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  • Articles  (38)
  • AMS(MOS): 65L05  (20)
  • nonlinear programming  (16)
  • Ecology
  • crystal structure
  • 1980-1984  (38)
  • Mathematics  (37)
  • Geosciences  (1)
  • 1
    Electronic Resource
    Electronic Resource
    Stability and accuracy of time discretizations for initial value problems (1982)
    Springer
    Numerische Mathematik 40 (1982), S. 245-296 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65L05
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary This paper continues earlier work by the same authors concerning the shape and size of the stability regions of general linear discretization methods for initial value problems. Here the treatment is extended to cover also implicit schemes, and by placing the accuracy of the schemes into a more central position in the discussion general ‘method-free’ statements are again obtained. More specialized results are additionally given for linear multistep methods and for the Taylor series method.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01400542
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  • 2
    Electronic Resource
    Electronic Resource
    Adams methods for neutral functional differential equations (1982)
    Springer
    Numerische Mathematik 39 (1982), S. 221-230 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65L05 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary In this paper Adams type methods for the special case of neutral functional differential equations are examined. It is shown thatk-step methods maintain orderk+1 for sufficiently small step size in a sufficiently smooth situation. However, when these methods are applied to an equation with a “non-smooth” solution the order of convergence is only one. Some computational considerations are given and numerical experiments are presented.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01408695
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  • 3
    Electronic Resource
    Electronic Resource
    Über die Effektivität einiger nichtlinearer Verfahren bei der numerischen Behandlung steifer Differentialgleichungssysteme (1982)
    Springer
    Numerische Mathematik 40 (1982), S. 169-177 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65L05 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The stability and accuracy of some explicit nonlinear methods for the numerical integration of stiff systems of ordinary differential equations are investigated. It is shown, that in the general case they can produce the essential error. The special class of stiff systems is singled out, for which these methods are highly efficient. Some numerical results are also presented.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01400537
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  • 4
    Electronic Resource
    Electronic Resource
    A semi-implicit mid-point rule for stiff systems of ordinary differential equations (1983)
    Springer
    Numerische Mathematik 41 (1983), S. 373-398 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65L05 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The paper introduces a new semi-implicit extrapolation method especially designed for the numerical solution of stiff systems of ordinary differential equations. The existence of a quadratic asymptotic expansion in terms of the stepsize is shown. Moreover, the new discretization is analyzed in the light of well-known stability models. The efficiency of the new integrator is clearly demonstrated by solving a series of challenging test problems including real life examples.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01418331
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  • 5
    Electronic Resource
    Electronic Resource
    On the uniform power-boundedness of a family of matrices and the applications to one-leg and linear multistep methods (1983)
    Springer
    Numerische Mathematik 42 (1983), S. 1-13 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65L05 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We study the difference equations obtained when a linear multistep method is applied to the scalar test equationdy/dt=λy and constant stepsizeh. LetS be the region of the absolute stability of the method, and letD be a closed subset ofS (on the Riemann sphere $$\mathbb{C}$$ ). It is shown that the solutions of these difference equations are bounded forn≧0, uniformly for λh∈D.S is itself closed in $$\mathbb{C}$$ iff ∂S is free of cusps. The question is studed by means of contractivity analysis and a matrix theorem, derived from the matrix theorem of Kreiss.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01400914
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  • 6
    Electronic Resource
    Electronic Resource
    Order and stepsize control in extrapolation methods (1983)
    Springer
    Numerische Mathematik 41 (1983), S. 399-422 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65L05 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The paper presents a new theory for joint order and stepsize control in extrapolation methods. This theory defines a locally optimal order that can be determined along any trajectory to be computed. In addition, Shannon's information theory is applied to derive some ideal convergence model that is expected to describe the behavior of an extrapolation method over a large set of test problems. Extensive numerical comparisons document a drastic acceleration in stiff integration and a mild acceleration in non-stiff integration by the new device. Moreover, a significant increase in reliability, robustness, and portability of the extrapolation codes is achieved.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01418332
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  • 7
    Electronic Resource
    Electronic Resource
    Split linear multistep methods for the numerical integration of stiff differential systems (1983)
    Springer
    Numerische Mathematik 42 (1983), S. 299-310 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65L05 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A new approach to the problem of numerically integrating stiff differential systems is described. In this approach a linear multistep method (the basic method) is split into a kind of predictor-corrector scheme, where the predictor is also implicit. If this splitting is done in an appropriate manner, the modified method has considerably better stability properties than the basic method. As a result, splitting methods are particularly useful for problems where conventional integration methods experience stability difficulties. In particular some highly stable split linear multistep methods based on backward differentiation formulae are derived and a highly stable variable step implementation is proposed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01389575
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  • 8
    Electronic Resource
    Electronic Resource
    On the integration of stiff systems of O.D.E.s using extended backward differentiation formulae (1980)
    Springer
    Numerische Mathematik 34 (1980), S. 235-246 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65L05 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A class of extended backward differentiation formulae suitable for the approximate numerical integration of stiff systems of first order ordinary differential equations is derived. An algorithm is described whereby the required solution is predicted using a conventional backward differentiation scheme and then corrected using an extended backward differentiation scheme of higher order. This approach allows us to developL-stable schemes of order up to 4 andL(α)-stable schemes of order up to 9. An algorithm based on the integration formulae derived in this paper is illustrated by some numerical examples and it is shown that it is often superior to certain existing algorithms.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01396701
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  • 9
    Electronic Resource
    Electronic Resource
    Asymptotic bounds on the errors of one-step methods (1984)
    Springer
    Numerische Mathematik 45 (1984), S. 201-206 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65L05 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Bulirsch and Stoer have shown how to construct asymptotic upper and lower bounds on the true (global) errors resulting from the solution by extrapolation of the initial value problem for a system of ordinary differential equations. It is shown here how to do this for any one-step method endowed with an asymptotically correct local error estimator. The one-step method can be changed at every step.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01389465
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  • 10
    Electronic Resource
    Electronic Resource
    A globally convergent, implementable multiplier method with automatic penalty limitation (1980)
    Springer
    Applied mathematics & optimization 6 (1980), S. 335-360 
    Details
    ISSN: 1432-0606
    Keywords: nonlinear programming ; multiplier methods ; penalty methods ; global convergence ; penalty limitation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper deals with penalty function and multiplier methods for the solution of constrained nonconvex nonlinear programming problems. Starting from an idea introduced several years ago by Polak, we develop a class of implementable methods which, under suitable assumptions, produce a sequence of points converging to a strong local minimum for the problem, regardless of the location of the initial guess. In addition, for sequential minimization type multiplier methods, we make use of a rate of convergence result due to Bertsekas and Polyak, to develop a test for limiting the growth of the penalty parameter and thereby prevent ill-conditioning in the resulting sequence of unconstrained optimization problems.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01442901
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