Convergence and Complexity of Newton Iteration for Operator Equations

Traub, Joseph F.; Woźniakowski, Henryk

doi:10.1145/322123.322130

Cited by 111 publications

(69 citation statements)

References 2 publications

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“…As was shown in [16] (see also, [19]) this is the best possible convergence radius for Newton's Method. Therefore, for vanishing residuals, Theorem 5 merges into the theory of Newton's Method and, as a consequence, Theorem 4 does too.…”

Section: Remarkmentioning

confidence: 68%

Local convergence analysis of inexact Newton-like methods under majorant condition

Ferreira¹,

Gonçalves²

2009

Comput Optim Appl

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We present a local convergence analysis of inexact Newton-like methods for solving nonlinear equations under majorant conditions. This analysis provides an estimate of the convergence radius and a clear relationship between the majorant function, which relaxes the Lipschitz continuity of the derivative, and the nonlinear operator under consideration. It also allow us to obtain some important special cases.

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Section: Remarkmentioning

confidence: 68%

Local convergence analysis of inexact Newton-like methods under majorant condition

Ferreira¹,

Gonçalves²

2009

Comput Optim Appl

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“…Such results are useful in the context of predictor-corrector continuation procedures ( [2]- [4], [6]- [7], [l 1]- [13]). Some of these results have previously been given ( [7], [15]). …”

Section: G(x) = X-(f'(x))-if(x)mentioning

confidence: 93%

“…• We also require an affine invariant form of the mean value theorem of [8, p. 73]. This result is implicit in [4], [7] and [15] …”

Section: Preliminary Lemmasmentioning

confidence: 99%

Affine invariant convergence results for Newton's method

Ypma

1982

BIT

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Abstract.We present a unified derivation of affine invariant convergence results for Newton's method. Initially we derive affine invariant forms of the perturbation lemma and a mean value theorem. With their aid we obtain an optimal radius of convergence for Newton's method, from which further radius of convergence estimates follow. From the Newton-Kantorovitch theorem we derive other estimates of the radius of convergence. We discuss estimation of the parameters in the expressions we have derived.

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“…For the particular case of Newton's method this can be improved [3,13,18,21] to 2 llao-a*ll < min {tr, 3//(G,a*)}' while if more rapid convergence is required the Newton-Kantorovitch theorem [1][2][3][4][5] is satisfied if min ,Ja, _2-x/2 ~ Ilao < { 2#(G, a*)~"…”

Section: '(A*)-~(g'(oo-g'(fl)h ~(G A*)lla-flllmentioning

confidence: 99%

Following paths through turning points

Ypma

1982

BIT

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Abstract.The use of the continuation principle in the solution of systems of nonlinear equations frequently leads to the need to follow trajectories through turning points. This can be done by using a different parametrization at every step along the trajectory. We show how to construct accurate predictors and adaptive steplength estimators for use in predictor-corrector algorithms which follow trajectories in this way.

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Convergence and Complexity of Newton Iteration for Operator Equations

Cited by 111 publications

References 2 publications

Local convergence analysis of inexact Newton-like methods under majorant condition

Local convergence analysis of inexact Newton-like methods under majorant condition

Affine invariant convergence results for Newton's method

Following paths through turning points

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