Diagonally Dominant Tridiagonal Matrices; Three Examples

2020 ◽  
pp. 17-33
Author(s):  
Tom Lyche ◽  
Georg Muntingh ◽  
Øyvind Ryan
Keyword(s):  
Tridiagonal Matrices ◽  
Diagonally Dominant

SECOND-ORDER METHOD FOR SOLVING 2D NONLINEAR PARABOLIC DIFFERENTIAL EQUATIONS BASED ON ADI METHOD

2010 ◽  
Vol 01 (01) ◽  
pp. 133-146 ◽  
Author(s):  
SADEGH AMIRI ◽  
S. MOHAMMAD HOSSEINI
Keyword(s):  
Parabolic Problems ◽  
Computationally Efficient ◽  
Parabolic Differential Equations ◽  
Adi Method ◽  
Tridiagonal Matrices ◽  
Nonlinear Parabolic ◽  
Alternating Direction ◽  
Method Of Solution ◽  
Diagonally Dominant ◽  
Bounded Sets

In this article, a new unconditionally stable method based on alternating direction implicit (ADI) method for solving nonlinear unsteady 2D parabolic problems is presented. The order of this method in both time and space is 2. For this development, the Peaceman–Rachford ADI method has been modified suitably to take care of nonlinear forcing term of the equation appropriately. The unconditional stability of the method is shown under discrete L2 norm on bounded sets. The method of solution consists of a number of strictly diagonally dominant tridiagonal matrices, which make the method computationally efficient. The accuracy of the proposed method is also demonstrated by some numerical examples.

Estimates for the Lower Bounds on the Inverse Elements of Strictly Diagonally Dominant Tridiagonal Matrices in Signal Processing

2010 ◽  
Vol 121-122 ◽  
pp. 929-933
Author(s):  
Chuan Dai Dong
Keyword(s):  
Signal Processing ◽  
Lower Bounds ◽  
Practical Applications ◽  
Tridiagonal Matrices ◽  
Diagonally Dominant

In the theory and practical applications, tridiagonal matrices play a very important role. In this paper, Motivated by the references, especially [2], we give the estimates for the lower bounds on the inverse elements of strictly diagonally dominant tridiagonal matrices.

scholarly journals Polyhedral approximations of the semidefinite cone and their application

2021 ◽  
Author(s):  
Yuzhu Wang ◽  
Akihiro Tanaka ◽  
Akiko Yoshise
Keyword(s):  
Initial Approximation ◽  
Semidefinite Relaxation ◽  
Stable Set ◽  
Cutting Plane Methods ◽  
Diagonally Dominant Matrices ◽  
Maximum Stable Set ◽  
Simple Expansion ◽  
Diagonally Dominant ◽  
Polyhedral Approximations ◽  
Semidefinite Cone

AbstractWe develop techniques to construct a series of sparse polyhedral approximations of the semidefinite cone. Motivated by the semidefinite (SD) bases proposed by Tanaka and Yoshise (Ann Oper Res 265:155–182, 2018), we propose a simple expansion of SD bases so as to keep the sparsity of the matrices composing it. We prove that the polyhedral approximation using our expanded SD bases contains the set of all diagonally dominant matrices and is contained in the set of all scaled diagonally dominant matrices. We also prove that the set of all scaled diagonally dominant matrices can be expressed using an infinite number of expanded SD bases. We use our approximations as the initial approximation in cutting plane methods for solving a semidefinite relaxation of the maximum stable set problem. It is found that the proposed methods with expanded SD bases are significantly more efficient than methods using other existing approximations or solving semidefinite relaxation problems directly.

scholarly journals Inversion of Tridiagonal Matrices Using the Dunford-Taylor’s Integral

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 870
Author(s):  
Diego Caratelli ◽  
Paolo Emilio Ricci
Keyword(s):  
Functional Analysis ◽  
Computer Algebra ◽  
Toeplitz Matrices ◽  
Test Cases ◽  
Recursive Procedure ◽  
Tridiagonal Matrices ◽  
Special Cases ◽  
The Matrix ◽  
Matrix Eigenvalues ◽  
Complex Valued

We show that using Dunford-Taylor’s integral, a classical tool of functional analysis, it is possible to derive an expression for the inverse of a general non-singular complex-valued tridiagonal matrix. The special cases of Jacobi’s symmetric and Toeplitz (in particular symmetric Toeplitz) matrices are included. The proposed method does not require the knowledge of the matrix eigenvalues and relies only on the relevant invariants which are determined, in a computationally effective way, by means of a dedicated recursive procedure. The considered technique has been validated through several test cases with the aid of the computer algebra program Mathematica©.

Sign in / Sign up

Export Citation Format

Share Document