We discuss matrix polynomials expressed in a Bernstein basis, and the associated polynomial eigenvalue problems. Using Möbius transformations of matrix polynomials, large new families of strong linearizations are generated. Matrix polynomials that are structured with respect to a Bernstein basis, together with their associated spectral symmetries, are also investigated. The results in this paper apply equally well to scalar polynomials, and include the development of new companion pencils for polynomials expressed in a Bernstein basis.
We discuss matrix polynomials expressed in a Newton basis, and the associated polynomial eigenvalue problems. Properties of the generalized ansatz spaces associated with such polynomials are proved directly by utilizing a novel representation of pencils in these spaces. Also, we show how the family of Fiedler pencils can be adapted to matrix polynomials expressed in a Newton basis. These new Newton-Fiedler pencils are shown to be strong linearizations, and some computational aspects related to them are discussed.
We discuss a simple, easily overlooked, explicit deflation procedure applied to Golub-Kahan-Lanczos Bidiagonalization (GKLB)-based methods to compute the next set of the largest singular triplets of a matrix from an already computed partial singular value decomposition. Our results here complement the vast literature on this topic, provide additional insight, and highlight the simplicity and the effectiveness of this procedure. We demonstrate how existing GKLB-based routines for the computation of the largest singular triplets can be easily adapted to take advantage of explicit deflation, thus making it more appealing to a wider range of users. Numerical examples are presented including an application of singular value thresholding.
Algorithms for ensemble methods (EM) based on bootstrap aggregation often perform copious amount of redundant computations (RC) thus limiting their practicality. Given this constraint, we propose a framework that views these algorithms as a collection of computational units (cu), a tightly coupled set of both mathematical operations and data. This view facilitates a reduction in RC (RRC), thereby allowing for faster execution plans. Inspired by the floor tiling approach in VLSI, we look to engineer solutions for RRC while possibly reconfiguring the underlying computing system's compiler technology stack. We start by showing that under the assumption that the computational system has unbounded but finite memory (i.e., the memory is large enough to hold all intermediate values) and that each cu has a uniform cost, our approach reduces to a well-studied directed bandwidth problem for the directed acyclic graphs (DAGs). Next, we consider a more realistic scenario where the computing system has limited memory and concurrent execution while still assuming a uniform cost. Using a new notion of (r,s) set cover of a DAG (nodes representing computational units and edges representing their interdependencies) we formulate the problem of reducing redundant computational steps in EM as a variation of a directed bandwidth problem. We show that the graph's minimum bandwidth is closely related to memory requirements for studying RRC. Finally, our preliminary experimental results are supportive of the proposed approach for RRC and promising that it can be applied to a broader set of algorithms in decision sciences.
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