Fast Rational Lanczos Method for the Toeplitz Symmetric Positive Semidefinite Matrix Functions

2021 ◽  
pp. 164-172
Author(s):  
Lei Chen ◽  
Lu Zhang ◽  
Mengjia Wu ◽  
Jianqiang Zhao
Keyword(s):  
Positive Semidefinite ◽  
Lanczos Method ◽  
Matrix Functions ◽  
Positive Semidefinite Matrix ◽  
Semidefinite Matrix

scholarly journals A trace bound for integer-diagonal positive semidefinite matrices

2020 ◽  
Vol 8 (1) ◽  
pp. 14-16
Author(s):  
Lon Mitchell
Keyword(s):  
Positive Semidefinite ◽  
Positive Semidefinite Matrix ◽  
Positive Semidefinite Matrices ◽  
Semidefinite Matrix

AbstractWe prove that an n-by-n complex positive semidefinite matrix of rank r whose graph is connected, whose diagonal entries are integers, and whose non-zero off-diagonal entries have modulus at least one, has trace at least n + r − 1.

scholarly journals Refined Upper Solution Bound of the Continuous Coupled Algebraic Riccati Equation

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Li Wang
Keyword(s):  
Riccati Equation ◽  
Positive Semidefinite ◽  
Algebraic Riccati Equation ◽  
Positive Semidefinite Matrix ◽  
Matrix Inequalities ◽  
Numerical Examples ◽  
Upper Solution ◽  
Solution Bound ◽  
Semidefinite Matrix ◽  
Two Parameter

The continuous coupled algebraic Riccati equation (CCARE) has wide applications in control theory and linear systems. In this paper, by a constructed positive semidefinite matrix, matrix inequalities, and matrix eigenvalue inequalities, we propose a new two-parameter-type upper solution bound of the CCARE. Next, we present an iterative algorithm for finding the tighter upper solution bound of CCARE, prove its boundedness, and analyse its monotonicity and convergence. Finally, corresponding numerical examples are given to illustrate the superiority and effectiveness of the derived results.

scholarly journals A New Algorithm for Positive Semidefinite Matrix Completion

2016 ◽  
Vol 2016 ◽  
pp. 1-5 ◽  
Author(s):  
Fangfang Xu ◽  
Peng Pan
Keyword(s):  
Matrix Completion ◽  
Positive Semidefinite ◽  
Alternating Direction Method ◽  
Low Rank ◽  
System Control ◽  
Positive Semidefinite Matrix ◽  
Linear Least Squares ◽  
Alternating Direction ◽  
Semidefinite Matrix Completion ◽  
Semidefinite Matrix

Positive semidefinite matrix completion (PSDMC) aims to recover positive semidefinite and low-rank matrices from a subset of entries of a matrix. It is widely applicable in many fields, such as statistic analysis and system control. This task can be conducted by solving the nuclear norm regularized linear least squares model with positive semidefinite constraints. We apply the widely used alternating direction method of multipliers to solve the model and get a novel algorithm. The applicability and efficiency of the new algorithm are demonstrated in numerical experiments. Recovery results show that our algorithm is helpful.

scholarly journals The Solvency II Standard Formula, Linear Geometry, and Diversification

2016 ◽  
Author(s):  
Joachim Paulusch
Keyword(s):  
Convex Cone ◽  
Risk Measure ◽  
Positive Semidefinite ◽  
Solvency Ii ◽  
Positive Semidefinite Matrix ◽  
Lp Norm ◽  
Standard Formula ◽  
Risk Aggregation ◽  
Semidefinite Matrix ◽  
Linear Geometry

We introduce the notions of monotony, subadditivity, and homogeneity for functions defined on a convex cone, call functions with these properties diversification functions and obtain the respective properties for the risk aggregation given by such a function. Examples of diversification functions are given by seminorms, which are monotone on the convex cone of non-negative vectors. Any Lp norm has this property, and any scalar product given by a non-negative positive semidefinite matrix as well. In particular, the Standard Formula is a diversification function, hence a risk measure that preserves homogeneity, subadditivity, and convexity.

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