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PDEs
Published in A. C. Faul, A Concise Introduction to Numerical Analysis, 2018
To achieve stability, the spectral radius of A has to be smaller or equal to one. This is satisfied if and only if µ ≤ 14. Thus, compared to the one-dimensional case where µ ≤ 12 we are further restricted in our choice of Δt ≤ 14(Δx)2. Generally, using the same discretization in space for all space dimensions and employing the forward Euler method, we have µ ≤ 12d where d is the number of space dimensions.
Matrices and Linear Algebra
Published in William S. Levine, The Control Handbook: Control System Fundamentals, 2017
The spectral radius of a square matrix (over C or R) is defined as ρ(A)=limk→∞‖Ak‖1/k. Any induced matrix norm can be used in the definition; the same limit value always exists. The spectral radius equals the largest eigenvalue magnitude; ρ(A) = max{|λ|: det(λI — A) = 0}. All eigenvalues of A lie within the disc {s ∈ ℂ :|s|≤ ρ(A)} in the complex plane. If A is invertible, all eigenvalues of A lie within the annulus {s ∈ ℂ : ρ(A—1) ≤ |s| ≤ ρ(A)} in the complex plane.
Linear Algebra for Quantum Mechanics
Published in Caio Lima Firme, Quantum Mechanics, 2022
The above condition is only met when the spectral radius of A, ρ(A), is less than one. The spectral radius of a square matrix is the largest value of its eigenvalues. ρ(A)=max{|λ1|,...,|λn|}
A General Modeling Framework for Network Autoregressive Processes
Published in Technometrics, 2023
Hang Yin, Abolfazl Safikhani, George Michailidis
Throughout the article, we use to denote the matrix induced infinity norm of matrix , that is, . We use , ǁAǁ and to denote the element-wise max norm, the operator norm and Frobenius norm of A, respectively. We use to denote the ith unit vector in . For matrices, we use to denote element-wise convergence in probability, and to denote convergence in distribution. For a symmetric or Hermitian matrix A, we denote its spectral radius by , where the spectral radius of a square matrix is the maximum of the absolute values of its eigenvalues.
Performance analysis and optimization of a retrial queue with working vacations and starting failures
Published in Mathematical and Computer Modelling of Dynamical Systems, 2019
and is a zero matrix of size . The spectral radius of the matrix is less than one if the stability condition is satisfied. We numerically compute matrix using the iterative method. Starting with the initial value , the following iteration is executed until the sum of the absolute value of all elements of the matrix is smaller than , i.e., .
Numerical analysis on the applicability of sorption isotherm models in aquifers and its correlation with recharged water movement
Published in ISH Journal of Hydraulic Engineering, 2022
Nitha Ayinippully Nalarajan, Suresh Kumar Govindarajan, Indumathi M Nambi
Also, Youssef (2012) explained the necessary and sufficient condition for the convergence of a given iterative method by ensuring the corresponding iteration matrix’s spectral radius (rs) to be less than one. Smaller rs offers faster convergence of the corresponding iteration scheme. Hence, rs were computed for the iteration matrices of Equation (1) and Equation (3) to obtain a value less than one.