scholarly journals Fast verification for the Perron pair of an irreducible nonnegative matrix

2021 ◽  
Vol 37 ◽  
pp. 402-415
Author(s):  
Shinya Miyajima
Keyword(s):  
Linear System ◽  
Error Bounds ◽  
Computational Efficiency ◽  
Nonnegative Matrix ◽  
Fast Algorithms ◽  
Coefficient Matrix ◽  
Numerical Results ◽  
Perron Root ◽  
Irreducible Nonnegative Matrix

Fast algorithms are proposed for calculating error bounds for a numerically computed Perron root and vector of an irreducible nonnegative matrix. Emphasis is put on the computational efficiency of these algorithms. Error bounds for the root and vector are based on the Collatz--Wielandt theorem, and estimating a solution of a linear system whose coefficient matrix is an $M$-matrix, respectively. We introduce a technique for obtaining better error bounds. Numerical results show properties of the algorithms.

A modified algorithm for the Perron root of a nonnegative matrix

2011 ◽  
Vol 217 (9) ◽  
pp. 4453-4458 ◽  
Author(s):  
Chengming Wen ◽  
Ting-Zhu Huang
Keyword(s):  
Nonnegative Matrix ◽  
Perron Root ◽  
Modified Algorithm

Numerical simulation of squeezing Cu–Water nanofluid flow by a kernel-based method

2021 ◽  
pp. 2250005
Author(s):  
Mojtaba Fardi ◽  
Yasir Khan
Keyword(s):  
Numerical Simulation ◽  
Hilbert Space ◽  
Linear System ◽  
Numerical Simulations ◽  
Numerical Results ◽  
Kernel Matrix ◽  
Nanofluid Flow ◽  
Parallel Disks ◽  
Engineering Problems ◽  
Well Conditioning

The main aim of this paper is to propose a kernel-based method for solving the problem of squeezing Cu–Water nanofluid flow between parallel disks. Our method is based on Gaussian Hilbert–Schmidt SVD (HS-SVD), which gives an alternate basis for the data-dependent subspace of “native” Hilbert space without ever forming kernel matrix. The well-conditioning linear system is one of the critical advantages of using the alternate basis obtained from HS-SVD. Numerical simulations are performed to illustrate the efficiency and applicability of the proposed method in the sense of accuracy. Numerical results obtained by the proposed method are assessed by comparing available results in references. The results demonstrate that the proposed method can be recommended as a good option to study the squeezing nanofluid flow in engineering problems.

Adaptive complex frequency with V-cycle GMRES for preconditioning 3D Helmholtz equation

Geophysics ◽  
2021 ◽  
pp. 1-40
Author(s):  
Wenhao Xu ◽  
Yang Zhong ◽  
Bangyu Wu ◽  
Jinghuai Gao ◽  
Qing Huo Liu
Keyword(s):  
Linear System ◽  
Helmholtz Equation ◽  
3D Models ◽  
Numerical Results ◽  
Spatial Sampling ◽  
Gmres Method ◽  
Complex Frequency ◽  
Iterative Solver ◽  
Helmholtz Equations ◽  
Sampling Density

Solving the Helmholtz equation has important applications in various areas, such as acoustics and electromagnetics. Using an iterative solver together with a proper preconditioner is key for solving large 3D Helmholtz equations. The performance of existing Helmholtz preconditioners usually deteriorates when the minimum spatial sampling density is small (approximately four points per wavelength [PPW]). To improve the efficiency of the Helmholtz preconditioner at a small minimum spatial sampling density, we have adopted a new preconditioner. In our scheme, the preconditioning matrix is constructed based on an adaptive complex frequency that varies with the minimum spatial sampling density in terms of the number of PPWs. Furthermore, the multigrid V-cycle with a GMRES smoother is adopted to effectively solve the corresponding preconditioning linear system. The adaptive complex frequency together with a GMRES smoother can work stably and efficiently at different minimum spatial sampling densities. Numerical results of three typical 3D models show that our scheme is more efficient than the multilevel GMRES method and shifted Laplacian with multigrid full V-cycle and a symmetric Gauss-Seidel smoother for preconditioning the 3D Helmholtz linear system, especially when the minimum spatial sampling density is large (approximately 120 PPW) or small (approximately 4 PPW).

Thruster and FPSO Hull Interaction

2014 ◽  
Author(s):  
Ye Tian ◽  
Spyros A. Kinnas
Keyword(s):  
Experimental Data ◽  
Hybrid Method ◽  
Computational Efficiency ◽  
Vortex Lattice ◽  
Numerical Results ◽  
Navier Stokes ◽  
Dynamic Positioning ◽  
Lattice Method ◽  
Vortex Lattice Method ◽  
Number Of Cells

A hybrid method which couples a Vortex-Lattice Method (VLM) solver and a Reynolds-Averaged Navier-Stokes (RANS) solver is applied to simulate the interaction between a Dynamic Positioning (DP) thruster and an FPSO hull. The hybrid method could significantly reduce the number of cells to fifth of that in a full blown RANS simulation and thus greatly enhance the computational efficiency. The numerical results are first validated with available experimental data, and then used to assess the significance of the thruster/hull interaction in DP systems.

Legendre Collocation Method to Solve the Riccati Equations with Functional Arguments

2020 ◽  
Vol 17 (10) ◽  
pp. 2050011
Author(s):  
Şuayip Yüzbaşı ◽  
Gamze Yıldırım
Keyword(s):  
Approximate Solutions ◽  
Coefficient Matrix ◽  
Numerical Results ◽  
Matrix Form ◽  
Algebraic Equations ◽  
Mixed Condition ◽  
Collocation Points ◽  
Nonlinear Algebraic Equations ◽  
The Matrix ◽  
Legendre Collocation Method

In this study, a method for numerically solving Riccatti type differential equations with functional arguments under the mixed condition is presented. For the method, Legendre polynomials, the solution forms and the required expressions are written in the matrix form and the collocation points are defined. Then, by using the obtained matrix relations and the collocation points, the Riccati problem is reduced to a system of nonlinear algebraic equations. The condition in the problem is written in the matrix form and a new system of the nonlinear algebraic equations is found with the aid of the obtained matrix relation. This system is solved and thus the coefficient matrix is detected. This coefficient matrix is written in the solution form and hence approximate solution is obtained. In addition, by defining the residual function, an error problem is established and approximate solutions which give better numerical results are obtained. To demonstrate that the method is trustworthy and convenient, the presented method and error estimation technique are explicated by numerical examples. Consequently, the numerical results are shown more clearly with the aid of the tables and graphs and also the results are compared with the results of other methods.

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