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  • Articles  (37)
  • Articles: DFG German National Licenses  (37)
  • CR: 5.15  (37)
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  • Mathematics  (37)
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  • Articles: DFG German National Licenses  (37)
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  • Mathematics  (37)
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  • 1
    Electronic Resource
    Electronic Resource
    Sublinear convergence of the Chord method at singular points (1983)
    Springer
    Numerische Mathematik 42 (1983), S. 147-154 
    Details
    ISSN: 0945-3245
    Keywords: AMS (MOS): 65H10 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We give sufficient conditions for convergence of the chord method for a class of singular problems. The rate of convergence is sublinear.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01395307
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  • 2
    Electronic Resource
    Electronic Resource
    A method of solving nonlinear variational problems by nonlinear transformation of the objective functional. Part I (1984)
    Springer
    Numerische Mathematik 43 (1984), S. 59-82 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65K10 ; CR: 5.15 ; 5.41
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A general globally convergent iterative method for solving nonlinear variational problems is introduced. The method is applied to a temperature control problem and to the minimal surface problem. Several aspects of finite element implementation of the method are discussed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01389638
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  • 3
    Electronic Resource
    Electronic Resource
    Das modifizierte Newton-Verfahren in verallgemeinerten Banach-Räumen (1984)
    Springer
    Numerische Mathematik 43 (1984), S. 91-104 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65J15 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A Newton-Kantorowitsch-analysis of the modified Newton's method in generalized Banach-spaces is given. The application of generalized norms — mappings from a linear space into an ordered Banach-space — improves convergence- and existence results as well as error estimates compared with the real-norm theory.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01389640
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  • 4
    Electronic Resource
    Electronic Resource
    A general convergence theorem for the reduced gradient method (1984)
    Springer
    Numerische Mathematik 43 (1984), S. 241-247 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 90C30 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Consider the problem of minimizing a real function subject to linear equality constraints and to nonnegativity bounds on the variables. A convergence theorem is established for a general algorithm model based on the reduced gradient method. The most meaningful assumptions that are made, concern two crucial points: the choice of the independent variables and that of the search direction.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01390125
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  • 5
    Electronic Resource
    Electronic Resource
    Error bounds of Newton type process on Banach spaces (1982)
    Springer
    Numerische Mathematik 39 (1982), S. 175-193 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS) (1980) ; Primary 65J05, 65J15 ; Secondary 47H10 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary For solving a nonlinear operator equationF(x)=0 in Banach spaces, the Newton's method or Newton type methods are important numerical techniques. We use the properties of real equationt=Φ(t) majorizing an operator equationx=Gx to find a fixed point ofG as a solution of equationF(x)=0. Various type of operatorsG are considered in this paper. For a nonlinear operatorG, we would find a real function Φ majorizing the operatorG and it will be related to a rate of convergence $$\omega (r) = \frac{{r^2 }}{{2(r^2 + d)^{1/2} }}.$$ It follows thatG has a fixed point as a solution ofF(x)=0. Practical limitations of error bounds like as in Potra and Pták [5] are discribed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01408692
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  • 6
    Electronic Resource
    Electronic Resource
    Some convergence results for asynchronous algorithms (1982)
    Springer
    Numerische Mathematik 39 (1982), S. 325-340 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65H10 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Description / Table of Contents: Summary In this paper we present convergence results for the asynchronous algorithms based essentially on the notion of classical contraction. We generalize, in particular, all convergence results for those algorithms which are based on the vectorial norm hypothesis, in wide spread use recently. Certain problems, for which the vectorial norm hypothesis can be difficult or even impossible to verify, can nontheless be tackled within the scope of the classical contraction that we adopte.
    Notes: Resumé Nous présentons dans cet article des résultats de convergence des algorithmes asynchrones basés essentiellement sur la notion classique de contraction. Nous généralisons, en particulier, tous les résultats de convergence de ces algorithmes qui font l'hypothèse de contraction en norme vectorielle qui récemment a été très souvant utilisée. Par ailleurs, l'hypothèse de contraction en norme vectorielle peut se trouver difficile, voire impossible à vérifier pour certains problèmes qui peuvent être cependant abordés dans le cadre de la contraction classique que nous adoptons.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01407866
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  • 7
    Electronic Resource
    Electronic Resource
    Local convergence analysis for partitioned quasi-Newton updates (1982)
    Springer
    Numerische Mathematik 39 (1982), S. 429-448 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65K05 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary This paper considers local convergence properties of inexact partitioned quasi-Newton algorithms for the solution of certain non-linear equations and, in particular, the optimization of partially separable objective functions. Using the bounded deterioration principle, one obtains local and linear convergence, which impliesQ-superlinear convergence under the usual conditions on the quasi-Newton updates. For the optimization case, these conditions are shown to be satisfied by any sequence of updates within the convex Broyden class, even if some Hessians are singular at the minimizer. Finally, local andQ-superlinear convergence is established for an inexact partitioned variable metric method under mild assumptions on the initial Hessian approximations.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01407874
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  • 8
    Electronic Resource
    Electronic Resource
    Bisection is optimal (1982)
    Springer
    Numerische Mathematik 40 (1982), S. 111-117 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65H10 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We seek an approximation to a zero of a continuous functionf:[a,b]→ℝ such thatf(a)≦0 andf(b)≧0. It is known that the bisection algorithm makes optimal use ofn function evaluations, i.e., yields the minimal error which is (b−a)/2 n+1, see e.g. Kung [2]. Traub and Wozniakowski [5] proposed using more general information onf by permitting the adaptive evaluations ofn arbitrary linear functionals. They conjectured [5, p. 170] that the bisection algorithm remains optimal even if these general evaluations are permitted. This paper affirmatively proves this conjecture. In fact we prove optimality of the bisection algorithm even assuming thatf is infinitely many times differentiable on [a, b] and has exactly one simple zero.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01459080
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  • 9
    Electronic Resource
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    On a numerical treatment for the curve-tracing of the homotopy method (1983)
    Springer
    Numerische Mathematik 42 (1983), S. 323-329 
    Details
    ISSN: 0945-3245
    Keywords: AMS (MOS): 65K05 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A Newton-like iteration scheme is proposed for the tracing of an implicitly defined smooth curve. This scheme originates from the study of the continuous Newton-Raphson method for underdetermined systems and, hence, inherits the characteristic property of orthogonality. Its domain of attraction is formed and makes it possible to trace this curve more efficiently.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01389577
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  • 10
    Electronic Resource
    Electronic Resource
    A generalized conjugate gradient algorithm for minimization (1980)
    Springer
    Numerische Mathematik 35 (1980), S. 223-230 
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    ISSN: 0945-3245
    Keywords: AMS (MOS): 65K05 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A generalized conjugate gradient algorithm which is invariant to a nonlinear scaling of a strictly convex quadratic function is described, which terminates after at mostn steps when applied to scaled quadratic functionsf: R n →R1 of the formf(x)=h(F(x)) withF(x) strictly convex quadratic andh∈C 1 (R1) an arbitrary strictly monotone functionh. The algorithm does not suppose the knowledge ofh orF but only off(x) and its gradientg(x).
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01396318
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  • 11
    Electronic Resource
    Electronic Resource
    On the rate of superlinear convergence of a class of variable metric methods (1980)
    Springer
    Numerische Mathematik 35 (1980), S. 293-313 
    Details
    ISSN: 0945-3245
    Keywords: AMS (MOS) ; 90C30 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary This paper considers a class of variable metric methods for unconstrained minimization. Without requiring exact line searches each algorithm in this class converges globally and superlinearly on convex functions. Various results on the rate of the superlinear convergence are obtained.
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    URL: http://dx.doi.org/10.1007/BF01396414
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  • 12
    Electronic Resource
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    Quotienten-Differenzen-Algorithmus: Beweis der Regeln von Rutishauser (1982)
    Springer
    Numerische Mathematik 40 (1982), S. 93-98 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65D15 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Description / Table of Contents: Zusammenfassung Der Quotienten-Differenzen-Algorithmus nach Rutishauser ist geeignet zur Bestimmung von Polen meromorpher Funktionen, gegeben durch eine Taylorreihe. Sind mehrere Pole betragsgleich, so kann eine Polynomfolge bestimmt werden, deren Grenzpolynom diese Pole als Nullstellen hat. Die Konvergenz wurde von Rutishauser jedoch nicht bewiesen. Ein Beweis wird in der vorliegenden Arbeit präsentiert.
    Notes: Summary The Quotient-Difference Algorithm of Rutishauser can be used for the determination of poles of a meromorphic function given by its power series. If some of the poles have same modulus, a sequence of polynomials can be determined such that the limiting polynomial has exactly these poles as zeros. The convergence has not been proved by Rutishauser, however. A proof is presented in this paper.
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    URL: http://dx.doi.org/10.1007/BF01459078
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  • 13
    Electronic Resource
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    On the steps of convergence of approximate eigenvectors in the Rayleigh-Schrödinger series (1983)
    Springer
    Numerische Mathematik 42 (1983), S. 31-50 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65J10 ; 41A25 ; 41A65 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Error bounds are given for the iterative computation of the eigenvectors in the Rayleigh-Schrödinger series. These bounds remove the discrepancy in the theoretical behaviour and numerical results, noted by Redont, under the assumption of collectively compact convergence. As a particular case, it follows that the eigenvector in the iterative Galerkin method proposed by Sloan improves upon the eigenvector in the Galerkin method. This is illustrated by numerical experiments.
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    URL: http://dx.doi.org/10.1007/BF01400916
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  • 14
    Electronic Resource
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    Imperfect conjugate gradient algorithms for extended quadratic functions (1983)
    Springer
    Numerische Mathematik 42 (1983), S. 97-105 
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    ISSN: 0945-3245
    Keywords: AMS (MOS): 65K05 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Generalized conjugate gradient algorithms which are invariant to a nonlinear scaling of a strictly convex quadratic function are described. The algorithms when applied to scaled quadratic functionsf∶R n →R 1 of the formf(x)=h(F(x)) withF(x) strictly convex quadratic andh∈C 1(R 1) an arbitrary strictly monotone functionh generate the same direction vectors as for the functionF without perfect steps.
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    URL: http://dx.doi.org/10.1007/BF01400920
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  • 15
    Electronic Resource
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    Sharp error bounds for Newton's process (1980)
    Springer
    Numerische Mathematik 34 (1980), S. 63-72 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65H10 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The method of nondiscrete mathematical induction is applied to the Newton process. The method yields a very simple proof of the convergence and sharp apriori estimates; it also gives aposteriori bounds which are, in general, better than those given in [1].
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01463998
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  • 16
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    Sur la convergence théorique de la méthode du gradient réduit généralisé (1980)
    Springer
    Numerische Mathematik 34 (1980), S. 73-85 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65K05 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The theoretical convergence of Generalized Reduced Gradient Method (GRG) has not been proved; the purpose of this paper is to propose two theoritical and general variants of the original method and a proof of their convergence.
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    URL: http://dx.doi.org/10.1007/BF01463999
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  • 17
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    The numerical solution of the inverse Stefan problem (1980)
    Springer
    Numerische Mathematik 34 (1980), S. 411-429 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65N10 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The inverse Stefan problem can be understood as a problem of nonlinear approximation theory which we solved numerically by a generalized Gauss-Newton method introduced by Osborne and Watson [19]. Under some assumptions on the parameter space we prove its quadratic convergence and demonstrate its high efficiency by three numerical examples.
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    URL: http://dx.doi.org/10.1007/BF01403678
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  • 18
    Electronic Resource
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    Starlike domains of convergence for Newton's method at singularities (1980)
    Springer
    Numerische Mathematik 35 (1980), S. 95-111 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65H10 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Given a solutionx * of a system of nonlinear equationsf with singular Jacobian ∇f(x *) we construct an open starlike domainR of initial points, from which Newton's method converges linearly tox *. Under certain conditions the union of those straight lines throughx *, that do not intersect withR is shown to form a closed set of measure zero, which is necessarily disjoint from any starlike domain of convergence. The results apply to first and higher order singularities.
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    URL: http://dx.doi.org/10.1007/BF01396373
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  • 19
    Electronic Resource
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    Algorithmes mixtes asynchrones. Etude de convergence monotone (1984)
    Springer
    Numerische Mathematik 44 (1984), S. 363-369 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65H10 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Description / Table of Contents: Summary In this paper we introduce two classes of iterative algorithms which we call “Asynchronous mixed algorithms” and we study their convergence under partial ordering. These algorithms can be implemented just as well on monoprocessors as on multiprocessors, and, along with their convergence study, constitute a generalization of the mixed classical “Newton-relaxation” algorithms.
    Notes: Résumé Nous introduisons dans cet article deux classes d'algorithmes itératifs que nous appelons «Algorithmes mixtes asynchrones» et nous en étudierons la convergence selon un ordre partiel. Ces algorithmes sont implémentables aussi bien sur les monoprocesseurs que sur les multiprocesseurs, et avec leur étude de convergence constituent une généralisation des algorithmes mixtes classiques «Newton-relaxation».
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    URL: http://dx.doi.org/10.1007/BF01405568
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  • 20
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    A quadratic approximation method for minimizing a class of quasidifferentiable functions (1984)
    Springer
    Numerische Mathematik 45 (1984), S. 411-430 
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    ISSN: 0945-3245
    Keywords: AMS MOS ; 90C30 ; 65K05 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary An unconstrained nonlinear programming problem with nondifferentiabilities is considered. The nondifferentiabilities arise from terms of the form max [f 1(x), ...,f n (x)], which may enter nonlinearly in the objective function. Local convex polyhedral upper approximations to the objective function are introduced. These approximations are used in an iterative method for solving the problem. The algorithm proceeds by solving quadratic programming subproblems to generate search directions. Approximate line searches ensure global convergence of the method to stationary points. The algorithm is conceptually simple and easy to implement. It generalizes efficient variable metric methods for minimax calculations.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01391417
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  • 21
    Electronic Resource
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    Root finding by divided differences (1981)
    Springer
    Numerische Mathematik 37 (1981), S. 93-104 
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    ISSN: 0945-3245
    Keywords: AMS (MOS): 65H05 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A recursive method is presented for computing a simple zero of an analytic functionf from information contained in a table of divided differences of its reciprocalh=1/f. A good deal of flexibility is permitted in the choice of ordinate and derivative values, and in the choice of the number of previous points upon which to base the next estimate of the required zero. The method is shown to be equivalent to a process of fitting rational functions with linear numerators to data sampled fromf. Asymptotic and regional convergence properties of such a process have already been studied; in particular, asymptotically quadratic convergence is easily obtained, at the expense of only one function evaluation and a moderate amount of “overhead” computation per step. In these respects the method is comparable with the Newton form of iterated polynomial inverse interpolation, while its regional convergence characteristics may be superior in certain circum-stances. It is also shown that the method is not unduly sensitive to round-off errors.
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    URL: http://dx.doi.org/10.1007/BF01396188
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  • 22
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    Superlinear convergence of symmetric Huang's class of methods (1981)
    Springer
    Numerische Mathematik 38 (1981), S. 209-218 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65K05 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary In this paper the problem of minimizing the functionalf:D∈R n →R is considered. Typical assumptions onf are assumed. A class of Quasi-Newton methods, namely Huang's class of methods is used for finding an optimal solution of this problem. A new theorem connected with this class is presented. By means of this theorem some convergence results known up till now only for the methods which satisfy Quasi-Newton condition are extended, that is the results of superlinear convergence of variable metric methods in the cases of exact and asymptotically exact minimization and the so-called direct-prediction case. This theorem allows to interpretate one of the parameters as the scaling parameter.
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    URL: http://dx.doi.org/10.1007/BF01397090
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  • 23
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    Quasi-Newton-Verfahren vom Rang-Eins-Typ zur Lösung unrestringierter Minimierungsprobleme (1981)
    Springer
    Numerische Mathematik 38 (1981), S. 219-228 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65K05 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary In this paper new Quasi-Newton methods of rank-one type for solving the unconstrained minimization problem are developed. The methods belong to the Oren-Luenberger class (for negative parametersΘ k ) and they generate always positive definite updating matrices. Moreover it is shown that these methods are invariant by scaling of the objective function.
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    URL: http://dx.doi.org/10.1007/BF01397091
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  • 24
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    Quasi-Newton-Verfahren vom Rang-Eins-Typ zur Lösung unrestringierter Minimierungsprobleme (1981)
    Springer
    Numerische Mathematik 38 (1981), S. 229-244 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65 K05 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary It is shown that the methods developed in part I of this paper [6] converge at leastn-step quadratically to the global minimum point of the objective functionf, if the Hessian off is uniformly positive definite, uniformly bounded and Lipschitz continuous and if the method is used with restart and asymptotically exact line search.
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    URL: http://dx.doi.org/10.1007/BF01397092
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  • 25
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    Some supplementary results on the 1+ $$\sqrt 2 $$ order method for the solution of nonlinear equations (1982)
    Springer
    Numerische Mathematik 38 (1982), S. 383-392 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65H10 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Recently an iterative method for the solution of systems of nonlinear equations having at leastR-order 1+ $$\sqrt 2 $$ for simple roots has been investigated by the author [7]; this method uses as many function evaluations per step as the classical Newton method. In the present note we deal with several properties of the method such as monotone convergence, asymptotic inclusion of the solution and convergence in the case of multiple roots.
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    URL: http://dx.doi.org/10.1007/BF01396439
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  • 26
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    Approximation of optimal distributed control problems governed by variational inequalities (1982)
    Springer
    Numerische Mathematik 38 (1982), S. 393-416 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 49D05, 49D07, 65K10 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A method for approximating the optimal control and the optimal state for a class of distributed control problems governed by variational inequalities is given. It uses a Rayleigh-Ritz-Galerkin scheme, regularising techniques and a gradient algorithm. A numerical example is given.
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    URL: http://dx.doi.org/10.1007/BF01396440
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  • 27
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    An error analysis for the secant method (1982)
    Springer
    Numerische Mathematik 38 (1982), S. 427-445 
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    ISSN: 0945-3245
    Keywords: AMS(MOS) 65J15, 65G99 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Sharp a priori and a posteriori error bounds are given for the secant method for solving non-linear equations in Banach spaces. The numerical stability of this method is also investigated. The stability results are analogous to those obtained by Lancaster for Newton's method.
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    URL: http://dx.doi.org/10.1007/BF01396443
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  • 28
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    Ein lokal überlinear konvergentes Verfahren zur Bestimmung von Rückkehrpunkten implizit definierter Raumkurven (1982)
    Springer
    Numerische Mathematik 38 (1982), S. 455-465 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65H10, 47H17 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A numerical technique is described for computing turning points of a space curveL implicitly defined by a nonlinear system ofn equations inn+1 variables. The basic idea is a local parametrization ofL where the parameter that gives the next iterate is determined by applying one step of the well-known method for minimizing a real function using cubic Hermite interpolation with two nodes. The method is shown to convergeQ-super-linearly and withR-order of at least two. A numerical example concerning the analysis of nonlinear resistive circuits shows the algorithm to work effectively on real life problems.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01396445
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    Optimal solution of nonlinear equations satisfying a Lipschitz condition (1984)
    Springer
    Numerische Mathematik 43 (1984), S. 225-240 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65 H 10 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary For a given nonnegative ε we seek a pointx * such that |f(x *)|≦ε wheref is a nonlinear transformation of the cubeB=[0,1] m into ℝ (or ℝ p ,p〉1) satisfying a Lipschitz condition with the constantK and having a zero inB. The information operator onf consists ofn values of arbitrary linear functionals which are computed adaptively. The pointx * is constructed by means of an algorithm which is a mapping depending on the information operator. We find an optimal algorithm, i.e., algorithm with the smallest error, which usesn function evaluations computed adaptively. We also exhibit nearly optimal information operators, i.e., the linear functionals for which the error of an optimal algorithm that uses them is almost minimal. Nearly optimal information operators consists ofn nonadaptive function evaluations at equispaced pointsx j in the cubeB. This result exhibits the superiority of the T. Aird and J. Rice procedure ZSRCH (IMSL library [1]) over Sobol's approach [7] for solving nonlinear equations in our class of functions. We also prove that the simple search algorithm which yields a pointx *=x k such that $$\left| {f(x_k )} \right| = \mathop {\min }\limits_{1 \leqq j \leqq n} \left| {f(x_j )} \right|$$ is nearly optimal. The complexity, i.e., the minimal cost of solving our problem is roughly equal to (K/ε) m .
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01390124
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    Solving systems of polynomial equations by bounded and real homotopy (1984)
    Springer
    Numerische Mathematik 43 (1984), S. 397-418 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65H10, 58C99, 55M25 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The homotopy method for solving systems of polynomial equations proposed by Chow, Mallet-Paret and Yorke is modified in two ways. The first modification allows to keep the homotopy solution curves bounded, the second one to work with real polynomials when solving a system of real equations. For the first method numerical results are presented.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01390182
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    On the convergence order of accelerated root iterations (1984)
    Springer
    Numerische Mathematik 44 (1984), S. 463-476 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65H05 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A Gauss-Seidel procedure for accelerating the convergence of the generalized method of the root iterations type of the (k+2)-th order (k∈N) for finding polynomial complex zeros, given in [7], is considered in this paper. It is shown that theR-order of convergence of the accelerated method is at leastk+1+σ n (k), where σ n (k)〉1 is the unique positive root of the equation σ n -σ-k-1 = 0 andn is the degree of the polynomial. The examples of algebraic equations in ordinary and circular arithmetic are given.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01405575
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  • 32
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    Une méthode numérique pour le calcul des points de retournement. Application à un problème aux limites non-linéaire (1981)
    Springer
    Numerische Mathematik 37 (1981), S. 433-444 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65H10 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary An iterative method is presented for determining a turning point in a branch of solutions of an algebraic system of equations depending on a parameter. One shows that the convergence is superlinear and that this method may constitute a complement for the Newton method. With a finite element method, numerical results are given testing the process on the discrete version of a mildly nonlinear boundary value problem to determine the turning point in the positive solution branch.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01400320
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    Nonlinear successive over-relaxation (1984)
    Springer
    Numerische Mathematik 44 (1984), S. 309-315 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65H10 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We study the convergence of Gauss-Seidel and nonlinear successive overrelaxation methods for finding the minimum of a strictly convex functional defined onR n .
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01410114
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    A generalization of regula falsi (1980)
    Springer
    Numerische Mathematik 36 (1980), S. 333-346 
    Details
    ISSN: 0945-3245
    Keywords: AMS (MOS) ; 65H10 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The method of nondiscrete mathematical induction is applied to a multistep variant of the secant method. Optimal conditions for convergence as well as error estimates, sharp in every step, are obtained.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01396659
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    Global approximate Newton methods (1981)
    Springer
    Numerische Mathematik 37 (1981), S. 279-295 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65H 10 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We derive a class of globally and quadratically converging algorithms for a system of nonlinear equations,g(u)=0, whereg is a sufficiently smooth homeomorphism. Particular attention is directed to key parameters which control the iteration. Several examples are given that have been successful in solving the coupled nonlinear PDEs which arise in semiconductor device modelling.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01398257
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    Une méthode numérique pour le calcul des points de retournement. Application à un problème aux limites non-linéaire (1981)
    Springer
    Numerische Mathematik 37 (1981), S. 445-452 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65H10 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A finite element method (P1) with numerical integration for approximating the boundary value problem −Δu=λe u is considered. It is shown that the discrete problem has a solution branch (with turning point) which converges uniformely to a solution branch of the continuous problem. Error estimates are given; for example it is found that $$\left| {\lambda _0 - \lambda _h^0 } \right| = 0(h^{2 - \varepsilon } )$$ , ɛ〉0, whereλ 0 andλ h 0 are critical values of the parameter λ for continuous and discrete problems.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01400321
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    Partitioned variable metric updates for large structured optimization problems (1982)
    Springer
    Numerische Mathematik 39 (1982), S. 119-137 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65H10 ; CR: 5.15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary This paper presents a minimization method based on the idea of partitioned updating of the Hessian matrix in the case where the objective function can be decomposed in a sum of convex “element” functions. This situation occurs in a large class of practical problems including nonlinear finite elements calculations. Some theoretical and algorithmic properties of the update are discussed and encouraging numerical results are presented.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01399316
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