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  • Artikel  (159)
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  • Numerical Methods  (135)
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  • 1
    Digitale Medien
    Digitale Medien
    A convergence theorem for Newton-like methods in Banach spaces (1987)
    Springer
    Numerische Mathematik 51 (1987), S. 545-557 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS) ; 65G99 ; 65J15 ; CR: G1.5
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary A convergence theorem for Newton-like methods in Banach spaces is given, which improves results of Rheinboldt [27], Dennis [4], Miel [15, 16] and Moret [18] and includes as a special case an updated (affine-invariant [6]) version of the Kantorovich theorem for the Newton method given in previous papers [35, 36]. Error bounds obtained in [34] are also improved. This paper unifies the study of finding sharp error bounds for Newton-like methods under Kantorovich type assumptions.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01400355
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  • 2
    Digitale Medien
    Digitale Medien
    Solving large nonlinear systems of equations by an adaptive condensation process (1986)
    Springer
    Numerische Mathematik 50 (1986), S. 633-653 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65H10 ; 65H15 ; 65K10 ; 65N20 ; 65N30 ; CR: G1.5
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary We present an algorithm which efficiently solves large nonlinear systems of the form $$Au = F(u), u \in \mathbb{R}^n $$ whenever an (iterative) solver “A −1” for the symmetric positive definite matrixA is available andF'(u) is symmetric. Such problems arise from the discretization of nonlinear elliptic partial differential equations. By means of an adaptive decomposition process we split the original system into a low dimensional system — to be treated by any sophisticated solver — and a remaining high-dimensional system, which can easily be solved by fixed point iteration. Specifically we choose a Newton-type trust region algorithm for the treatment of the small system. We show global convergence under natural assumptions on the nonlinearity. The convergence results typical for trust-region algorithms carry over to the full iteration process. The only large systems to be solved are linear ones with the fixed matrixA. Thus existing software for positive definite sparse linear systems can be used.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01398377
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  • 3
    Digitale Medien
    Digitale Medien
    Comparison of Brown's and Newton's method in the monotone case (1987)
    Springer
    Numerische Mathematik 52 (1987), S. 511-521 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65H10 ; CR: G1.5
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary In many cases when Newton's method, applied to a nonlinear sytemF(x)=0, produces a monotonically decreasing sequence of iterates, Brown's method converges monotonically, too. We compare the iterates of Brown's and Newton's method in these monotone cases with respect to the natural partial ordering. It turns out that in most of the cases arising in applications Brown's method then produces “better” iterates than Newton's method.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01400889
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  • 4
    Digitale Medien
    Digitale Medien
    A reliable argument principle algorithm to find the number of zeros of an analytic function in a bounded domain (1988)
    Springer
    Numerische Mathematik 53 (1988), S. 143-163 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65H05 ; CR: G1.5
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary The argument principle is a natural and simple method to determine the number of zeros of an analytic functionf(z) in a bounded domainD. N, the number of zeros (counting multiplicities) off(z), is 1/2π times the change in Argf(z) asz moves along the closed contour σD. Since the range of Argf(z) is (−π, π] a critical point in the computational procedure is to assure that the discretization of σD, {z i ,i=1, ...,M}, is such that $$|\Delta _{{\text{[z}}_i {\text{,}} {\text{z}}_{i + 1} {\text{]}}} Arg f(z)| \leqq \pi $$ . Discretization control which may violate this inequality can lead to an unreliable algorithm. Mathematical theorems derived for the discretization of σD lead to a completely reliable algorithm to computeN. This algorithm also treats in an elementary way the case when a zero is on or near the contour σD. Numerical examples are given for the reliable algorithm formulated here and it is pointed out in these examples how inadequate discretization control can lead to failure of other algorithms.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01395882
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  • 5
    Digitale Medien
    Digitale Medien
    A consturction of monotonically convergent sequences from successive approximations in certain Banach spaces (1989)
    Springer
    Numerische Mathematik 56 (1989), S. 67-71 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65D15 ; CR: G1.5
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary LetA: X→X (whereX=C q [a, b] orL p [a, b]) be a contraction having the fixed pointf. In this note, using ideas from [1–8], we obtain a modified successive approximation sequence which approximatesf and which has certain properties regarding monotonicity too.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01395778
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  • 6
    Digitale Medien
    Digitale Medien
    Enlargement procedure for resolution of singularities at simple singular solutions of nonlinear equations (1988)
    Springer
    Numerische Mathematik 52 (1988), S. 401-411 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65J15 ; CR: G1.5
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary A solution of a nonlinear equation in Hilbert spaces is said to be a simple singular solution if the Fréchet derivative at the solution has one-dimensional kernel and cokernel. In this paper we present the enlargement procedure for resolution of singularities at simple singular solutions of nonlinear equations. Once singularities are resolved, we can compute accurately the singular solution by Newton's method. Conditions for which the procedure terminates in finite steps are given. In particular, if the equation defined in ℝn is analytic and the simple singular solution is geometrically isolated, the procedure stops in finite steps, and we obtain the enlarged problem with an isolated solution. Numerical examples are given.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01462236
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  • 7
    Digitale Medien
    Digitale Medien
    A projected Newton method for minimization problems with nonlinear inequality constraints (1988)
    Springer
    Numerische Mathematik 53 (1988), S. 377-409 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65H 10 ; CR: G1.5
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary Recently developed projected Newton methods for minimization problems in polyhedrons and Cartesian products of Euclidean balls are extended here to general convex feasible sets defined by finitely many smooth nonlinear inequalities. Iterate sequences generated by this scheme are shown to be locally superlinearly convergent to nonsingular extremals $$\bar u$$ , and more specifically, to local minimizers $$\bar u$$ satisfying the standard second order Kuhn-Tucker sufficient conditions; moreover, all such convergent iterate sequences eventually enter and remain within the smooth manifold defined by the active constraints at $$\bar u$$ . Implementation issues are considered for large scale specially structured nonlinear programs, and in particular, for multistage discrete-time optimal control problems; in the latter case, overall per iteration computational costs will typically increase only linearly with the number of stages. Sample calculations are presented for nonlinear programs in a right circular cylinder in ℝ3.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01396325
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  • 8
    Digitale Medien
    Digitale Medien
    On theR-order of coupled sequences arising in single-step type methods (1988)
    Springer
    Numerische Mathematik 53 (1988), S. 653-661 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65H10 ; 15A48 ; CR: G1.5
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary As shown in preceding papers of the authors, the verification of anR-convergence order τ for sequences coupled by a system (1.1) of basic inequalities can be reduced to the positive solvability of system (3.3) of linear inequalities. Further, the bestR-order $$\bar \tau$$ implied by (1.1) is equal to the minimal spectral radius of certain matrices composed from the exponents occuring in (1.1). Now, these results are proven in a unified and essentially simpler way. Moreover, they are somewhat extended in order to facilitate applications to concrete methods.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01397134
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  • 9
    Digitale Medien
    Digitale Medien
    On the computation of multi-dimensional solution manifolds of parametrized equations (1988)
    Springer
    Numerische Mathematik 53 (1988), S. 165-181 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS):65H10 ; CR: G1.5
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary A new algorithm is presented for computing vertices of a simplicial triangulation of thep-dimensional solution manifold of a parametrized equationF(x)=0, whereF is a nonlinear mapping fromR n toR m ,p=n−m〉1. An essential part of the method is a constructive algorithm for computing moving frames on the manifold; that is, of orthonormal bases of the tangent spaces that vary smoothly with their points of contact. The triangulation algorithm uses these bases, together with a chord form of the Gauss-Newton process as corrector, to compute the desired vertices. The Jacobian matrix of the mapping is not required at all the vertices but only at the centers of certain local “triangulation patches”. Several numerical examples show that the method is very efficient in computing triangulations, even around singularities such as limit points and bifurcation points. This opens up new possibilities for determining the form and special features of such solution manifolds.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01395883
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  • 10
    Digitale Medien
    Digitale Medien
    The random product homotopy and deficient polynomial systems (1987)
    Springer
    Numerische Mathematik 51 (1987), S. 481-500 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65H10 ; 65H15 ; CR: G1.5
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary Most systems ofn polynomial equations inn unknowns arising in applications aredeficient, in the sense that they have fewer solutions than that predicted by the total degree of the system. We introduce the random product homotopy, an efficient homotopy continuation method for numerically determining all isolated solutions of deficient systems. In many cases, the amount of computation required to find all solutions can be made roughly proportional to the number of solutions.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01400351
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