A weakly stable algorithm for general Toeplitz systems

We show that a fast algorithm for the QR factorization of a Toeplitz or Hankel matrix A is weakly stable in the sense that R^T.R is close to A^T.A. Thus, when the algorithm is used to solve the semi-normal equations R^T.Rx = A^Tb, we obtain a weakly stable method for the solution of a nonsingular Toeplitz or Hankel linear system Ax = b. The algorithm also applies to the solution of the full-rank Toeplitz or Hankel least squares problem.Comment: 17 pages. An old Technical Report with postscript added. For further details, see http://wwwmaths.anu.edu.au/~brent/pub/pub143.htm

On the stability of the Bareiss and related Toeplitz factorization algorithms

This paper contains a numerical stability analysis of factorization algorithms for computing the Cholesky decomposition of symmetric positive definite matrices of displacement rank 2. The algorithms in the class can be expressed as sequences of elementary downdating steps. The stability of the factorization algorithms follows directly from the numerical properties of algorithms for realizing elementary downdating operations. It is shown that the Bareiss algorithm for factorizing a symmetric positive definite Toeplitz matrix is in the class and hence the Bareiss algorithm is stable. Some numerical experiments that compare behavior of the Bareiss algorithm and the Levinson algorithm are presented. These experiments indicate that in general (when the reection coefficients are not all positive) the Levinson algorithm can give much larger residuals than the Bareiss algorithm

Superfast Inference for Stationary Gaussian Processes in Particle Tracking Microrheology

Particle tracking of passive microscopic species has become the experimental measurement of choice in diverse applications, where either the material volumes are limited, or the materials themselves are so soft that they deform uncontrollably under the stresses and strains of traditional instruments. As such, the results of countless biological and rheological analyses hinge pivotally on extracting reliable dynamical information from large datasets of particle trajectory recordings. However, to do this in a statistically and computationally efficient manner presents a number of important challenges. Addressing some of these challenges is the focus of the present work. In Chapter 2, we present a superfast set of tools for parametric inference in single-particle tracking. Parametric likelihoods for particle trajectory measurements typically consist of stationary Gaussian time series, for which traditional fast inference algorithms scale as N-square in the number of observations. We present a superfast algorithm for parametric inference for stationary Gaussian processes and propose novel superfast algorithms for score and Hessian calculations. This effectively enables superfast inference for stationary Gaussian process via a wide array of frequentist and Bayesian methods. In Chapters 3 and 4, we use the superfast toolkit to address two outstanding problems prevalent in many particle tracking analyses. The first is that particle position measurements are generally contaminated by various forms of high-frequency errors. Failure to account for these errors leads to considerable bias in estimation results. In Chapter 3 we propose a novel strategy to filter high-frequency noise from measurements of particle positions. Our filters are shown theoretically to cover a vast range of high-frequency noise regimes and lead to an efficient computational estimator of model coefficients. Analyses of numerous experimental and simulated datasets suggest that our filtering approach performs remarkably well. The second problem we address is the considerable heterogeneity of typical biological fluids in which particle tracking experiments are conducted. In Chapter 4, we propose a simple metric by which to quantify the degree of heterogeneity of a fluid, along with a computationally efficient estimator and statistical test against the hypothesis that the fluid is homogeneous. The thesis is concluded by outlining several directions for future research

On recursive least-squares filtering algorithms and implementations

In many real-time signal processing applications, fast and numerically stable algorithms for solving least-squares problems are necessary and important. In particular, under non-stationary conditions, these algorithms must be able to adapt themselves to reflect the changes in the system and take appropriate adjustments to achieve optimum performances. Among existing algorithms, the QR-decomposition (QRD)-based recursive least-squares (RLS) methods have been shown to be useful and effective for adaptive signal processing. In order to increase the speed of processing and achieve high throughput rate, many algorithms are being vectorized and/or pipelined to facilitate high degrees of parallelism. A time-recursive formulation of RLS filtering employing block QRD will be considered first. Several methods, including a new non-continuous windowing scheme based on selectively rejecting contaminated data, were investigated for adaptive processing. Based on systolic triarrays, many other forms of systolic arrays are shown to be capable of implementing different algorithms. Various updating and downdating systolic algorithms and architectures for RLS filtering are examined and compared in details, which include Householder reflector, Gram-Schmidt procedure, and Givens rotation. A unified approach encompassing existing square-root-free algorithms is also proposed. For the sinusoidal spectrum estimation problem, a judicious method of separating the noise from the signal is of great interest. Various truncated QR methods are proposed for this purpose and compared to the truncated SVD method. Computer simulations provided for detailed comparisons show the effectiveness of these methods. This thesis deals with fundamental issues of numerical stability, computational efficiency, adaptivity, and VLSI implementation for the RLS filtering problems. In all, various new and modified algorithms and architectures are proposed and analyzed; the significance of any of the new method depends crucially on specific application