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  • 1
    Electronic Resource
    Electronic Resource
    Springer
    Numerische Mathematik 51 (1987), S. 209-227 
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65F10 ; CR:G1.3
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A generalizeds-term truncated conjugate gradient method of least square type, proposed in [1a, b], is extended to a form more suitable for proving when the truncated version is identical to the full-term version. Advantages with keeping a control term in the truncated version is pointed out. A computationally efficient new algorithm, based on a special inner product with a small demand of storage is also presented. We also give simplified and slightly extended proofs of termination of the iterative sequence and of existence of ans-term recursion, identical to the full-term version. Important earlier results on this latter topic are found in [15, 16, 8 and 11].
    Type of Medium: Electronic Resource
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  • 2
    Electronic Resource
    Electronic Resource
    Springer
    Numerische Mathematik 69 (1994), S. 1-15 
    ISSN: 0945-3245
    Keywords: Mathematics Subject Classification (1991): 65H10; 65F10
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary. The Generalized Conjugate Gradient method (see [1]) is an iterative method for nonsymmetric linear systems. We obtain generalizations of this method for nonlinear systems with nonsymmetric Jacobians. We prove global convergence results.
    Type of Medium: Electronic Resource
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  • 3
    Electronic Resource
    Electronic Resource
    s.l. : American Chemical Society
    The @journal of physical chemistry 〈Washington, DC〉 99 (1995), S. 7028-7035 
    Source: ACS Legacy Archives
    Topics: Chemistry and Pharmacology , Physics
    Type of Medium: Electronic Resource
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  • 4
    Electronic Resource
    Electronic Resource
    s.l. : American Chemical Society
    The @journal of physical chemistry 〈Washington, DC〉 97 (1993), S. 7652-7659 
    Source: ACS Legacy Archives
    Topics: Chemistry and Pharmacology , Physics
    Type of Medium: Electronic Resource
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  • 5
    ISSN: 0022-4731
    Source: Elsevier Journal Backfiles on ScienceDirect 1907 - 2002
    Topics: Biology , Chemistry and Pharmacology
    Type of Medium: Electronic Resource
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  • 6
    Electronic Resource
    Electronic Resource
    Amsterdam : Elsevier
    Journal of Computational Physics 35 (1980), S. 284-289 
    ISSN: 0021-9991
    Source: Elsevier Journal Backfiles on ScienceDirect 1907 - 2002
    Topics: Computer Science , Physics
    Type of Medium: Electronic Resource
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  • 7
    Electronic Resource
    Electronic Resource
    Springer
    Computing 46 (1991), S. 233-252 
    ISSN: 1436-5057
    Keywords: Preconditioned iterative methods ; generalized SSOR methods ; wavefront methods ; 15-point difference methods ; mesh-connected computer architectures
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Klassische Wellenfront-vorkonditionierte iterative Methoden für Differenzmatrizen verwenden Wellenfronten, die auf diagonalen (Linien- oder Flächen-) Ordnungen der Gitterpunkte basieren. Da solche Wellenfronten keine konstante Breite haben, ist es nicht möglich, sie effizient auf Parallelrechner-Architekturen auszuführen. Wir diskutieren verschiedene Methoden, wie man Wellenfronten mit konstanter Breite für elliptische Probleme zweiter Ordnung erhalten kann. Insbesondere diskutieren wir die Anwendung dieser Methoden für neun- (2D) und fünfzehnpunktige (3D) Differenzapproximationen des Laplaceoperators, die für geeignete Wahl der Koeffizienten von vierter Ordnung sind. Wir erhalten vorkonditionierte Methoden mit Wellenfronten als vertikale oder horizontale Linien sowohl in 2D als auch in 3D, die die KonditionszahlO(h −1) haben. Die Methoden benutzen nur Verbindungen zu benachbarten Knoten. Infolgedessen können sie nicht nur auf Rechner-Architekturen mit geteiltem Speicher, sondern auch auf verteilten Systemen, Z.B. vernetzten Parallelrechner-Architekturen, effizient ausgeführt werden.
    Notes: Abstract Classical wavefront preconditioned iteration methods for difference matrices on a rectangular or on a rectangular parallelepipedal domain use wavefronts based on diagonal (line or plane, respectively) orderings of the meshpoint. Since such wavefronts do not have constant widths, they cannot be implemented efficiently on parallel computers. We discuss various methods to get wavefronts with constant width for difference matrices for second order elliptic problems. In particular, we discuss their applications for the nine-point (2D) and 15-point (3D) difference approximations for the Laplacian, which are fourth order accurate for proper choices of the coefficients. It turns out that we can easily get preconditioning methods with wavefronts in the form of vertical or horizontal lines both in 2D and 3D, which have condition numberO(h −1), but for general three space dimensional problems no simple ordering leading to constant plane wavefronts seems to exist in general, for which the corresponding preconditioner has such a small condition number. A crucial property we make use of in the methods is the spectral equivalence between the nine-point and the standard five-point difference matrices and between the 15-point and the standard seven-point difference matrices in two and three space dimensions, respectively. The methods use only nearest neighbor connections and can therefore be implemented efficiently not only on shared memory computers but also on distributed memory computer architectures, such as mesh-connected computer architectures.
    Type of Medium: Electronic Resource
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  • 8
    Electronic Resource
    Electronic Resource
    Springer
    BIT 24 (1984), S. 413-424 
    ISSN: 1572-9125
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Classical discretization error estimates for systems of ordinary differential equations contain a factor exp (Lt), whereL is the Lipschitz constant. For strongly monotone operators, however, one may prove that for aϑ-method, 0〈ϑ〈1/2, the errors are bounded uniformly in time and with errorO(Δt)2, ifϑ=1/2−|O(Δt)|. This was done by this author (1977), for an operator in a reflexive Banach space and includes the case of systems of differential equations as a special case. In the present paper we restate this result as it may have been overlooked and consider also the monotone (inclusive of the conservative) and unbounded cases. We also discuss cases where the truncation errors are bounded by a constant independent of the stiffness of the problem. This extends previous results in [6] and [7]. Finally we discuss a boundary value technique in the context above.
    Type of Medium: Electronic Resource
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  • 9
    Electronic Resource
    Electronic Resource
    Springer
    BIT 14 (1974), S. 279-287 
    ISSN: 1572-9125
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The numerical solution of systems of differential equations of the formB dx/dt=σ(t)Ax(t)+f(t),x(0) given, whereB andA (withB and —(A+A T) positive definite) are supposed to be large sparse matrices, is considered.A-stable methods like the Implicit Runge-Kutta methods based on Radau quadrature are combined with iterative methods for the solution of the algebraic systems of equations.
    Type of Medium: Electronic Resource
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  • 10
    ISSN: 1572-9125
    Keywords: 65F10 ; 65N30
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Standard Galerkin finite element methods or finite difference methods for singular perturbation problems lead to strongly unsymmetric matrices, which furthermore are in general notM-matrices. Accordingly, preconditioned iterative methods such as preconditioned (generalized) conjugate gradient methods, which have turned out to be very successful for symmetric and positive definite problems, can fail to converge or require an excessive number of iterations for singular perturbation problems. This is not so much due to the asymmetry, as it is to the fact that the spectrum can have both eigenvalues with positive and negative real parts, or eigenvalues with arbitrary small positive real parts and nonnegligible imaginary parts. This will be the case for a standard Galerkin method, unless the meshparameterh is chosen excessively small. There exist other discretization methods, however, for which the corresponding bilinear form is coercive, whence its finite element matrix has only eigenvalues with positive real parts; in fact, the real parts are positive uniformly in the singular perturbation parameter. In the present paper we examine the streamline diffusion finite element method in this respect. It is found that incomplete block-matrix factorization methods, both on classical form and on an inverse-free (vectorizable) form, coupled with a general least squares conjugate gradient method, can work exceptionally well on this type of problem. The number of iterations is sometimes significantly smaller than for the corresponding almost symmetric problem where the velocity field is close to zero or the singular perturbation parameter ε=1.
    Type of Medium: Electronic Resource
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