Effects Of Finite Precision Arithmetic Research Articles

Adaptive noise reduction using numerically stable fast recursive least squares algorithm

AbstractThis paper is concerned with adaptive noise reduction based on the fast recursive least squares (FRLS) algorithm. It is well known that the fast recursive least squares (FRLS) algorithm suffers from numerical instability when operating under the effects of finite precision arithmetic. Several numerical solutions of stabilization were proposed in the case of stationary signals. In this work a new version of a numerically stable FRLS algorithm (NS‐FRLS) is proposed. The stability characteristics of this new stabilized algorithm are analysed. The analysis is based on a linear model for the errors in the states of the adaptive filter. Experimental results confirm the merits of adaptive filtering with the NS‐FRLS algorithm over optimum filtering using the solution provided by Wiener–Hopf equations. Copyright © 2006 John Wiley & Sons, Ltd.

Effects of Finite-Precision Arithmetic on Interior-Point Methods for Nonlinear Programming

We show that the effects of finite-precision arithmetic in forming and solving the linear system that arises at each iteration of primal-dual interior-point algorithms for nonlinear programming are benign, provided that the iterates satisfy centrality and feasibility conditions of the type usually associated with path-following methods. When we replace the standard assumption that the active constraint gradients are independent by the weaker Mangasarian--Fromovitz constraint qualification, rapid convergence usually is attainable, even when cancellation and roundoff errors occur during the calculations. In deriving our main results, we prove a key technical result about the size of the exact primal-dual step. This result can be used to modify existing analysis of primal-dual interior-point methods for convex programming, making it possible to extend the superlinear local convergence results to the nonconvex case.

Relations between Galerkin and Norm-Minimizing Iterative Methods for Solving Linear Systems

Several iterative methods for solving linear systems $Ax = b$ first construct a basis for a Krylov subspace and then use the basis vectors, together with the Hessenberg (or tridiagonal) matrix generated during that construction, to obtain an approximate solution to the linear system. To determine the approximate solution, it is necessary to solve either a linear system with the Hessenberg matrix as coefficient matrix or an extended Hessenberg least squares problem. In the first case, referred to as a Galerkin method, the residual is orthogonal to the Krylov subspace, whereas in the second case, referred to as a norm-minimizing method, the residual (or a related quantity) is minimized over the Krylov subspace. Examples of such pairs include the full orthogonalization method (FOM) (Arnoldi) and generalized minimal residual (GMRES) algorithms, the biconjugate gradient (BCG) and quasi-minimal residual (QMR) algorithms, and their symmetric equivalents, the Lanczos and minimal residual (MINRES) algorithms. A relationship between the solution of the linear system and that of the least squares problem is used to relate the residual norms in Galerkin processes to the norms of the quantities minimized in the corresponding norm-minimizing processes. It is shown that when the norm-minimizing process is converging rapidly, the residual norms in the corresponding Galerkin process exhibit similar behavior, whereas when the norm-minimizing process is converging very slowly, the residual norms in the corresponding Galerkin process are significantly larger. This is a generalization of the relationship established between Arnoldi and GMRES residual norms in P. N. Brown, A theoretical comparison of the Arnoldi and GMRES algorithms, SIAM J. Sci. Statist. Comput., 12, 1991, pp. 58–78. For MINRES and Lanczos, and for two nonsymmetric bidiagonalization procedures, we extend the arguments to incorporate the effects of finite precision arithmetic.

Physical and numerical aspects in Lanczos and modified Lanczos calculations

The use of recurrence relations in Lanczos procedure is often limited in practice by loss of numerical stability. In many applications this fact severely limits the numerical reliability of the results and hence the interpretation of the physical models proposed. In this paper we focus on the effects of finite precision arithmetic in the Lanczos algorithm and examine some methods for assessing and testing the reliability of the tridiagonal matrix generated. In addition to the residual vector test we propose an initial vector test, and an orthogonalization test to assess the quality of the eigenvalues. We also implement the use of multiple precision to obtain a significant improvement in the tridiagonal chain transformation. As a final resort, we consider the modified Lanczos procedure to obtain one by one, energies and wavefunctions of a system, within any desired energy range and with any desired accuracy.

Computing Fourier Transforms and Convolutions on the 2-Sphere

This paper considers the problem of efficient computation of the spherical harmonic expansion, or Fourier transform, of functions defined on the two dimensional sphere, S 2. The resulting algorithms are applied to the efficient computation of convolutions of functions on the sphere. We begin by proving convolution theorems generalizing well known and useful results from the abelian case. These convolution theorems are then used to develop a sampling theorem on the sphere. which reduces the calculation of Fourier transforms and convolutions of band-limited functions to discrete computations. We show how to perform these efficiently, starting with an O( n(log n) 2) time algorithm for computing the Legendre transform of a function defined on the interval [−1,1] sampled at n points there. Theoretical and experimental results on the effects of finite precision arithmetic are presented. The Legendre transform algorithm is then generalized to obtain an algorithm for the Fourier transform, requiring O( n(log n) 2) time, and an algorithm for its inverse in O( n 1.5) time, where n is the number of points on the sphere at which the function is sampled. This improves the naive O( n 2) bound, which is the best previously known. These transforms give an O( n 1.5) algorithm for convolving two functions on the sphere.

QR-decomposition decision feedback equalisation and finite-precision results

The QRD LS algorithm is generally recognised for its superior numerical properties under finite-precision implementation. Furthermore, its systolic architecture is well suited for VLSI implementation. In DFE applications the inherent and implementation nonlinearities make it impossible to analyse precisely all effects of finite-precision arithmetic. The paper presents finite precision results of a QRD LS DFE using the TMS320C25 16-bit precision DSP processors as a simulation platform. Using the bit error rate as a performance measure, results are presented for 2-PSK, 4-PSK and Pi/4-DQPSK modulation formats. Also presented are the numerical accuracy and convergence sensitivity for the filter weights. These results may serve as a basis for practical implementation of QRD LS DFEs.

Computer arithmetic, chaos and fractals

In this paper we explore aspects of computer arithmetic from the viewpoint of dynamical systems. We demonstrate the effects of finite precision arithmetic in three uniformly hyperbolic chaotic dynamical systems: Bernoulli shifts, cat maps, and pseudorandom number generators. We show that elementary floating-point operations in binary computer arithmetic possess an inherently fractal structure. Each of these dynamical systems allows us to compare the exact results in integer arithmetic with those obtained by using floating-point arithmetic.

Fixed-point error analysis of stochastic gradient adaptive lattice filters

A theoretical analysis is presented of the stochastic gradient adaptive lattice filter used as a linear, one-step predictor, when the effects of finite-precision arithmetic are taken into account. Only the fixed-point implementation is considered. Both the unnormalized and normalized adaptation algorithms are analyzed. Expressions of the steady-state mean-squared values of the accumulated numerical errors in the computation of the reflection coefficients and the prediction errors of different orders have been developed. The results show that the dominant term in the expressions for the mean-squared values of the numerical errors is inversely proportional to the convergence parameter. Furthermore, they indicate that the quantization errors associated with the reflection coefficients are more critical than those associated with representing the prediction-error sequences. Signals with high correlation among samples produce larger numerical errors in the adaptive lattice filter. The authors present several simulation examples that show close agreement with the theoretical results. They also show that the gradient adaptive lattice filters have better numerical properties than their transversal counterparts. >