A new multi-step method for solving nonlinear systems with high efficiency indices

Erfanifar, Raziyeh; Hajarian, Masoud

doi:10.1007/s11075-023-01735-2

Cited by 8 publications

(6 citation statements)

References 30 publications

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“…Examples complement the theory. Due to its generality, this article's technique can be applied on other method with inverses along the same lines [6,14,19,32,39,43,[45][46][47][48][49]. It is worth noting that the method (8) should be used for sufficiently small m. Otherwise, if m is very large, it may be as expensive to implement as method (4).…”

Section: Discussionmentioning

confidence: 99%

On the Kantorovich Theory for Nonsingular and Singular Equations

Argyros,

George,

Regmi

et al. 2024

Axioms

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We develop a new Kantorovich-like convergence analysis of Newton-type methods to solve nonsingular and singular nonlinear equations in Banach spaces. The outer or generalized inverses are exchanged by a finite sum of linear operators making the implementation of these methods easier than in earlier studies. The analysis uses relaxed generalized continuity of the derivatives of operators involved required to control the derivative and on real majorizing sequences. The same approach can also be implemented on other iterative methods with inverses. The examples complement the theory by verifying the convergence conditions and demonstrating the performance of the methods.

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Section: Discussionmentioning

confidence: 99%

On the Kantorovich Theory for Nonsingular and Singular Equations

Argyros,

George,

Regmi

et al. 2024

Axioms

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“…Table 5 demonstrates that the convergence of the proposed methods closely corresponds with the convergence of Newton's method, particularly for values of k ranging from 4 to 5 with the convergence order closely approximating 2. Then, the definition of the derivative according to Fréchet [1,8,11,27,30] is given for the mapping F F ′ (a) =   1 0 0 0 e a 2 0 0 0 (e − 1)a 3 + 1   .…”

Section: Methods (2)mentioning

confidence: 99%

“…The conditions of the Theorem 3 hold provided that ℓ 0 = e − 1, k = 1, a = a(t) = ℓ 0 t. Then, we can have ρ ∈ (0, 0.2909883534). Then, the definition of the derivative according to Fréchet [1,8,11,27,30] is given below for the operator E E ′ (z(w))(ξ) = w(ξ) − 18 1 0 ξτz(τ) 2 w(τ)dτ for each w ∈ K[0, 1]. Therefore, the conditions are validated, since for x * = 0, E ′ (x * (ξ)) = I provided that ℓ 0 = 9, a = a(t) = ℓ 0 t, and k = 1.…”

Section: Methods (2)mentioning

confidence: 99%

Asymptotically Newton-Type Methods without Inverses for Solving Equations

Argyros,

George,

Shakhno

et al. 2024

Mathematics

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The implementation of Newton’s method for solving nonlinear equations in abstract domains requires the inversion of a linear operator at each step. Such an inversion may be computationally very expensive or impossible to find. That is why alternative iterative methods are developed in this article that require no inversion or only one inversion of a linear operator at each step. The inverse of the operator is replaced by a frozen sum of linear operators depending on the Fréchet derivative of an operator. The numerical examples illustrate that for all practical purposes, the new methods are as effective as Newton’s but much cheaper to implement. The same methodology can be used to create similar alternatives to other methods using inversions of linear operators such as divided differences or other linear operators.

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“…They are especially important in chaos theory and complexity, where they can model systems with great sensitivity to initial conditions, e.g., in meteorology, population dynamics, and in the financial sector [9,10]. According to Abel's impossibility theorem [11], there is no algebraic solution to polynomials of degree greater than four expressed in terms of a finite number of additions, subtractions, multiplications, divisions, and root extractions; thus, we need to turn to numerical iterative methods such as Newton's method and fixed-point iteration [12][13][14] to approximate the roots of a general Equation (1) one at a time. Single root finding methods are highly sensitive to the initial guess values used, though, and their local convergence behavior diverges as f ′ (x) approaches zero.…”

Section: Introductionmentioning

confidence: 99%

Q-Analogues of Parallel Numerical Scheme Based on Neural Networks and Their Engineering Applications

Shams,

Carpentieri

2024

Applied Sciences

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Quantum calculus can provide new insights into the nonlinear behaviour of functions and equations, addressing problems that may be difficult to tackle by classical calculus due to high nonlinearity. Iterative methods for solving nonlinear equations can benefit greatly from the mathematical theory and tools provided by quantum calculus, e.g., using the concept of q-derivatives, which extends beyond classical derivatives. In this paper, we develop parallel numerical root-finding algorithms that approximate all distinct roots of nonlinear equations by utilizing q-analogies of the function derivative. Furthermore, we utilize neural networks to accelerate the convergence rate by providing accurate initial guesses for our parallel schemes. The global convergence of the q-parallel numerical techniques is demonstrated using random initial approximations on selected biomedical applications, and the efficiency, stability, and consistency of the proposed hybrid numerical schemes are analyzed.

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A new multi-step method for solving nonlinear systems with high efficiency indices

Cited by 8 publications

References 30 publications

On the Kantorovich Theory for Nonsingular and Singular Equations

On the Kantorovich Theory for Nonsingular and Singular Equations

Asymptotically Newton-Type Methods without Inverses for Solving Equations

Q-Analogues of Parallel Numerical Scheme Based on Neural Networks and Their Engineering Applications

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