Effect on Spectral Properties by the Splitting Correction Preconditioning for Linear Systems that Arise from Periodic Boundary Problems

2002 ◽  
pp. 234-243
Author(s):  
Shoji Itoh ◽  
Yoshio Oyanagi ◽  
Shao-Liang Zhang ◽  
Makoto Natori
Keyword(s):  
Linear Systems ◽  
Spectral Properties ◽  
Periodic Boundary ◽  
Boundary Problems

On the spectral properties and positivity of solutionsof a periodic boundary value problemfor a second-order functional differential equation

2020 ◽  
pp. 123-130
Author(s):  
Manuel J. Alves ◽  
Sergey M. Labovskiy
Keyword(s):  
Differential Equation ◽  
Green Function ◽  
Differential Operator ◽  
Boundary Value Problem ◽  
Functional Differential Equation ◽  
Spectral Properties ◽  
Periodic Boundary ◽  
Boundary Value ◽  
Functional Differential ◽  
The Green Function

For a functional-differential operator Lu = (1/ρ)(-(pu')' + ∫_0^l▒〖u(s)d_s r(x,s)〗) with symmetry, the completeness and orthogonality of the eigenfunctions is shown. Thepositivity conditions of the Green function of the periodic boundary value problem areobtained.

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.

scholarly journals A Comparison of Two Equivalent Real Formulations for Complex-Valued Linear Systems Part 2: Results

2003 ◽  
Vol 1 (4) ◽  
Author(s):  
Abnita Munankarmy ◽  
Michael Heroux
Keyword(s):  
Linear Systems ◽  
Iterative Methods ◽  
Complex Plane ◽  
Spectral Properties ◽  
Complex Problem ◽  
Linear Solver ◽  
The Real ◽  
Preconditioned Iterative ◽  
Preconditioned Iterative Methods ◽  
Complex Valued

Many iterative linear solver packages focus on real-valued systems and do not deal well with complex-valued systems, even though preconditioned iterative methods typically apply to both real and complex-valued linear systems. Instead, commonly available packages such as PETSc and Aztec tend to focus on the real-valued systems, while complex-valued systems are seen as a late addition. At the same time, by changing the complex problem into an equivalent real formulation (ERF), a real valued solver can be used. In this paper we consider two ERF’s that can be used to solve complex-valued linear systems. We investigate the spectral properties of each and show how each can be preconditioned to move eigenvalues in a cloud around the point (1,0) in the complex plane. Finally, we consider an interleaved formulation, combining each of the previously mentioned approaches, and show that the interleaved form achieves a better outcome than either separate ERF. [This article is the second part of a sequence of reports. See the December 2002 issue for Part 1—Editor.]

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