TWO NEW LOWER BOUNDS FOR THE SPARK OF A MATRIX

2017 ◽  
Vol 96 (3) ◽  
pp. 353-360
Author(s):  
HAIFENG LIU ◽  
JIHUA ZHU ◽  
JIGEN PENG
Keyword(s):  
Lower Bounds ◽  
Coherence Function ◽  
Weyl’S Theorem ◽  
Combinatorial Search ◽  
Mutual Coherence ◽  
Numerical Examples ◽  
The Matrix ◽  
Np Hard Problem ◽  
Minimisation Problem ◽  
Better Than

The $l_{0}$-minimisation problem has attracted much attention in recent years with the development of compressive sensing. The spark of a matrix is an important measure that can determine whether a given sparse vector is the solution of an $l_{0}$-minimisation problem. However, its calculation involves a combinatorial search over all possible subsets of columns of the matrix, which is an NP-hard problem. We use Weyl’s theorem to give two new lower bounds for the spark of a matrix. One is based on the mutual coherence and the other on the coherence function. Numerical examples are given to show that the new bounds can be significantly better than existing ones.

scholarly journals Lower bounds to eigenvalues of the Schrödinger equation by solution of a 90-y challenge

2020 ◽  
Vol 117 (28) ◽  
pp. 16181-16186
Author(s):  
Rocco Martinazzo ◽  
Eli Pollak
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Upper Bound ◽  
Spectral Features ◽  
Numerical Examples ◽  
Tunneling Splittings ◽  
Order Of Magnitude ◽  
Bound Theorem ◽  
Math Phys ◽  
Better Than

The Ritz upper bound to eigenvalues of Hermitian operators is essential for many applications in science. It is a staple of quantum chemistry and physics computations. The lower bound devised by Temple in 1928 [G. Temple,Proc. R. Soc. A Math. Phys. Eng. Sci.119, 276–293 (1928)] is not, since it converges too slowly. The need for a good lower-bound theorem and algorithm cannot be overstated, since an upper bound alone is not sufficient for determining differences between eigenvalues such as tunneling splittings and spectral features. In this paper, after 90 y, we derive a generalization and improvement of Temple’s lower bound. Numerical examples based on implementation of the Lanczos tridiagonalization are provided for nontrivial lattice model Hamiltonians, exemplifying convergence over a range of 13 orders of magnitude. This lower bound is typically at least one order of magnitude better than Temple’s result. Its rate of convergence is comparable to that of the Ritz upper bound. It is not limited to ground states. These results complement Ritz’s upper bound and may turn the computation of lower bounds into a staple of eigenvalue and spectral problems in physics and chemistry.

On the lower bounds for the minimum eigenvalue of the Hadamard product of an M-matrix and its inverse

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Mustafa Özel ◽  
Dilek Varol
Keyword(s):  
Lower Bounds ◽  
Hadamard Product ◽  
Minimum Eigenvalue ◽  
Numerical Examples ◽  
Better Than

AbstractRecently, some authors have established a number of inequalities involving the minimum eigenvalue for the Hadamard product of M-matrices. In this paper, we improve these results and give some new lower bounds on the minimum eigenvalue for the Hadamard product of an M-matrix A and its inverse {A^{-1}}. Finally, it is shown by the numerical examples that our bounds are also better than some previous results.

scholarly journals New error bounds for linear complementarity problems of Σ-SDD matrices and SB-matrices

2019 ◽  
Vol 17 (1) ◽  
pp. 1599-1614
Author(s):  
Zhiwu Hou ◽  
Xia Jing ◽  
Lei Gao
Keyword(s):  
Linear Algebra ◽  
Error Bound ◽  
Linear Complementarity Problem ◽  
Error Bounds ◽  
Complementarity Problems ◽  
Linear Complementarity Problems ◽  
Linear Complementarity ◽  
Numerical Examples ◽  
Numer Algor ◽  
Better Than

Abstract A new error bound for the linear complementarity problem (LCP) of Σ-SDD matrices is given, which depends only on the entries of the involved matrices. Numerical examples are given to show that the new bound is better than that provided by García-Esnaola and Peña [Linear Algebra Appl., 2013, 438, 1339–1446] in some cases. Based on the obtained results, we also give an error bound for the LCP of SB-matrices. It is proved that the new bound is sharper than that provided by Dai et al. [Numer. Algor., 2012, 61, 121–139] under certain assumptions.

scholarly journals A New Iterative Algorithm for Solving a Class of Matrix Nearness Problem

2012 ◽  
Vol 2012 ◽  
pp. 1-6
Author(s):  
Xuefeng Duan ◽  
Chunmei Li
Keyword(s):  
Iterative Algorithm ◽  
Projection Algorithm ◽  
Frobenius Norm ◽  
Matrix Equations ◽  
Von Neumann ◽  
Numerical Examples ◽  
Matrix Nearness Problem ◽  
Least Frobenius Norm Solution ◽  
Alternating Projection ◽  
The Matrix

Based on the alternating projection algorithm, which was proposed by Von Neumann to treat the problem of finding the projection of a given point onto the intersection of two closed subspaces, we propose a new iterative algorithm to solve the matrix nearness problem associated with the matrix equations AXB=E, CXD=F, which arises frequently in experimental design. If we choose the initial iterative matrix X0=0, the least Frobenius norm solution of these matrix equations is obtained. Numerical examples show that the new algorithm is feasible and effective.

scholarly journals Practical data correlation of flashpoints of binary mixtures by a reciprocal function: The concept and numerical examples

2011 ◽  
Vol 15 (3) ◽  
pp. 905-910 ◽  
Author(s):  
Mariana Hristova ◽  
Dimitar Damgaliev ◽  
Jordan Hristov
Keyword(s):  
Binary Mixture ◽  
Binary Mixtures ◽  
Alcohol Solution ◽  
Aqueous Alcohol ◽  
Thermodynamic Models ◽  
Data Correlation ◽  
Numerical Examples ◽  
Aqueous Alcohol Solution ◽  
Better Than

Simple data correlation of flashpoint data of binary mixture has been developed on a basic of rational reciprocal function. The new approximation requires has only two coefficients and needs the flashpoint temperature of the pure flammable component to be known. The approximation has been tested by literature data concerning aqueous-alcohol solution and compared to calculations performed by several thermodynamic models predicting flashpoint temperatures. The suggested approximation provides accuracy comparable and to some extent better than that of the thermodynamic methods.

Martin Shapiro: An Appreciation

The Forum ◽  
2016 ◽  
Vol 14 (2) ◽  
Author(s):  
R. Shep Melnick
Keyword(s):  
Institutional Environment ◽  
Empirical Studies ◽  
The Political ◽  
Public Officials ◽  
Thick Description ◽  
Past Half Century ◽  
The Us ◽  
The Matrix ◽  
The One ◽  
Better Than

AbstractOver the past half century no judicial politics scholar has been more respected or influential than Martin Shapiro. Yet it is hard to identify a school of thought one could call “Shapiroism.” Rather than offer convenient methodologies or grand theories, Shapiro provides rich empirical studies that show us how to think about the relationship between law and courts on the one hand and politics and governing on the other. Three key themes run through Shapiro’s impressive oevre. First, rather than study courts in isolation, political scientists should view them as “one government agency among many,” and seek to “integrate the judicial system in the matrix of government and politics in which it actually operates.” Law professors may understand legal doctrines better than political scientists, but we know (or should know) the rest of the political system better than they do. Second, although judges inevitably make political decisions, their institutional environment leads them to act differently from other public officials. Most importantly, their legitimacy rests on their perceived impartiality within the plaintiff-defendant-judge triad. The conflict between judges’ role as impartial arbiter and enforcer of the laws of the regime can never be completely resolved and places powerful constraints on their actions. Third, the best way to understand the complex relationship between courts and other elements of the regime is comparative analysis. Shapiro played a major role in resuscitating comparative law, especially in his work comparing the US and the EU. All this he did with a rare combination of thick description and crisp, jargon-free analysis, certainly a rarity the political science of our time.

Machinability and Microstructure of SiC/h-BN Nano-Composites

2008 ◽  
Vol 569 ◽  
pp. 45-48
Author(s):  
Hai Yun Jin ◽  
Guan Jun Qiao ◽  
Zong Ren Peng ◽  
Ji Qiang Gao
Keyword(s):  
Fracture Strength ◽  
Nano Composites ◽  
Weak Interface ◽  
X Ray Diffraction ◽  
X Ray ◽  
Sic Particles ◽  
N2 Atmosphere ◽  
The Matrix ◽  
Better Than

SiC particles coated with nano-BN were synthesized and the machinable SiC/BN ceramic nano-composites were fabricated by Plasma Active Sintering (PAS) in N2 atmosphere. The existing and distribution of h-BN phase were revealed by X-ray diffraction (XRD), and SEM. For the existing of weak interface between h-BN and SiC grains, the machinability of both SiC/BN micro-composites and nano-composites were improved obviously. Because the nano-sized h-BN crystals were homogeneously dispersed around the SiC grains of the matrix, the fracture strength of the nano-composites was better than the SiC/h-BN micro-composite.

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