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  • Articles  (3)
  • Consistent orderings  (2)
  • Algebraic multilevel  (1)
  • Wiley-Blackwell  (3)
  • Sage Publications
  • 2020-2023
  • 2015-2019
  • 2000-2004
  • 1990-1994  (3)
  • Mathematics  (3)
  • Energy, Environment Protection, Nuclear Power Engineering
Collection
  • Articles  (3)
Source
Keywords
Publisher
  • Wiley-Blackwell  (3)
  • Sage Publications
Years
  • 2020-2023
  • 2015-2019
  • 2000-2004
  • 1990-1994  (3)
Year
Topic
  • Mathematics  (3)
  • Energy, Environment Protection, Nuclear Power Engineering
  • 1
    Electronic Resource
    Electronic Resource
    Algebraic multilevel iteration method for Stieltjes matrices (1994)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 1 (1994), S. 213-236 
    Details
    ISSN: 1070-5325
    Keywords: Optimal order preconditioners ; Algebraic multilevel ; Chebyshev polynomial approximation ; Diagonal compensation ; Approximate inverses ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The numerical solution of elliptic selfadjoint second-order boundary value problems leads to a class of linear systems of equations with symmetric, positive definite, large and sparse matrices which can be solved iteratively using a preconditioned version of some algorithm. Such differential equations originate from various applications such as heat conducting and electromagnetics. Systems of equations of similar type can also arise in the finite element analysis of structures.We discuss a recursive method constructing preconditioners to a symmetric, positive definite matrix. An algebraic multilevel technique based on partitioning of the matrix in two by two matrix block form, approximating some of these by other matrices with more simple sparsity structure and using the corresponding Schur complement as a matrix on the lower level, is considered.The quality of the preconditioners is improved by special matrix polynomials which recursively connect the preconditioners on every two adjoining levels. Upper and lower bounds for the degree of the polynomials are derived as conditions for a computational complexity of optimal order for each level and for an optimal rate of convergence, respectively.The method is an extended and more accurate algebraic formulation of a method for nine-point and mixed five- and nine-point difference matrices, presented in some previous papers.
    Additional Material: 9 Tab.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nla.1680010302
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  • 2
    Electronic Resource
    Electronic Resource
    Approximate factorizations with modified S/P consistently ordered M-factors (1994)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 1 (1994), S. 3-17 
    Details
    ISSN: 1070-5325
    Keywords: M-matrices ; Preconditioning ; Incomplete factorizations ; Consistent orderings ; Matrix graphs ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Preconditioned iterative methods are widely used to solve linear systems such as those arising from the finite element formulation of boundary value problems and approximate factorizations are widely used as preconditioners. The ordering of the unknowns is therefore an important issue because it has a strong influence on the convergence behaviour of the iteration method while it is also a decisive aspect for their parallel implementation. Consistent orderings are attractive for parallel implementations and it has been shown that some subclasses of these orderings also enhance the convergence behaviour of the associated iteration methods. This has in particular been shown for the so-called S/P consistent orderings. A wider definition of this class of orderings has recently been proposed and we investigate here how approximate factorizations should be implemented when using such more general orderings (still called S/P consistent) in order to keep their expected high convergence properties. A simple practical conclusion is suggested, supported by both theoretical and numerical arguments.
    Additional Material: 1 Tab.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nla.1680010103
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  • 3
    Electronic Resource
    Electronic Resource
    S/P images of upper triangular M-matrices (1994)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 1 (1994), S. 19-31 
    Details
    ISSN: 1070-5325
    Keywords: M-matrices ; Preconditioning ; Incomplete factorizations ; Consistent orderings ; Matrix graphs ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Preconditioning by approximate factorizations is widely used in iterative methods for solving linear systems such as those arising from the finite element formulation of many engineering problems. The influence of the ordering of the unknowns on their convergence behaviour has been the subject of recent investigations because of its particular relevance for the parallel implementation of these methods. Consistent orderings are attractive for parallel implementations and subclasses of these orderings have been shown to also enhance the convergence properties of the associated preconditioned iteration scheme. The present contribution is concerned with one such class of orderings, called S/P consistent orderings. More precisely, we review here their known properties and we propose a new definition which enlarges their scope of application. A device, called S/P image of an upper triangular M-matrix, provides a criterion for checking S/P consistency and a means to compute a relevant parameter, called maximal reduction ratio. All known properties of S/P consistent orderings are generalized to the new definition.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nla.1680010104
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