behavior of one algorithm due to B.W. Dickinson [IEEE Trans. Acoust. Speech Signal Process. 27:421-423 (1979)] and two algorithms due to A.K. Jain [IEEE Trans. Acoust. Speech Signal Process. 26:121-126 (1978)]. We show that none of the three algorithms is weakly stable when used to invert symmetric positive definite systems. One of Jain's algorithms is shown to be weakly stable when used to invert a symmetric banded Toeplitz matrix with a well-conditioned positive definite infinite extension. We present a new algorithm which is weakly stable under a more general condition and can be modified to invert certain Toeplitz-like matrices. © 1992.