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 Some new bounds of the minimum eigenvalue for the Hadamard product of an M-matrix and an inverse M-matrix

2016 ◽  
Vol 14 (1) ◽  
pp. 81-88
Author(s):  
Jianxing Zhao ◽  
Caili Sang
Keyword(s):  
Lower Bounds ◽  
Hadamard Product ◽  
Minimum Eigenvalue ◽  
Numerical Examples ◽  
True Value ◽  
Convergent Sequences

AbstractSome convergent sequences of the lower bounds of the minimum eigenvalue for the Hadamard product of a nonsingular M-matrix B and the inverse of a nonsingular M-matrix A are given by using Brauer’s theorem. It is proved that these sequences are monotone increasing, and numerical examples are given to show that these sequences could reach the true value of the minimum eigenvalue in some cases. These results in this paper improve some known results.

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.

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 Some Bounds on Eigenvalues of the Hadamard Product and the Fan Product of Matrices

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 147
Author(s):  
Qianping Guo ◽  
Jinsong Leng ◽  
Houbiao Li ◽  
Carlo Cattani
Keyword(s):  
Lower Bound ◽  
Spectral Radius ◽  
Upper Bound ◽  
Hadamard Product ◽  
Simple Type ◽  
Nonnegative Matrices ◽  
Minimum Eigenvalue ◽  
Numerical Examples ◽  
Product Of Matrices

In this paper, an upper bound on the spectral radius ρ ( A ∘ B ) for the Hadamard product of two nonnegative matrices (A and B) and the minimum eigenvalue τ ( C ★ D ) of the Fan product of two M-matrices (C and D) are researched. These bounds complement some corresponding results on the simple type bounds. In addition, a new lower bound on the minimum eigenvalue of the Fan product of several M-matrices is also presented. These results and numerical examples show that the new bounds improve some existing results.

scholarly journals Some Bounds on Eigenvalues of the Hadamard Product and the Fan Product of Matrices

2018 ◽  
Author(s):  
Qianping Guo ◽  
Jinsong Leng ◽  
Houbiao Li ◽  
Carlo Cattani
Keyword(s):  
Lower Bound ◽  
Spectral Radius ◽  
Mixed Type ◽  
Hadamard Product ◽  
Simple Type ◽  
Nonnegative Matrices ◽  
Minimum Eigenvalue ◽  
Numerical Examples ◽  
Special Cases ◽  
Product Of Matrices

In this paper, some mixed type bounds on the spectral radius $\rho(A\circ B)$ for the Hadamard product of two nonnegative matrices ($A$ and $B$) and the minimum eigenvalue $\tau(C\star D)$ of the Fan product of two $M$-matrices ($C$ and $D$) are researched. These bounds complement some corresponding results on the simple type bounds. In addition, a new lower bound on the minimum eigenvalue of the Fan product of several $M$-matrices is also presented: $$ \tau(A_{1}\star A_{2}\cdots\star A_{m})\geq \min_{1\leq i\leq n}\{\prod^{m}_{k=1}A_{k}(i,i)-\prod^{m}_{k=1}[A_{k}(i,i)^{P_{k}}-\tau(A_{k}^{(P_{k})})]^\frac{1}{P_{k}}\}, $$ where $A_{1},\ldots, A_{k}$ are $n\times n$ $M$-matrices and $P_{1},\ldots, P_{k}>0$ satisfy $\sum^{m}_{k=1}\frac{1}{P_{k}}\geq 1$. Some special cases of the above result and numerical examples show that this new bound improves some existing results.

scholarly journals Lower bounds for the minimum eigenvalue of Hadamard product of an M-matrix and its inverse

2007 ◽  
Vol 420 (1) ◽  
pp. 235-247 ◽  
Author(s):  
Hou-Biao Li ◽  
Ting-Zhu Huang ◽  
Shu-Qian Shen ◽  
Hong Li
Keyword(s):  
Lower Bounds ◽  
Hadamard Product ◽  
Minimum Eigenvalue

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 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.

Tail of compound distributions and excess time

1996 ◽  
Vol 33 (01) ◽  
pp. 184-195 ◽  
Author(s):  
Xiaodong Lin
Keyword(s):  
Size Distribution ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Claim Size ◽  
Compound Distributions ◽  
Fundamental Identity ◽  
Claim Size Distribution ◽  
Simple Bounds ◽  
New Better Than Used ◽  
Better Than

Bounds on the tail of compound distributions are considered. Using a generalization of Wald's fundamental identity, we derive upper and lower bounds for various compound distributions in terms of new worse than used (NWU) and new better than used (NBU) distributions respectively. Simple bounds are obtained when the claim size distribution is NWUC, NBUC, NWU, NBU, IMRL, DMRL, DFR and IFR. Examples on how to use these bounds are given.

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