Numerical analysis of mappings associated with positive definite Toeplitz matrices

Associated with an $(n+1)\times(n+1)$ positive definite Toeplitz matrix $A=\{a\sb{\vert i-j\vert}\}\sbsp{i,j=0}{n}$, there are three groups of parameters, which are called reflection coefficients, autocorrelation sequence and AR parameters. They appear in a wide variety of applications in science and engineering such as theory of analytic functions, geophysics, speech processing, statistics, transmission lines and others. These three sets of parameters are in one to one correspondence with each other. In this dissertation, the numerical behavior of the maps from autocorrelation sequence to reflection coefficients, from autocorrelation sequence to AR parameters and from reflection coefficients to autocorrelation sequence are studied. The error analysis of Levinson algorithm, Schur algorithm and Direct algorithm for computing these maps is given. It is found that Schur and Levinson algorithms are forward stable to compute both the AR parameters and reflection coefficients. We also show that Direct algorithm is a stable algorithm for computing autocorrelation sequence. The formulas and efficient algorithms for the condition numbers of these three maps are given.