Quadrature based Broyden-like method for systems of nonlinear equations

Osinuga, Idowu Ademola; Yusuff, Saidat Olaide

doi:10.19139/soic.v6i1.471

Cited by 9 publications

(10 citation statements)

References 15 publications

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“…where and its first and second derivatives, are calculated at , can be used to derive the Newton's method. For multiple vector variable function , an analogous expression for it [18], as in ( 4 Approximating the integral in (4) by the average of the Trapezoidal and the Simpson 3/8 quadrature rules yields:…”

Section: Derivation Of Trapezoidal-simpson 3/8 Methods (Ts -3/8)mentioning

confidence: 99%

“…For a given + using initial matrix c + = : , an approximated solution for ? can be computed by the iterative schemes as in [18];…”

Section: Derivation Of Trapezoidal-simpson 3/8 Methods (Ts -3/8)mentioning

confidence: 99%

“…A leading trend of new methods developed for the computation of solutions of systems of nonlinear equations for the past few years has been to formulate using the quadrature rules. Some references in relation to developed methods using quadrature rules include [16][17][18][19][20] Newton Cotes quadrature rules are a group of formulas for numerical integration based on evaluating the integrand at equally spaced points. Named after Isaac Newton and Roger Cotes [7], they fit data to local order k polynomial approximants.…”

Section: Introductionmentioning

confidence: 99%

“…where , and its first and second derivatives are evaluated at . In the case of a multiple variable function : → , (2) can be shown [18], to equivalently give:…”

Section: Introductionmentioning

confidence: 99%

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A New Trapezoidal-Simpson 3/8 Method for Solving Systems of Nonlinear Equations

Azure¹,

Stephen²,

Seidu³

2021

AJMCM

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Since its introduction, the Broyden method has been used as the foundation to develop several other Broyden-like methods (or hybrid Broyden methods) which in many cases have turned out to be improved forms of the original method. The modified classical Broyden methods developed by many authors to solve system of nonlinear equations have been effective in overcoming the deficiency of the classical Newton Raphson method, however there are new trends of methods proposed by authors, which have proven to be more efficient than some already existing ones. This work introduces two Broyden-like method developed from a weighted combination of quadrature rules, namely the Trapizoidal, Simpson 3/8 and Simpson 1/3 quadrature rules. Hence the new Broyden-like methods named by the authors as TS-3/8 and TS -1/3 methods have been developed from these rules. After subjecting the proposed methods together with some other existing Broyden-like methods to solve four bench-mark problems, the results of numerical test confirm that the TS-3/8 method is promising (in terms of speed and in most cases accuracy) when compared with other proposed Broyden-like methods. Results gathered after the comparison of TS -3/8 with the other methods revealed that TS -3/8 method performed better than all the methods in terms of speed and the number of iterations needed to reach a solution. On the other hand, TS -1/3 method yielded results for all the benchmark problems but with a relatively higher number of iterations compared with the other methods selected for comparison.

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Section: Derivation Of Trapezoidal-simpson 3/8 Methods (Ts -3/8)mentioning

confidence: 99%

“…For a given + using initial matrix c + = : , an approximated solution for ? can be computed by the iterative schemes as in [18];…”

Section: Derivation Of Trapezoidal-simpson 3/8 Methods (Ts -3/8)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…where , and its first and second derivatives are evaluated at . In the case of a multiple variable function : → , (2) can be shown [18], to equivalently give:…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A New Trapezoidal-Simpson 3/8 Method for Solving Systems of Nonlinear Equations

Azure¹,

Stephen²,

Seidu³

2021

AJMCM

View full text Add to dashboard Cite

show abstract

“…In fact, the exact step length is difficult or even impossible to seek in practical computation. Therefore the most frequently used line search in practice is inexact line search [3,9,11]. Brown and Saad [5] proposed the following line search rule to obtain the step length α k…”

Section: Introductionmentioning

confidence: 99%

Inexact Double Step Length Method For Solving Systems Of Nonlinear Equations

Halilu

Waziri

Musa

2020

Stat., optim. inf. comput.

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In this paper, a single direction with double step length method for solving systems of nonlinear equations is presented. Main idea used in the algorithm is to approximate the Jacobian via acceleration parameter. Furthermore, the two step lengths are calculated using inexact line search procedure. This method is matrix-free, and so is advantageous when solving large-scale problems. The proposed method is proven to be globally convergent under appropriate conditions. The preliminary numerical results reported in this paper using a large-scale benchmark test problems show that the proposed method is practically quite effective.

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