Compatible rate-of-convergence bounds for the grid method in the eigenvalue problem for the Helmholtz equation in a right triangle

1993 ◽  
Vol 66 (2) ◽  
pp. 2165-2171
Author(s):  
Yu. I. Rybak
Keyword(s):  
Eigenvalue Problem ◽  
Helmholtz Equation ◽  
Rate Of Convergence ◽  
Grid Method

The Modified Power Method for Solving the Eigenvalue Problem with the Use of Idempotent Matrix

2015 ◽  
Vol 712 ◽  
pp. 43-48
Author(s):  
Rafał Palej ◽  
Artur Krowiak ◽  
Renata Filipowska
Keyword(s):  
Eigenvalue Problem ◽  
Rate Of Convergence ◽  
Rayleigh Quotient ◽  
Frobenius Norm ◽  
Dominant Eigenvalue ◽  
Power Method ◽  
New Approach ◽  
Idempotent Matrix ◽  
The Given ◽  
Scaling Coefficient

The work presents a new approach to the power method serving the purpose of solving the eigenvalue problem of a matrix. Instead of calculating the eigenvector corresponding to the dominant eigenvalue from the formula , the idempotent matrix B associated with the given matrix A is calculated from the formula , where m stands for the method’s rate of convergence. The scaling coefficient ki is determined by the quotient of any norms of matrices Bi and or by the reciprocal of the Frobenius norm of matrix Bi. In the presented approach the condition for completing calculations has the form. Once the calculations are completed, the columns of matrix B are vectors parallel to the eigenvector corresponding to the dominant eigenvalue, which is calculated from the Rayleigh quotient. The new approach eliminates the necessity to use a starting vector, increases the rate of convergence and shortens the calculation time when compared to the classic method.

A Two-Grid Method of Nonconforming Element Based on the Shifted-Inverse Power Method for the Steklov Eigenvalue Problem

2013 ◽  
Vol 694-697 ◽  
pp. 2918-2921
Author(s):  
Hai Bi
Keyword(s):  
Eigenvalue Problem ◽  
High Efficiency ◽  
Grid Method ◽  
Inverse Power ◽  
Power Method ◽  
Discretization Scheme ◽  
Nonconforming Element ◽  
Steklov Eigenvalue ◽  
Steklov Eigenvalue Problem ◽  
Numer Anal

This paper establishes a new kind of two-grid discretization scheme of nonconforming Crouzeix-Raviart element based on the shifted-inverse power method for the Steklov eigenvalue problem. The error estimates are provided from the work of Yang and Bi (SIAM J. Numer. Anal., 49, pp.1602-1624, 2011). Finally, numerical experiments are reported to illustrate the high efficiency of the two-grid discretization scheme proposed in this paper.

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