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Iterative Methods
Published in Jeffery J. Leader, Numerical Analysis and Scientific Computation, 2022
(i=1,…,n) that is, if the diagonal element of each row equals or exceeds, in absolute value, the sum of the absolute values of all other entries in that row. If the inequality is strict we say that the matrix is strictly diagonally dominant and if it is weak we say that the matrix is weakly diagonally dominant. If A is strictly diagonally dominant then M=−D−1(U+L) is convergent and Jacobi iteration will converge. In fact, the method will frequently converge if A is weakly diagonally dominant. Diagonally dominant matrices occur often in applications involving the numerical solution of partial differential equations. Symmetric diagonally dominant matrices are positive semi-definite, and symmetric strictly diagonally dominant matrices are positive definite.
The Drift-Diffusion Equations and Their Numerical Solution
Published in Dragica Vasileska, Stephen M. Goodnick, Gerhard Klimeck, Computational Electronics, 2017
Dragica Vasileska, Stephen M. Goodnick, Gerhard Klimeck
4.3 To obtain a diagonally dominant coefficient matrix when using a finite difference scheme for the discretization of the Poisson equation, it is necessary to use some linearization scheme. The simplest way to achieve this is to use ψ → ψ + δ, where δ is small. Write down (derive) the linearized Poisson equation using this linearization scheme.Write down (derive) the scaled version of the result obtained in (a).Write the finite-difference approximation for the scaled Poisson equation.If one solves (c) for the improvement δ, show that the resultant coefficient matrix A is diagonally dominant.(Note: Matrix A is diagonally dominant if the absolute value of the sum of the off-diagonal elements in each row is smaller than the absolute value of the corresponding diagonal term.)
Special Matrices
Published in Ravi P. Agarwal, Cristina Flaut, An Introduction to Linear Algebra, 2017
Ravi P. Agarwal, Cristina Flaut
An n × n matrix A = (aij) is said to be diagonally dominant if for every row of the matrix, the magnitude of the diagonal entry in a row is larger than or equal to the sum of the magnitudes of all the other (non-diagonal) entries in that row, i.e., |aii|≥∑j=1,j≠in|aij|for all1≤i≤n. $$ |a_{ii}| \geq \mathop \sum \limits_{j = 1,j \ne i}^{n} |a_{ij} | \text{for all} 1 \leq i \leq n. $$
Multiple-Coil Magnetic Resonance Image Denoising and Deblurring With Nonlocal Total Bounded Variation
Published in IETE Technical Review, 2020
K. Shivarama Holla, P. Jidesh, A. A. Bini
Now taking the first variation of in the above u-subproblem, we get: since the matrix is diagonally dominant, one can opt Gauss-Seidel method for solving it. Here and denote the non-local gradient and Laplacian, respectively. Similarly, the d-subproblem where