The Wiener–Levinson algorithm and ill-conditioned normal equations

SUMMARY

Treitel & Wang (1976) noted that, when autocorrelation matrices are ill-conditioned, elements of Wiener filters are significantly different when the normal equations are solved on different computers. They presented an example in which the Wiener–Levinson algorithm produced a prediction filter exhibiting significant error. In recent years there has been controversy in mathematical literature relating to stability of algorithms, such as the Wiener–Levinson algorithm, for solving linear systems of equations with a Toeplitz coefficient matrix. In this paper, it is argued that poor-quality results produced by the Wiener–Levinson algorithm, when applied to problems exhibiting an ill-conditioned autocorrelation matrix, may be attributed to stability properties of that algorithm. An example is presented, comparing the results of Gaussian elimination, the Wiener–Levinson algorithm, and the conjugate gradient algorithm. The use of intermediate results of the Wiener–Levinson algorithm to detect ill-conditioned normal equations is discussed.