Using noncommutative Gröbner bases in solving partially prescribed matrix inverse completion problems

Given the m lowest eigenvalues, we seek to recover an approximation to the density function ρ in the weighted Helmholtz equation -Δ=λρu on a rectangle with Dirchlet boundary conditions. The density ρ is assumed to be symmetric with respect to the midlines of the rectangle. Projection of the boundary value problem and the unknown density function onto appropriate vector spaces leads to a matrix inverse problem. Solutions of the matrix inverse problem exist provided that the reciprocals of the prescribed eigenvalues are close to the reciprocals of the simple eigenvalues of the base problem with ρ = 1. The matrix inverse problem is solved by a fixed—point iterative method and a density function ρ* is constructed which has the same m lowest eigenvalues as the unknown ρ. The algorithm can be modified when multiple base eigenvalues arise, although the success of the modification depends on the symmetry properties of the base eigenfunctions.

In this paper, we give a perturbation bound for the solution of the Jacobi matrix inverse eigenvalue problem.

Solving matrix inversion is very useful in many areas of science (e.g., in physics and engineering, such as chemical processes, robotics, electronic circuits, engineered materials, and other natural sciences). Various methods exist to solve matrix inversion problems. Most of them are very good algorithms, which, however, have the drawback of being efficient only when implemented on single-processor systems. Therefore, those algorithms are very inefficient when implemented on multiprocessor platforms; thus, they lack sufficient parallelizability. The main root of the problem lies in the nature of the algorithms, since they were originally designed for implementations on single-processor systems. Some novel concepts involving neurocomputing, however, have the potential for more efficiency in multicore environments. This chapter provides a comprehensive overview of both traditional and neurocomputing-based methods for solving the matrix inversion problem (i.e., analytical, heuristics, dynamical-system-based methods, etc.). These methods are compared based on some important criteria including convergence, parallelizability/scalability, accuracy, and applicability to time-varying matrices. Finally, we propose a new concept based on neurocomputing for solving the matrix inversion problem. The main advantage of this concept is the possibility of efficiently satisfying all the important criteria.

This paper studies the problem of the restoration of images corrupted by mixed Gaussian-impulse noise. In recent years, low-rank matrix reconstruction has become a research hotspot in many scientific and engineering domains such as machine learning, image processing, computer vision and bioinformatics, which mainly involves the problem of matrix completion and robust principal component analysis, namely recovering a low-rank matrix from an incomplete but accurate sampling subset of its entries and from an observed data matrix with an unknown fraction of its entries being arbitrarily corrupted, respectively. Inspired by these ideas, we consider the problem of recovering a low-rank matrix from an incomplete sampling subset of its entries with an unknown fraction of the samplings contaminated by arbitrary errors, which is defined as the problem of matrix completion from corrupted samplings and modeled as a convex optimization problem that minimizes a combination of the nuclear norm and the -norm in this paper. Meanwhile, we put forward a novel and effective algorithm called augmented Lagrange multipliers to exactly solve the problem. For mixed Gaussian-impulse noise removal, we regard it as the problem of matrix completion from corrupted samplings, and restore the noisy image following an impulse-detecting procedure. Compared with some existing methods for mixed noise removal, the recovery quality performance of our method is dominant if images possess low-rank features such as geometrically regular textures and similar structured contents; especially when the density of impulse noise is relatively high and the variance of Gaussian noise is small, our method can outperform the traditional methods significantly not only in the simultaneous removal of Gaussian noise and impulse noise, and the restoration ability for a low-rank image matrix, but also in the preservation of textures and details in the image.

The International Symposium on Symbolic and Algebraic Computation (ISSAC) is the premier conference for research in symbolic computation and computer algebra. ISSAC 2012 is the 37th meeting in the series, started in 1966 and held annually since 1981, in North America, Europe and Asia. The conference presents a range of invited speakers, tutorials, poster sessions, software demonstrations and vendor exhibits with a centerpiece of contributed research papers.

This is the second of a planned sequence of papers on inverse problems for canonical systems of differential equations. It is devoted to the inverse monodromy problem for canonical integral and differential systems. In this part, which focuses on the case of a diagonal signature matrixJ, a parametrization is obtained of the set of all solutionsM (t) for the inverse problem for integral systems in terms of two chains of entire matrix valued inner functions. Special classes of solutions correspond to special choices of these chains. This theme will be elaborated upon further in a third part of this paper which will be published in a subsequent issue of this journal. There the emphasis will be on symmetries and growth conditions all of which serve to specify or restrict the chains alluded to above, from the outside, so to speak.

A Review of Arithmetic Circuit Verification Using Symbolic Computer Algebra

We derive the full set of beta functions for the marginal essential couplings of projectable Ho\ifmmode \check{r}\else \v{r}\fi{}ava gravity in ($3+1$)-dimensional spacetime. To this end we compute the divergent part of the one-loop effective action in static background with an arbitrary spatial metric. The computation is done in several steps: reduction of the problem to three dimensions, extraction of an operator square root from the spatial part of the fluctuation operator, and evaluation of its trace using the method of universal functional traces. This provides us with the renormalization of couplings in the potential part of the action which we combine with the results for the kinetic part obtained previously. The calculation uses symbolic computer algebra and is performed in four different gauges yielding identical results for the essential beta functions. We additionally check the calculation by evaluating the effective action on a special background with spherical spatial slices using an alternative method of spectral summation. We conclude with a preliminary discussion of the properties of the beta functions and the resulting renormalization group flow, identifying several candidate asymptotically free fixed points.

Multiplier circuit covers a more extensive area of embedded system application in digital signal processing, cryptography, and multimedia. Nonstandard implementations and custom optimization are being done to reduce the size of multipliers. The circuit became prone to a buggy, and hence the demand for verification increased. Formal verification methods, such as satisfiability (SAT), symbolic computer algebra (SCA), and binary decision diagrams (BDDs) have made massive progress over the last few decades. However, these methods are insufficient to verify the optimized multipliers. SAT-based equivalence checking is computationally expensive. SCA-based backward rewriting is limited to algebraic-friendly multipliers. The complexity of BDDs is exponential with the input size. Although, by allowing an additional variable method, the size of the BDD is limited to 4th degree polynomial of the number of the inputs, this method is not explored to verify optimized multipliers. This article focus on verifying integer multipliers with diverse architectures. We propose an algorithm for the direct construction of BDDs without traversing circuits and generate BDDs up to 1024 bits. We utilize the additional variable method and constructing BDDs using high-to-low variable ordering. We reduce the complexity of BDD size to a 3rd degree polynomial. We generate BDDs and verify the multipliers with various architectures up to 64 bits. We propose a method to verify optimized multipliers by checking equivalence and verifying up to 32-bits optimized multipliers. We do the error tolerance analysis of our approach by inserting bugs in a circuit at various locations.

A Jacobi matrix inverse eigenvalue problem with mixed data

On robust matrix completion with prescribed eigenvalues

In this paper, the P0,1+-matrix completion is introduced. A real n × n matrix is a P0,1+-matrix if it is a P0+-matrix with all positive diagonal entries. Firstly some necessary conditions for P0,1+-matrix completion are obtained and later on some sufficient conditions are also shown. Also the relationship between the P0,1+-matrix completion problem and other completion problems of p-matrix classes are discussed. Finally most of the digraphs of order at most 4 having P0,1+-completion are classified.

Phased-microphone arrays associated with beamforming have become a standard technique to localize aeroacoustic sources. The limitations of beamforming have been overcome thanks to deconvolution technique (DAMAS or CLEAN-SC) or iterative process (L1-GIB). However, the computational cost of these methods can be large or assumptions on the source coherence have to be done. In this paper we present a technique based on inverse methods initially developed for sound eld extrapolation. The aim is to use a beamforming regularization matrix to penalize the non-signal region in the inverse problem. First, this Hybrid Method is applied to laboratory experiments to demonstrate its eectiveness. Then noise data of an aero-engine measured over a half circular, far-eld microphone array are used. The source maps obtained show that the Hybrid Method provides better spatial resolution than beamforming, similar to Clean-SC and results in less iterations of DAMAS.

On efficient computation of rational {1,2}-pseudo-inverses for multivariable rational matrix-valued functions and their applications

In this chapter, we discuss the general theory of quadratic forms and the notion of congruence and its invariants, as well as the applications of the theory to the analysis of special forms. We will focus primarily on quadratic forms induced by rational functions, most notably the Hankel and Bezout forms, because of their connection to system-theoretic problems such as stability and signature symmetric realizations. These forms use as their data different representations of rational functions, namely power series and coprime factorizations respectively. But we will discuss also the partial fraction representation in relation to the computation of the Cauchy index of a rational function and the proof of the Hermite–Hurwitz theorem and the continued fraction representation as a tool in the computation of signatures of Hankel matrices as well as in the problem of Hankel matrix inversion. Thus, different representations of rational functions, i.e., different encodings of the information carried by a rational function, provide efficient starting points for different methods. The results obtained for rational functions will be applied to root-location problems for polynomials in the next chapter.