High Order Iterative Methods for Nonlinear Egvations

Vaarmann, O

doi:10.1007/978-3-642-46955-8_128

Cited by 2 publications

(5 citation statements)

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“…The performance of the combination of the cubically convergent method (14) with Newton method was superior both in speed and accuracy than single Newton method. These result can partially be found in [6]. These promising results encourage us to carry out the investigation of properties of polyalgorithmic procedures.…”

Section: Resultsmentioning

confidence: 68%

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High Order Iterative Methods for Decomposition‐coordination Problems

Vaarmann

2006

Technological and Economic Development of Economy

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Many real‐life optimization problems are of the multiobjective type and highdimensional. Possibilities for solving large scale optimization problems on a computer network or multiprocessor computer using a multi‐level approach are studied. The paper treats numerical methods in which procedural and rounding errors are unavoidable, for example, those arising in mathematical modelling and simulation. For the solution of involving decomposition‐coordination problems some rapidly convergent interative methods are developed based on the classical cubically convergent method of tangent hyperbolas (Chebyshev‐Halley method) and the method of tangent parabolas (Euler‐Chebyshev method). A family of iterative methods having the convergence order equal to four is also considered. Convergence properties and computational aspects of the methods under consideration are examined. The problems of their global implementation and polyalgorithmic strategy are discussed as well.

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Section: Resultsmentioning

confidence: 68%

“…Therefore (12) has the similar computational costs as Newton method. It can be shown that (12) with k k A Γ = remains faster than Newton method [6]. Using in (8) the approximatioń´2…”

Section: Methodsmentioning

confidence: 99%

High Order Iterative Methods for Decomposition‐coordination Problems

Vaarmann

2006

Technological and Economic Development of Economy

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“…Such methods are appropriate for problems where the other costs dominate the ones of the second derivative evaluation [3,4]. For avoiding the evaluation of second derivatives F can be replaced by a discretization formula containing one additional value of F and F [1,8]. In such a way we get from (2.3) the midpoint method [8] and from (2.4) the method…”

Section: Methodsmentioning

confidence: 99%

“…having the order of convergence order equal to 4. It can be shown that the higher order of convergence then more accurately, in general, the related linear subproblems are to be solved to preserve the order of convergence intrinsic to those standard methods [8]. Assume now that there exist the uniformly bounded inverse operator Γ k as well as the constants M, K, λ, Λ and sequences {γ k } and {b k } satisfying the following inequalities…”

Section: Methodsmentioning

confidence: 99%

“…The method (3.2) -(3.5) converges quadratically. The application of the formula (2.5) in combination with (2.2) for q = 3 to the equation (3.1) can be found in [8]. This variant of the method converges cubically but the execution of the formula for q = 3 is too expensive and it is desirable to find a reasonable compromise between the needed accuracy of approximation and computational cost.…”

Section: Some Rapidly Convergent Methods For Nonlinear Fredholm Integmentioning

confidence: 99%

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Some Rapidly Convergent Methods for Nonlinear Fredholm Integral Equation

Kaldo

Vaarmann

2005

Mathematical Modelling and Analysis

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Many problems in modelling can be reduced to the solution of a nonlinear equation F(x) = 0, where F is a Frechet‐differentiable (as many times as necessary) mapping between Banach spaces X and Y. For solving this equation we consider high order iteration methods of the type xk +1 =xk ‐ Q(xk, Ai k ), i ∈ I, I = {1,…, r}, r ≥ 1, k = 0, 1, …, where Q(x, Ai k ) is an operator from X into itself and Ai k, i ∈ I, are some approximations to the inverse operator(s) occurring in the associated exact method. In particular, this set of methods contains methods with successive approximation of the inverse operator(s) and those based on the use of iterative methods to obtain a cheap solution of limited accuracy for corresponding linear equation(s) at each iteration step. A convergence theorem is proved and computational aspects of the methods under consideration are examined. The solution of nonlinear Fredholm integral equation by means of methods with convergence order p ≥ 2 are considered and possibilities of organizing parallel computation in iteration process are also briefly discussed. Daug modeliavimo problemu galima suformuluoti netiesines lygties F(x) = 0 pavidalu. Čia F yra Banacho erdves X atvaizdavimas i Banacho erdve Y, turintis visas reikalingas Freshe išvestines. Lygčiai F(x) = 0 spresti taikomas aukštosios eiles iteracinis procesas tokio tipo xk + 1 =xk ‐ Q(xk, Ai k ), i ∈ {1,…, r}, k = 0, 1, …, Čia Q(x, Ai k ) yra tam tikras operatorius X → X, Ai k , yra atvirkštinio atvaizdavimo aproksimacijos. Irodyta konvergavimo teorema ir išnagrineti metodu taikymo skaičiavimo aspektai. Aptariamos skaičiavimu lygiagretinimo galimybes, taikant si ulomus metodus netiesinei Fredholmo integralinei lygčiai.

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High Order Iterative Methods for Nonlinear Egvations

Cited by 2 publications

References 2 publications

High Order Iterative Methods for Decomposition‐coordination Problems

High Order Iterative Methods for Decomposition‐coordination Problems

Some Rapidly Convergent Methods for Nonlinear Fredholm Integral Equation

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