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  • 1
    Electronic Resource
    Electronic Resource
    Optimum first and second order extrapolations of Successive Overrelaxation type methods for certain classes of matrices (1989)
    Springer
    BIT 29 (1989), S. 477-490 
    Details
    ISSN: 1572-9125
    Keywords: Primary 65F10 ; 5.14
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper deals with the iterative solution of the linear systemx=Bx+c when its Jacobi matrixB is weakly 2-cyclic consistently ordered and has a complex eigenvalue spectrum which lies on a straight-line segment. The optimization problem of the following three methods is considered and solved: i) The extrapolation of the optimum Successive Overrelaxation (SOR) ii) The second order extrapolation of a “good” SOR and iii) The second order extrapolation of the Gauss-Seidel method. In addition a variant of the second order methods considered, suitable for the solution of the system even ifB isnot necessarily weakly 2-cyclic consistently ordered, is proposed. Finally a reference to a theoretical comparison of the various optimum methods in the paper is made and their asymptotic convergence factors for selected eigenvalue spectra are illustrated in a Table in support of the theory developed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02219233
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    Paper (German National Licenses)
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  • 2
    Electronic Resource
    Electronic Resource
    On the optimization of a class of second order iterative schemes (1983)
    Springer
    BIT 23 (1983), S. 50-64 
    Details
    ISSN: 1572-9125
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper analyzes and studies the second order scheme of de Pillis' type, which is used for the solution of a linear system. This study leads to a monoparametric family of second order schemes of the aforementioned type and a method of selecting the optimal one is presented. In addition a number of concluding remarks is made and various applications and examples are given, which effectively show in some cases the superiority of our optimal second order scheme over the fastest first order ones like the SOR and AOR schemes. Many points are also made for the possibility of improving on the convergence rates of the optimal scheme of this paper, which suggest further research in this area.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01937325
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    Paper (German National Licenses)
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  • 3
    Electronic Resource
    Electronic Resource
    On an SSOR matrix relationship and its consequences (1989)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 27 (1989), S. 559-570 
    Details
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A matrix relationship connecting the Jacobi and the Symmetric Successive Overrelaxation (SSOR) matrices associated with a k-cyclic consistently ordered matrix A is presented. Next the equivalence of the SSOR method and a certain monoparametric k-step one for the solution of the linear algebraic system Ax = b is established. The aforementioned equivalence can be exploited to derive regions of convergence, optimum parameters involved, etc. of the two iterative methods above. This is done by studying the simplest of the two methods that is the monoparametric k-step one. To show how the idea works the case k = 2 is very briefly discussed.
    Additional Material: 5 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nme.1620270309
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    Paper (German National Licenses)
    Fulltext
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