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  • Articles  (3)
  • AMS(MOS): 65F10  (3)
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  • Articles  (3)
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  • 1
    Electronic Resource
    Electronic Resource
    On the sharpness of some upper bounds for the spectral radii of S.O.R. iteration matrices (1980)
    Springer
    Numerische Mathematik 35 (1980), S. 69-79 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65F10 ; CR: 5.14
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Sharpness is shown for three upper bounds for the spectral radii of point S.O.R. iteration matrices resulting from the splitting (i) of a nonsingularH-matrixA into the ‘usual’D−L−U, and (ii) of an hermitian positive definite matrixA intoD−L−U, whereD is hermitian positive definite andL=1/2(A−D+S) withS some skew-hermitian matrix. The first upper bound (which is related to the splitting in (i)) is due to Kahan [6], Apostolatos and Kulisch [1] and Kulisch [7], while the remaining upper bounds (which are related to the splitting in (ii)) are due to Varga [11]. The considerations regarding the first bound yield an answer to a question which, in essence, was recently posed by Professor Ridgway Scott: What is the largest interval in ω, ω≧0, for which the point S.O.R. iterative method is convergent for all strictly diagonally dominant matrices of arbitrary order? The answer is, precisely, the interval (0, 1].
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01396371
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  • 2
    Electronic Resource
    Electronic Resource
    A note on comparison theorems for nonnegative matrices (1985)
    Springer
    Numerische Mathematik 47 (1985), S. 427-434 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65F10 ; CR: G.1.3
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary In a recent paper, [4], Csordas and Varga have unified and extended earlier theorems, of Varga in [10] and Woźnicki in [11], on the comparison of the asymptotic rates of convergence of two iteration matrices induced by two regular splittings. The main purpose of this note is to show a connection between the Csordas-Varga paper and a paper by Beauwens, [1], in which a comparison theorem is developed for the asymptotic rate of convergence of two nonnegative iteration matrices induced by two splittings which are not necessarily regular. Monotonic norms already used in [1] play an important role in our work here.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01389590
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  • 3
    Electronic Resource
    Electronic Resource
    Generalizations of the projection method with applications to SOR theory for hermitian positive semidefinite linear systems (1987)
    Springer
    Numerische Mathematik 51 (1987), S. 123-141 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65F10 ; CR: G1.3
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Ann×n complex matrixB is calledparacontracting if ‖B‖2≦1 and 0≠x∈[N(I-B)]⊥⇒‖Bx‖2〈‖x‖2. We show that a productB=B k B k−1 ...B 1 ofk paracontracting matrices is semiconvergent and give upper bounds on the subdominant eigenvalue ofB in terms of the subdominant singular values of theB i 's and in terms of the angles between certain subspaces. Our results here extend earlier results due to Halperin and due to Smith, Solomon and Wagner. We also determine necessary and sufficient conditions forn numbers in the interval [0, 1] to form the spectrum of a product of two orthogonal projections and hence characterize the subdominant eigenvalue of such a product. In the final part of the paper we apply the upper bounds mentioned earlier to provide an estimate on the subdominant eigenvalue of the SOR iteration matrix ℒω associated with ann×n hermitian positive semidefinite matrixA none of whose diagonal entries vanish.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01396746
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