Fast linear algebra is stable

In an earlier paper, we showed that a large class of fast recursive matrix multiplication algorithms is stable in a normwise sense, and that in fact if multiplication of $n$-by-$n$ matrices can be done by any algorithm in $O(n^{\omega + \eta})$ operations for any $\eta > 0$, then it can be done stably in $O(n^{\omega + \eta})$ operations for any $\eta > 0$. Here we extend this result to show that essentially all standard linear algebra operations, including LU decomposition, QR decomposition, linear equation solving, matrix inversion, solving least squares problems, (generalized) eigenvalue problems and the singular value decomposition can also be done stably (in a normwise sense) in $O(n^{\omega + \eta})$ operations.Comment: 26 pages; final version; to appear in Numerische Mathemati

Dissipation time and decay of correlations

We consider the effect of noise on the dynamics generated by volume-preserving maps on a d-dimensional torus. The quantity we use to measure the irreversibility of the dynamics is the dissipation time. We focus on the asymptotic behaviour of this time in the limit of small noise. We derive universal lower and upper bounds for the dissipation time in terms of various properties of the map and its associated propagators: spectral properties, local expansivity, and global mixing properties. We show that the dissipation is slow for a general class of non-weakly-mixing maps; on the opposite, it is fast for a large class of exponentially mixing systems which include uniformly expanding maps and Anosov diffeomorphisms.Comment: 26 Pages, LaTex. Submitted to Nonlinearit

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

An ecological future for weed science to sustain crop production and the environment. A review

Sustainable strategies for managing weeds are critical to meeting agriculture's potential to feed the world's population while conserving the ecosystems and biodiversity on which we depend. The dominant paradigm of weed management in developed countries is currently founded on the two principal tools of herbicides and tillage to remove weeds. However, evidence of negative environmental impacts from both tools is growing, and herbicide resistance is increasingly prevalent. These challenges emerge from a lack of attention to how weeds interact with and are regulated by the agroecosystem as a whole. Novel technological tools proposed for weed control, such as new herbicides, gene editing, and seed destructors, do not address these systemic challenges and thus are unlikely to provide truly sustainable solutions. Combining multiple tools and techniques in an Integrated Weed Management strategy is a step forward, but many integrated strategies still remain overly reliant on too few tools. In contrast, advances in weed ecology are revealing a wealth of options to manage weedsat the agroecosystem levelthat, rather than aiming to eradicate weeds, act to regulate populations to limit their negative impacts while conserving diversity. Here, we review the current state of knowledge in weed ecology and identify how this can be translated into practical weed management. The major points are the following: (1) the diversity and type of crops, management actions and limiting resources can be manipulated to limit weed competitiveness while promoting weed diversity; (2) in contrast to technological tools, ecological approaches to weed management tend to be synergistic with other agroecosystem functions; and (3) there are many existing practices compatible with this approach that could be integrated into current systems, alongside new options to explore. Overall, this review demonstrates that integrating systems-level ecological thinking into agronomic decision-making offers the best route to achieving sustainable weed management

Stability Analysis of Difference Methods for Parabolic Initial Value Problems

Abstract A decomposition of the numerical solution can be defined by the normal mode representation, that generalizes further the spatial eigenmode decomposition of the von Neumann analysis by taking into account the boundary conditions which are not periodic. In this paper we present some new theoretical results on normal mode analysis for a linear and parabolic initial value problem. Furthermore we suggest an algorithm for the calculation of stability regions based on the normal mode theory