Fast Hessenberg reduction of some rank structured matrices

We develop two fast algorithms for Hessenberg reduction of a structured matrix $A = D + UV^H$ where $D$ is a real or unitary $n \times n$ diagonal matrix and $U, V \in\mathbb{C}^{n \times k}$. The proposed algorithm for the real case exploits a two--stage approach by first reducing the matrix to a generalized Hessenberg form and then completing the reduction by annihilation of the unwanted sub-diagonals. It is shown that the novel method requires $O(n^2k)$ arithmetic operations and it is significantly faster than other reduction algorithms for rank structured matrices. The method is then extended to the unitary plus low rank case by using a block analogue of the CMV form of unitary matrices. It is shown that a block Lanczos-type procedure for the block tridiagonalization of $\Re(D)$ induces a structured reduction on $A$ in a block staircase CMV--type shape. Then, we present a numerically stable method for performing this reduction using unitary transformations and we show how to generalize the sub-diagonal elimination to this shape, while still being able to provide a condensed representation for the reduced matrix. In this way the complexity still remains linear in $k$ and, moreover, the resulting algorithm can be adapted to deal efficiently with block companion matrices.Comment: 25 page

Two-way bidiagonalization scheme for downdating the singular-value decomposition

AbstractWe present a method that transforms the problem of downdating the singular-value decomposition into a problem of diagonalizing a diagonal matrix bordered by one column. The first step in this diagonalization involves bidiagonalization of a diagonal matrix bordered by one column. For updating the singular-value decomposition, a two-way chasing scheme has been recently introduced, which reduces the total number of rotations by 50% compared to previously developed one-way chasing schemes. Here, a two-way chasing scheme is introduced for the bidiagonalization step in downdating the singular-value decomposition. We show how the matrix elements can be rearranged and how the nonzero elements can be chased away towards two corners of the matrix. The newly proposed scheme saves nearly 50% of the number of plane rotations required by one-way chasing schemes

Spectrum Adaptation in Cognitive Radio Systems with Operating Constraints

The explosion of high-data-rate-demanding wireless applications such as smart-phones and wireless Internet access devices, together with growth of existing wireless services, are creating a shortage of the scarce Radio Frequency (RF) spectrum. However, several spectrum measurement campaigns revealed that current spectrum usage across time and frequency is inefficient, creating the artificial shortage of the spectrum because of the traditional exclusive command-and-control model of using the spectrum. Therefore, a new concept of Cognitive Radio (CR) has been emerging recently in which unlicensed users temporarily borrow spectrum from the licensed Primary Users (PU) based on the Dynamic Spectrum Access (DSA) technique that is also known as the spectrum sharing concept. A CR is an intelligent radio system based on the Software Defined Radio platform with artificial intelligence capability which can learn, adapt, and reconfigure through interaction with the operating environment. A CR system will revolutionize the way people share the RF spectrum, lowering harmful interference to the licensed PU of the spectrum, fostering innovative DSA technology and giving people more choices when it comes to using the wireless-communication-dependent applications without having any spectrum congestion problems. A key technical challenge for enabling secondary access to the licensed spectrum adaptation is to ensure that the CR does not interfere with the licensed incumbent users. However, incumbent user behavior is dynamic and requires CR systems to adapt this behavior in order to maintain smooth information transmission. In this context, the objective of this dissertation is to explore design issues for CR systems focusing on adaptation of physical layer parameters related to spectrum sensing, spectrum shaping, and rate/power control. Specifically, this dissertation discusses dynamic threshold adaptation for energy detector spectrum sensing, spectrum allocation and power control in Orthogonal Frequency Division Multiplexing-(OFDM-)based CR with operating constraints, and adjacent band interference suppression techniques in turbo-coded OFDM-based CR systems

Compressive and Noncompressive Power Spectral Density Estimation from Periodic Nonuniform Samples

This paper presents a novel power spectral density estimation technique for band-limited, wide-sense stationary signals from sub-Nyquist sampled data. The technique employs multi-coset sampling and incorporates the advantages of compressed sensing (CS) when the power spectrum is sparse, but applies to sparse and nonsparse power spectra alike. The estimates are consistent piecewise constant approximations whose resolutions (width of the piecewise constant segments) are controlled by the periodicity of the multi-coset sampling. We show that compressive estimates exhibit better tradeoffs among the estimator's resolution, system complexity, and average sampling rate compared to their noncompressive counterparts. For suitable sampling patterns, noncompressive estimates are obtained as least squares solutions. Because of the non-negativity of power spectra, compressive estimates can be computed by seeking non-negative least squares solutions (provided appropriate sampling patterns exist) instead of using standard CS recovery algorithms. This flexibility suggests a reduction in computational overhead for systems estimating both sparse and nonsparse power spectra because one algorithm can be used to compute both compressive and noncompressive estimates.Comment: 26 pages, single spaced, 9 figure

Multi-Processor Architectures for Solving Sparse Linear Systems. Application to the Load Flow Problem

IFAC SYMPOSIUM ON COMPONENTS,INSTRUMENTS AND TECHNIQUES FOR LOW COST AUTOMATION AND APPLICATIONS.1986 (.1986.VALENCIA)Three heuristic algorithms for solving a cluster problem associated with the tearing of a symmetric matrix are presented. Based on these partitioning procedures, a method for the parallel solution of the fast decoupled load flow has been developed, although it is also suitable for the parallel solution of any linear equations system. Experimental results of applying such algorithms on several test systems have been obtained using a multiprocessor architecture

LINEAR SYSTEMS OF DIFFERENTIAL EQUATIONS IN ARROWHEAD FORM

This paper deals with different approaches for solving linear systems of the first order differential equations with the system matrix in the symmetric arrowhead form.Some needed algebraic properties of the symmetric arrowhead matrix are proposed.We investigate the form of invariant factors of the arrowhead matrix.Also the entries of the adjugate matrix of the characteristic matrix of the arrowhead matrix are considered. Some reductions techniques for linear systems of differential equations with the system matrix in the arrowhead form are presented

Adaptive Path Planning for Depth Constrained Bathymetric Mapping with an Autonomous Surface Vessel

This paper describes the design, implementation and testing of a suite of algorithms to enable depth constrained autonomous bathymetric (underwater topography) mapping by an Autonomous Surface Vessel (ASV). Given a target depth and a bounding polygon, the ASV will find and follow the intersection of the bounding polygon and the depth contour as modeled online with a Gaussian Process (GP). This intersection, once mapped, will then be used as a boundary within which a path will be planned for coverage to build a map of the Bathymetry. Methods for sequential updates to GP's are described allowing online fitting, prediction and hyper-parameter optimisation on a small embedded PC. New algorithms are introduced for the partitioning of convex polygons to allow efficient path planning for coverage. These algorithms are tested both in simulation and in the field with a small twin hull differential thrust vessel built for the task.Comment: 21 pages, 9 Figures, 1 Table. Submitted to The Journal of Field Robotic

Row Compression and Nested Product Decomposition of a Hierarchical Representation of a Quasiseparable Matrix

This research introduces a row compression and nested product decomposition of an nxn hierarchical representation of a rank structured matrix A, which extends the compression and nested product decomposition of a quasiseparable matrix. The hierarchical parameter extraction algorithm of a quasiseparable matrix is efficient, requiring only O(nlog(n))operations, and is proven backward stable. The row compression is comprised of a sequence of small Householder transformations that are formed from the low-rank, lower triangular, off-diagonal blocks of the hierarchical representation. The row compression forms a factorization of matrix A, where A = QC, Q is the product of the Householder transformations, and C preserves the low-rank structure in both the lower and upper triangular parts of matrix A. The nested product decomposition is accomplished by applying a sequence of orthogonal transformations to the low-rank, upper triangular, off-diagonal blocks of the compressed matrix C. Both the compression and decomposition algorithms are stable, and require O(nlog(n)) operations. At this point, the matrix-vector product and solver algorithms are the only ones fully proven to be backward stable for quasiseparable matrices. By combining the fast matrix-vector product and system solver, linear systems involving the hierarchical representation to nested product decomposition are directly solved with linear complexity and unconditional stability. Applications in image deblurring and compression, that capitalize on the concepts from the row compression and nested product decomposition algorithms, will be shown

Thermonuclear Kinetics in Astrophysics

Over the billions of years since the Big Bang, the lives, deaths and afterlives of stars have enriched the Universe in the heavy elements that make up so much of ourselves and our world. This review summarizes the methods used to evolve these nuclear abundances within astrophysical simulations. These methods fall into 2 categories; evolution via rate equations and via equilibria. Because the rate equations in nucleosynthetic applications involve a wide range of timescales, implicit methods have proven mandatory, leading to the need to solve matrix equations. Efforts to improve the performance of such rate equation methods are focused on efficient solution of these matrix equations, in particular by making best use of the sparseness of these matrices, and finding methods that require less frequent matrix solutions. Recent work to produce hybrid schemes which use local equilibria to reduce the computational cost of the rate equations is also discussed. Such schemes offer significant improvements in the speed of reaction networks and are accurate under circumstances where calculations which assume complete equilibrium fail.Comment: 27 pages, 2 figures, a review for a special issue of Nuclear Physics