scholarly journals A Solution Approach for Lower Hessenberg Linear Systems

2011 ◽  
Vol 2011 ◽  
pp. 1-6 ◽  
Author(s):  
Anthony A. Ruffa
Keyword(s):  
Linear System ◽  
Linear Systems ◽  
Parallel Processors ◽  
Solution Vector ◽  
Solution Approach ◽  
Parallel Solution ◽  
Solution Algorithms ◽  
Right Hand ◽  
The Right

An approach is developed to obtain solutions to lower Hessenberg linear systems with general entries. The approach involves developing solution vectors for an extended lower Hessenberg linear system (having an extra column and an extra introduced unknown) for each nonzero term on the right hand side. The overall solution is then found through superposition and determination of the extra introduced unknown. The approach supports parallel solution algorithms without communication between processors, since each solution vector is computed independently of the others. The number of parallel processors needed will be equal to the number of nonzero right hand side terms.

scholarly journals GMRES Convergence Bounds for Eigenvalue Problems

2018 ◽  
Vol 18 (2) ◽  
pp. 203-222 ◽  
Author(s):  
Melina A. Freitag ◽  
Patrick Kürschner ◽  
Jennifer Pestana
Keyword(s):  
Linear System ◽  
Invariant Subspace ◽  
Linear Systems ◽  
Eigenvalue Problems ◽  
Sharp Decrease ◽  
Inverse Iteration ◽  
Subspace Iteration ◽  
Right Hand ◽  
Polynomial Preconditioner ◽  
The Right

AbstractThe convergence of GMRES for solving linear systems can be influenced heavily by the structure of the right-hand side. Within the solution of eigenvalue problems via inverse iteration or subspace iteration, the right-hand side is generally related to an approximate invariant subspace of the linear system. We give detailed and new bounds on (block) GMRES that take the special behavior of the right-hand side into account and explain the initial sharp decrease of the GMRES residual. The bounds motivate the use of specific preconditioners for these eigenvalue problems, e.g., tuned and polynomial preconditioners, as we describe. The numerical results show that the new (block) GMRES bounds are much sharper than conventional bounds and that preconditioned subspace iteration with either a tuned or polynomial preconditioner should be used in practice.

On the determination of the right-hand side in a parabolic equation

2012 ◽  
Vol 62 (11) ◽  
pp. 1672-1683 ◽  
Author(s):  
A. Ashyralyev ◽  
A.S. Erdogan ◽  
O. Demirdag
Keyword(s):  
Parabolic Equation ◽  
Right Hand ◽  
The Right

scholarly journals (M,β)-Stability of Positive Linear Systems

2016 ◽  
Vol 2016 ◽  
pp. 1-11 ◽  
Author(s):  
Octavian Pastravanu ◽  
Mihaela-Hanako Matcovschi
Keyword(s):  
Linear System ◽  
Linear Systems ◽  
Transient Behavior ◽  
Short Term ◽  
Time Dynamics ◽  
The Right ◽  
Long Term Behavior ◽  
Stability Concept

The main purpose of this work is to show that the Perron-Frobenius eigenstructure of a positive linear system is involved not only in the characterization of long-term behavior (for which well-known results are available) but also in the characterization of short-term or transient behavior. We address the analysis of the short-term behavior by the help of the “(M,β)-stability” concept introduced in literature for general classes of dynamics. Our paper exploits this concept relative to Hölder vectorp-norms,1≤p≤∞, adequately weighted by scaling operators, focusing on positive linear systems. Given an asymptotically stable positive linear system, for each1≤p≤∞, we prove the existence of a scaling operator (built from the right and left Perron-Frobenius eigenvectors, with concrete expressions depending onp) that ensures the best possible values for the parametersMandβ, corresponding to an “ideal” short-term (transient) behavior. We provide results that cover both discrete- and continuous-time dynamics. Our analysis also captures the differences between the cases where the system dynamics is defined by matrices irreducible and reducible, respectively. The theoretical developments are applied to the practical study of the short-term behavior for two positive linear systems already discussed in literature by other authors.

Approximations by multivariate perturbed neural network operators

2017 ◽  
Vol 15 (03) ◽  
pp. 413-432 ◽  
Author(s):  
George A. Anastassiou
Keyword(s):  
Neural Network ◽  
Rate Of Convergence ◽  
Activation Function ◽  
Jackson Type ◽  
Partial Derivatives ◽  
Right Hand ◽  
Hidden Layer ◽  
The Right ◽  
Derivatives Of

This article deals with the determination of the rate of convergence to the unit of each of three newly introduced here multivariate perturbed normalized neural network operators of one hidden layer. These are given through the multivariate modulus of continuity of the involved multivariate function or its high-order partial derivatives and that appears in the right-hand side of the associated multivariate Jackson type inequalities. The multivariate activation function is very general, especially it can derive from any multivariate sigmoid or multivariate bell-shaped function. The right-hand sides of our convergence inequalities do not depend on the activation function. The sample functionals are of multivariate Stancu, Kantorovich and quadrature types. We give applications for the first partial derivatives of the involved function.

scholarly journals A Breakdown-Free Block COCG Method for Complex Symmetric Linear Systems with Multiple Right-Hand Sides

Symmetry ◽  
2019 ◽  
Vol 11 (10) ◽  
pp. 1302
Author(s):  
Hong-Xiu Zhong ◽  
Xian-Ming Gu ◽  
Shao-Liang Zhang
Keyword(s):  
Linear Systems ◽  
Spectral Properties ◽  
Coefficient Matrix ◽  
Complex Symmetric Linear Systems ◽  
Right Hand ◽  
Orthogonality Conditions ◽  
Free Block ◽  
Complex Symmetric ◽  
The Right ◽  
Conjugate Residual

The block conjugate orthogonal conjugate gradient method (BCOCG) is recognized as a common method to solve complex symmetric linear systems with multiple right-hand sides. However, breakdown always occurs if the right-hand sides are rank deficient. In this paper, based on the orthogonality conditions, we present a breakdown-free BCOCG algorithm with new parameter matrices to handle rank deficiency. To improve the spectral properties of coefficient matrix A, a precondition version of the breakdown-free BCOCG is proposed in detail. We also give the relative algorithms for the block conjugate A-orthogonal conjugate residual method. Numerical results illustrate that when breakdown occurs, the breakdown-free algorithms yield faster convergence than the non-breakdown-free algorithms.

Sign in / Sign up

Export Citation Format

Share Document