Secant Method

The conjugated gradient methods can solve smooth functions with large-scale variables in the specified number of iterations for that they are highly important methods compared to concerning other iterative methods. In this paper, we propose two new conjugate gradient methods, namely the PMDL-1 and PMDL-2. However, for non-smooth functions, which are called conjugate gradient-free derivative methods depending on the projection technique. The two methods give great results compared to the basic PDL method. Moreover, we provide theorems that prove the global convergence between these two methods.

 The main purpose of this paper is to derive two higher order iterative methods for solving nonlinear equations as variants of Mir, Ayub and Rafiq method. These methods are free from higher order derivatives. We obtain these methods by amalgamating Mir, Ayub and Rafiq method with standard secant method and modified secant method given by Amat and Busquier. The order of convergence of new variants are four and six. Also, numerical examples are given to compare the performance of newly introduced methods with the similar existing methods. 2010 AMS Subject Classification: 65H05 Journal of the Institute of Engineering, 2018, 14(1): 179-187

The properties of multilevel optimization problems defined on a hierarchy of discretization grids can be used to define approximate secant equations, which describe the second-order behaviour of the objective function. Following earlier work by Gratton and Toint (2009), we introduce a quasi-Newton method (with a linesearch) and a nonlinear conjugate gradient method that both take advantage of this new second-order information. We then present numerical experiments with these methods and formulate recommendations for their practical use. Keywords nonlinear optimization • multilevel problems • quasi-Newton methods • nonlinear conjugate gradient methods • limited-memory algorithms.

In this paper, we show that the main results of the local convergence theory for least-change secant update methods of Dennis and Walker (SIAM J. Numer. Anal. 18 (1981), 949-987) can be proved using the theory introduced recently by Martinez (Math. Comp. 55 (1990), 143-167). In addition, we exhibit two generalizations of well-known methods whose local convergence can be easily proved using Martínez’s theory.

In this paper we analyze the use of structured quasi-Newton formulae as preconditioners of iterative linear methods when the inexact-Newton approach is employed for solving nonlinear systems of equations. We prove that superlinear convergence and bounded work per iteration is obtained if the preconditioners satisfy a Dennis-Moré condition. We develop a theory of Least-Change Secant Update preconditioners and we present an application concerning a structured BFGS preconditioner.

We construct two optimal Newton-Secant like iterative methods for solving non-linear equations. The proposed classes have convergence order four and eight and cost only three and four function evaluations per iteration, respectively. These methods support the Kung and Traub conjecture and possess a high computational efficiency. The new methods are illustrated by numerical experiments and a comparison with some existing optimal methods. We conclude with an investigation of the basins of attraction of the solutions in the complex plane.

The mathematical modeling of solar cell device is used to demonstrate the equivalent circuit of single-diode solar cell model operating under environmental conditions. In this work, bisection (BM) and secant (SM) Methods currently exists to demonstrate the non-linear electrical behavior of this device. The proposed method is tested in order to solve non-linear equation of PV cell with various values of load resistance are characterized. Comparisons between these numerical examples have been compared. The main idea of the present work is to calculate the PV parameters numerically using different methods with the comparison between them. The results showed that secant method is the best because it has least iteration to obtain the optimal values of the PV parameters.

In general, nonlinear problems cannot be solved analytically. A special theory or method is needed to simplify calculations. Many problems that are too complex, an exact solution is needed to support numerical solution. There are many numerical methods that can be used to solve nonlinear problems, including the Bisection, Secant and Newton methods, also known as the Newton-Raphson method. However, these methods cannot be used for large-scale of nonlinear programming problems; for example the Newton-Raphson method which does not always converge if it takes the wrong initial value. The Newton-Raphson method is widely used to find approximations to the roots of real functions. However, the Newton-Raphson method does not always converge if it takes the wrong initial value. Therefore, it is necessary to develop the Newton-Raphson method without using other methods in order to have a higher convergence. This research is a literature study compiled based on literature references with the i...

Abstract:This paper used the Newton-type Methodfor estimating a single root of nonlinear equations. This method is iterative method and also known as one of the open methods. Open method are fast converging method as compared to closed method but the convergence is not guaranteed. Since open methods are fast convergence methods that is why they are widely used in Applied Mathematics. The proposed numerical technique is second order of convergence, and which is based on Newton Raphson method. The developed algorithm is compared with the well-known Newton Raphson Method and results show that our developed method is much better than the well-known method. Furthermore, examples are also give in order to give more detailed about the present work and reader will come to know that why developed method is much better as compare to well-known method.

Puji syukur kami panjatkan kehadirat Allah SWT atas rahmat dan karuniaNya sehingga penyusun dapat menyelesaikan makalah Metode Newton Raphson dan Metode Secant dengan harapan dapat bermanfaat dalam menambah ilmu dan wawasan. Makalah ini dibuat dalam rangka untuk memenuhi tugas Mata Kuliah Metode Numerik. Dalam membuat makalah ini, dengan keterbatasan ilmu pengetahuan yang kami miliki, kami berusaha mencari sumber data dari berbagai sumber informasi, terutama dari buku dan media internet. Kami juga ingin mengucapkan terima kasih kepada seluruh pihak yang telah ikut serta membantu dalam pembuatan makalah ini dan beberapa sumber yang kami pakai sebagai data dan acuan. Dalam penulisan Makalah ini kami merasa masih banyak kekurangankekurangan baik pada teknis penulisan maupun materi, mengingat akan keterbatasan kemampuan yang kami miliki. Tidak semua bahasan dapat dideskripsikan dengan sempurna dalam makalah ini. Untuk itu kritik dan saran dari semua pihak sangat kami harapkan demi penyempurnaan pembuatan makalah ini. Akhirnya kami selaku penyusun berharap semoga makalah ini dapat memberikan manfaat bagi seluruh pembaca. Yogyakarta,

The secant method is a very effective numerical procedure used for solving nonlinear equations of the form f (x) = 0. In a recent work (A. Sidi, Generalization of the secant method for nonlinear equations. Appl. Math. E-Notes, 8:115-123, 2008), we presented a generalization of the secant method that uses only one evaluation of f (x) per iteration, and we provided a local convergence theory for it that concerns real roots. For each integer k, this method generates a sequence {x n } of approximations to a real root of f (x), where, for n ≥ k, being the polynomial of degree k that interpolates f (x) at x n , x n-1 , . . . , x n-k , the order s k of this method satisfying 1 < s k < 2. Clearly, when k = 1, this method reduces to the secant method with s 1 = (1 + √ 5)/2. In addition, In this note, we study the application of this method to simple complex roots of a function f (z). We show that the local convergence theory developed for real roots can be extended almost as is to complex roots, provided suitable assumptions and justifications are made. We illustrate the theory with two numerical examples.

The cubic convergence of a method inspired by a Hummel and Seebeck for solving variational inclusions, has been showed when the second order Fréchet derivative of some function f satisfies a Lipschitz condition. Here, we prove the superquadratic convergence of this method whenever this second order Fréchet derivative satisfies a Hölder condition.

This paper deals with variational inclusions of the form 0 ∈ Kf (x)g(x) where f is a smooth function from a reflexive Banach space X into a Banach space Y , g is a function from X into Y admitting divided differences and K is a nonempty closed convex cone in the space Y . We show that the previous problem can be solved by a combination of two methods: the Newton and the Secant methods. We show that the order of the semilocal method obtained is equal to . Numerical results are also given to illustrate the convergence at the end of the paper. The variational inclusions were introduced by Robinson as an abstract model for various problems encountered in different fields such as engineering, economy, transport theory, etc.

Dokumen ini membahas metode iterasi titik tetap sebagai bagian dari kajian mata kuliah Analisis Numerik. Penjelasan mencakup syarat kekonvergenan, langkah-langkah pengerjaan, serta implementasi metode untuk mencari akar persamaan non-linear. Selain itu, dokumen ini membandingkan metode tertutup (bagi dua dan posisi palsu) dengan metode terbuka (Newton-Raphson, secant, dan iterasi titik tetap) berdasarkan kecepatan konvergensi, kestabilan, dan kebutuhan informasi seperti turunan fungsi. Kesimpulan menyoroti pentingnya memilih metode yang sesuai dengan karakteristik masalah, akurasi, serta tebakan awal untuk mencapai hasil yang optimal.

Dokumen ini membahas penerapan metode Newton-Raphson dan metode Secant dalam mencari hampiran akar persamaan nonlinier. Penjelasan dimulai dengan langkah-langkah prosedural dari masing-masing metode, di mana metode Newton-Raphson menggunakan informasi turunan fungsi untuk mempercepat konvergensi, sedangkan metode Secant mengeliminasi kebutuhan akan turunan fungsi dengan menggunakan dua titik awal. Hasil analisis menunjukkan bahwa metode Newton-Raphson memiliki konvergensi kuadratik yang lebih cepat, namun memerlukan tebakan awal yang cukup dekat dengan akar agar tidak divergen. Di sisi lain, metode Secant menawarkan fleksibilitas lebih tinggi dengan tingkat konvergensi yang lebih rendah dibandingkan metode Newton-Raphson. Dokumen ini menyimpulkan perbandingan efisiensi dan penerapan kedua metode, terutama pada fungsi-fungsi yang sulit dihitung turunannya.

Newton's method is used to find the roots of a system of equations () 0 f x =. It is one of the most important procedures in numerical analysis, and its applicability extends to differential equations and integral equations. Analysis of the method shows a quadratic convergence under certain assumptions. For several years, researchers have improved the method by proposing modified Newton methods with salutary efforts. A modification of the Newton's method was proposed by McDougall and Wotherspoon [1] with an order of convergence of 1 2 +. On a new type of methods with cubic convergence was proposed by H. H. H. Homeier [2]. In this article, we present a new modification of Newton method based on secant method. Analysis of convergence shows that the new method is cubically convergent. Our method requires an evaluation of the function and one of its derivatives.

Newton's method is used to find the roots of a system of equations () 0 f x =. It is one of the most important procedures in numerical analysis, and its applicability extends to differential equations and integral equations. Analysis of the method shows a quadratic convergence under certain assumptions. For several years, researchers have improved the method by proposing modified Newton methods with salutary efforts. A modification of the Newton's method was proposed by McDougall and Wotherspoon [1] with an order of convergence of 1 2 +. On a new type of methods with cubic convergence was proposed by H. H. H. Homeier [2]. In this article, we present a new modification of Newton method based on secant method. Analysis of convergence shows that the new method is cubically convergent. Our method requires an evaluation of the function and one of its derivatives.

l. INTRODUCTION. Sometimes problems naturally occur in pairs, and it's best to tackle both at the same time. For instance, consider the problem of finding the nth derivative of tan x. It's not hard 'to see that there are polynomials Pn of degree n + 1 for n = 0, 1, ... so that dn dxn tan x = Pn(tan x). This problem has a natural companion: find the nth derivative of sec x. Here there are polynomials Qn of degree n so that dn dxn sec x = Qn(tan x)sec x. The Pn and Qn are different sequences of polynomials, but they are evidently related. The numbers Pn(O)jn! and Q"(O)jn! are the coefficients of xn in the Maclaurin series for tan x and sec x respectively, and their computation is a classical problem.

The notion of least-change secant updates is extended to apply to nonsquare matrices in a way appropriate for quasi-Newton methods used to solve systems of nonlinear equations that depend on parameters. Extensions of the widely used least-change secant updates for square matrices are given. A local convergence analysis for certain paradigm iterations is outlined as motivation for the use of these updates, and numerical experiments involving these iterations are discussed. Key words, least-change secant updates, quasi-Newton updates, parameter-dependent systems AMS(MOS) subject classification. 65H10 1. Introduction. Quasi-Newton methods are very widely used iterative methods for solving systems of nonlinear algebraic equations. The basic form of a quasi-Newton method for solving F(x) 0, F R n-. Rn, is (1.1) Xk+: =xk-B:F(xk), in which Bk ,. F(xk) E Rnn, the Jacobian (matrix) of F at xk. For practical success, it is usually essential to augment this basic form with procedures for modifying the step-B:F(xk) to ensure progress from bad starting points, but we need not consider such procedures here. For a general reference on all aspects of quasi-Newton methods, see Dennis and Schnabel [11]. The most effective quasi-Newton methods are those in which each successive Bk+: is determined as a least-change secant update of its predecessor Bk. As the name suggests, Bk+l is determined as a least-change secant update of Bk by making the least possible change in Bk (as measured by a suitable matrix norm) which incorporates current secant information (usually expressed in terms of successive x-and F-values) and other available information about the structure of F. There are also notable updates which, strictly speaking, are least-change inverse secant updates obtained in an analogous way by making the least possible change in B-:. When speaking generically of least-change secant updates, we intend to include these. In [10], Dennis and Schnabel precisely formalize the notion of a least-change secant update and show how the most widely used updates can be derived as least-change secant updates. In [12], Dennis and Walker show that least-change secant update methods, i.e., quasi-Newton methods which use least-change secant updates, have desirable convergence properties in general.

The notion of least-change secant updates is extended to apply to nonsquare matrices in a way appropriate for quasi-Newton methods used to solve systems of nonlinear equations that depend on parameters. Extensions of the widely used least-change secant updates for square matrices are given. A local convergence analysis for certain paradigm iterations is outlined as motivation for the use of these updates, and numerical experiments involving these iterations are discussed. Key words, least-change secant updates, quasi-Newton updates, parameter-dependent systems AMS(MOS) subject classification. 65H10 1. Introduction. Quasi-Newton methods are very widely used iterative methods for solving systems of nonlinear algebraic equations. The basic form of a quasi-Newton method for solving F(x) 0, F R n-. Rn, is (1.1) Xk+: =xk-B:F(xk), in which Bk ,. F(xk) E Rnn, the Jacobian (matrix) of F at xk. For practical success, it is usually essential to augment this basic form with procedures for modifying the step-B:F(xk) to ensure progress from bad starting points, but we need not consider such procedures here. For a general reference on all aspects of quasi-Newton methods, see Dennis and Schnabel [11]. The most effective quasi-Newton methods are those in which each successive Bk+: is determined as a least-change secant update of its predecessor Bk. As the name suggests, Bk+l is determined as a least-change secant update of Bk by making the least possible change in Bk (as measured by a suitable matrix norm) which incorporates current secant information (usually expressed in terms of successive x-and F-values) and other available information about the structure of F. There are also notable updates which, strictly speaking, are least-change inverse secant updates obtained in an analogous way by making the least possible change in B-:. When speaking generically of least-change secant updates, we intend to include these. In [10], Dennis and Schnabel precisely formalize the notion of a least-change secant update and show how the most widely used updates can be derived as least-change secant updates. In [12], Dennis and Walker show that least-change secant update methods, i.e., quasi-Newton methods which use least-change secant updates, have desirable convergence properties in general.

In this paper, we construct an iterative method with memory based on the Newton-Secants method to solve nonlinear equations. This proposed method has fourth order convergence and costs only three functions evaluation per iteration and without any evaluation of the derivative function. Acceleration of the convergence speed is achieved by an appropriate variation of a free parameter in each step. This self-accelerator parameter is estimated using Newtons interpolation polynomial of third degree. The order of convergence is increased from 4 to 5.23 without any extra function evaluation. This method has the efficiency index equal to 5.2313≈1.7358. We describe the analysis of the proposed method along with numerical experiments including comparison with the existing methods. Finally, the attraction basins of the proposed method are shown and compared with other existing methods.

In this paper, we modify the Newton-Secant method with third order of convergence for finding multiple roots of nonlinear equations. Per iteration this method requires two evaluations of the function and one evaluation of its first derivative. This method has the efficiency index equal to 3 1 3 ≈ 1.44225. We describe the analysis of the proposed method along with numerical experiments including comparison with existing methods. Moreover, the dynamics of the proposed method are shown with some comparisons to the other existing methods. Keywords Multi-point iterative methods • Newton-Secant method • multiple roots • Basin of attraction.

We construct two optimal Newton-Secant like iterative methods for solving non-linear equations. The proposed classes have convergence order four and eight and cost only three and four function evaluations per iteration, respectively. These methods support the Kung and Traub conjecture and possess a high computational efficiency. The new methods are illustrated by numerical experiments and a comparison with some existing optimal methods. We conclude with an investigation of the basins of attraction of the solutions in the complex plane.

Using chain rule, we propose a modified secant equation to get a more accurate approximation of the second curvature of the objective function. Then, based on this modified secant equation we present a new BFGS method for solving unconstrained optimization problems. The proposed method makes use of both gradient and function values, and utilizes information from two most recent steps, while the usual secant relation uses only the latest step information. Under appropriate conditions, we show that the proposed method is globally convergent without convexity assumption on the objective function. Comparative numerical results show computational efficiency of the proposed method in the sense of the Dolan-Moré performance profiles.

Using Taylor's series we propose a modified secant relation to get a more accurate approximation of the second curvature of the objective function. Then, based on this modified secant relation we present a new BFGS method for solving unconstrained optimization problems. The proposed method make use of both gradient and function values while the usual secant relation uses only gradient values. Under appropriate conditions, we show that the proposed method is globally convergent without needing convexity assumption on the objective function. Comparative results show computational efficiency of the proposed method in the sense of the Dolan-Moré performance profiles.

In order to get a higher order accuracy of approximating the Hessian matrix of the objective function, we use the chain rule and propose two modified secant equations. An interesting property of the proposed methods is that these utilize information from two most recent steps where the usual secant equation uses only the latest step information. The other point of interest to one of the proposed methods is that it makes use of both gradient and function value information. We show that the modified BFGS methods based on the new secant equations are globally convergent. The presented experimental results illustrate that the proposed methods are efficient.

Using Taylor&#39;s series we propose a modified secant relation to get a more accurate approximation of the second curvature of the objective function. Then, based on this modified secant relation we present a new BFGS method for solving unconstrained optimization problems. The proposed method make use of both gradient and function values while the usual secant relation uses only gradient values. Under appropriate conditions, we show that the proposed method is globally convergent without needing convexity assumption on the objective function. Comparative results show computational efficiency of the proposed method in the sense of the Dolan-More performance profiles.

Numerical solution of nonlinear least-squares problems is an important computational task in science and engineering. Effective algorithms have been developed for solving nonlinear least squares problems. The structured secant method is a class of efficient methods developed in recent years for optimization problems in which the Hessian of the objective function has some special structure. A primary and typical application of the structured secant method is to solve the nonlinear least squares problems. We present an exact penalty method for solving constrained nonlinear leastsquares problems, when the structured projected Hessian is approximated by a projected version of the structured BFGS formula and give its local two-step Q-superlinear convergence. For robustness, we employ a special nonsmooth line search strategy, taking account of the least squares objective. We discuss the comparative results of the testing of our programs and three nonlinear programming codes from KNITRO on some randomly generated test problems due to Bartels and Mahdavi-Amiri. Numerical results also confirm the practical relevance of our special considerations for the inherent structure of the least squares.

Abstract: A system of simultaneous multi-variable nonlinear equations can be solved by Newton’s method with local q-quadratic convergence if the Jacobian is analytically available. If this is not the case, then quasi-Newton methods with local q-superlinear convergence give solutions by approximating the Jacobian in some way. Unfortunately, the quasi-Newton condition (Secant equation) does not
completely specify the Jacobian approximate in multi-dimensional cases, so its full-rank update is not possible with classic variants of the method. The suggested new iteration strategy (“T-Secant”) allows for a full-rank update of the Jacobian approximate in each iteration by determining two independent approximates for the solution. They are used to generate a set of new independent trial approximates;
then, the Jacobian approximate can be fully updated. It is shown that the T-Secant approximate is in the vicinity of the classic quasi-Newton approximate, providing that the solution is evenly surrounded by the new trial approximates. The suggested procedure increases the superlinear convergence of the Secant method (φS = 1.618 . . .) to super-quadratic (φT = φS + 1 = 2.618 . . .) and the quadratic convergence of the Newton method (φN = 2) to cubic (φT = φN + 1 = 3) in one-dimensional cases.
In multi-dimensional cases, the Broyden-type efficiency (mean convergence rate) of the suggested method is an order higher than the efficiency of other classic low-rank-update quasi-Newton methods, as shown by numerical examples on a Rosenbrock-type test function with up to 1000 variables. The geometrical representation (hyperbolic approximation) in single-variable cases helps explain the basic operations, and a vector-space description is also given in multi-variable cases.

A new secant-method based numerical procedure (T-secant method) with super-quadratic convergence (convergence rate 2.618) has been developed for least-squares solving of simultaneous multi-variable nonlinear equations. The basic idea of the procedure is that the original secant equations are modified by a non-uniform scaling transformation "T", an additional new estimate is determined in each iteration and a completely new set of trial solutions is constructed for the next iteration. The suggested method seems to eliminate the unstable behavior of the traditional method and the efficiency is an order higher than Broyden's methods in case of the numerical example. The vector-based interpretation and its geometrical representation helps to explain the basic operations. The performance of the new method is demonstrated by the results of numerical tests with a Rosenbrock-type test-function with up to 1000 unknown variables.

A new secant-method based numerical procedure (T-secant method) with super-quadratic convergence (convergence rate 2.618) has been developed for least-squares solving of simultaneous multi-variable nonlinear equations. The basic idea of the procedure is that the original secant equations are modified by a non-uniform scaling transformation “T”, an additional new estimate is determined in each iteration and a completely new set of trial solutions is constructed for the next iteration. The suggested method seems to eliminate the unstable behavior of the traditional method and the efficiency is an order higher than Broyden’s methods in case of the numerical example. The vector-based interpretation and its geometrical representation helps to explain the basic operations. The performance of the new method is demonstrated by the results of numerical tests with a Rosenbrock-type test-function with up to 1000 unknown variables.

This paper presents a new diagonal quasi-Newton method for solving unconstrained optimization problems based on the weak secant equation. The new method uses new criteria to generate the Hessian approximation to control the diagonal elements. We establish the global convergence of the proposed method with the Armijo line search. Numerical results on a collection of standard test problems show that the proposed method is superior to several certain diagonal methods.

This paper presents a modified quasi-Newton method for structured unconstrained optimization. The usual SQN equation employs only the gradients, but ignores the available function value information. Several researchers paid attention to other secant conditions to get a better approximation of the Hessian matrix of the objective function. Recently Yabe et al. (2007) [6] proposed the modified secant condition which uses both gradient and function value information in order to get a higher-order accuracy in approximating the second curvature of the objective function. In this paper, we derive a new progressive modified SQN equation, with a vector parameter which use both available gradient and function value information, that maintains most properties of the usual and modified structured quasi-Newton methods. Furthermore, local and superlinear convergence of the algorithm is obtained under some reasonable conditions.

This ‎ paper ‎ deals with the study of relaxed conditions for semi-local convergence for a general iterative method, k-step Newton&#39;s method, using ‎majorizing‎ sequences. Dynamical behavior of the mentioned method is also analyzed via Julia set and basins of attraction. Numerical examples of nonlinear systems of equations will be examined to clarify the given theory.

The point of this short note concerns with two facts on the scheme of secant loci. The first one is an attempt to describe the tangent cone of these schemes globally and the second one is a comparison on the dimension of the tangent spaces of various schemes of secant Loci. Let C be a smooth projective algebraic curve of genu g; W 0 g-1 (C) its theta divisor and L ∈ W 0 g-1 (C) be a multiple point of the theta divisor. Based on a classical and nice result of Bernhard Riemann, the tangent cone of ). Subsequently, Arbarello, Cornalba, Griffiths and Harris used the scheme of linear series, G r d (C)'s, to give a global description of the tangent cone of the Brill-Noether schemes, W r d , at their multiple points when r and d ranges in 1 ≤ 2r ≤ d ≤ g -1. The scheme of secant loci of globally generated line bundles on C, being as a generalization of the classical Brill-Noether varieties, was under focus of some authors beginning by M. Coppens in 1990's to recently by M. Aprodu and E. Sernesi. Marc Coppens, M. E. Huibregetse and T. Johnsen, studying the local behavior of these schemes, have given descriptions of their tangent space and tangent cones at their various points, in terms of their local defining equations. The first aim of this note is to describe the tangent cone of the scheme of secant loci', globally. In order to do so the method of [3] in constructing linear series G r d , goes verbatim to construct analogous schemes on the varieties of secant divisors. The resulting spaces enjoy a powerful universal property. Based on this property; these schemes, so called "the scheme of divisor series" would be used to obtain a global description for the tangent cones of the scheme of secant divisors. W. Fulton and etal., established inequalities within the dimension of various Brill-Noether varieties in . The relations have been extended recently to the varieties of secant loci by M. Aprodu and E. Sernesi in . Inspired by their results, we report in Theorem 3.1 similar inequalities within dim , where x is a general point of C. This is the second aim of this paper. As a corollary to this result, the smoothness of V r d (Γ), when V r d (Γ) is of expected dimension, implies the same property for V r d+1 (Γ) and V r d (Γ(-x)).

We present a novel way of reusing the Krylov information generated by GMRES for solving the linear system arising within a Newton method. Our approach departs from the theory of ,ecant preconditioners developed by Martinez and then combines secant updates of the Hessenberg matrix generated by the Arnoldi process in GMRES, the Richardson iteration and limited memory quasi-Newton compact representations to generate descent directions for each Newton step. The proposed method allows to reflect-without explicitly computing them-secant updates of the Jacobian matrix that lead us to skip GM RES for the benefit of satisfying the Dembo-Eisenstat-Steihaug condition. Hence, the resulting method turns out to be computationally more economical than traditional inexact Newton implementations. Computational experiments reveal the suitability of this approach for large scale problems in several application contexts.

m this paper, we use the Secant method to find a solution of a nonlinear operator equation in Banach spaces. A semilocal convergence result is obtained. For that, we consider a condition for divided differences which generalizes the usual once, i.e., Lipschitz continuous or Hiilder continuous conditions. Besides, we apply our results to approximate the solution of a nonlinear equation.

The secant method is a very effective numerical procedure used for solving nonlinear equations of the form f (x) = 0. It is derived via a linear interpolation procedure and employs only values of f (x) at the approximations to the root of f (x) = 0, hence it computes f (x) only once per iteration. In this note, we generalize it by replacing the relevant linear interpolant by a suitable (k + 1)-point polynomial of interpolation, where k is an integer at least 2. Just as the secant method, this generalization too enjoys the property that it computes f (x) only once per iteration. We provide its error in closed form and analyze its order of convergence s k. We show that this order of convergence is greater than that of the secant method, and it increases towards 2 as k → ∞. (Indeed, s 7 = 1.9960 • • • , for example.) This is true for the efficiency index of the method too. We also confirm the theory via an illustrative example.

In this paper, we propose a novel modified secant method to compute the flow fair share rate within the framework of the core-stateless fair queueing 1]. The geometric explanation and numerical results demonstrate that the proposed method possesses better performance in terms of accuracy and convergence than that proposed in 1].

Multipoint secant and interpolation methods are effective tools for solving systems of nonlinear equations. They use quasi-Newton updates for approximating the Jacobian matrix. Owing to their ability to more completely utilize the information about the Jacobian matrix gathered at the previous iterations, these methods are especially efficient in the case of expensive functions. They are known to be local and superlinearly convergent. We combine these methods with the nonmonotone line search proposed by Li and Fukushima (2000), and study global and superlinear convergence of this combination. Results of numerical experiments are presented. They indicate that the multipoint secant and interpolation methods tend to be more robust and efficient than Broyden's method globalized in the same way.

Abstract:This paper used the Newton-type Methodfor estimating a single root of nonlinear equations. This method is iterative method and also known as one of the open methods. Open method are fast converging method as compared to closed method but the convergence is not guaranteed. Since open methods are fast convergence methods that is why they are widely used in Applied Mathematics. The proposed numerical technique is second order of convergence, and which is based on Newton Raphson method. The developed algorithm is compared with the well-known Newton Raphson Method and results show that our developed method is much better than the well-known method. Furthermore, examples are also give in order to give more detailed about the present work and reader will come to know that why developed method is much better as compare to well-known method.

This paper presents an improved diagonal Secant-like method using two-step approach for solving large scale systems of nonlinear equations. In this scheme, instead of using direct updating matrix in every iteration to construct the interpolation curves, we chose to use an implicit updating approach to obtain an enhanced approximation of the Jacobian matrix which only requires a vector storage. The fact that the proposed method solves systems of nonlinear equations without the cost of storing the weighted matrix can be considered as a clear advantage of this method over some variants of Newton's methods.

This research was mainly conducted to explore the possibility of formulating an efficient algorithm to find roots of nonlinear equations without using the derivative of the function. The Weerakoon-Fernando method had been taken as the base in this project to find a new method without the derivative since Weerakoon-Fernando method gives 3rd order convergence. After several unsuccessful attempts we were able to formulate the Finite Difference Weerakoon-Fernando Method (FDWFM) presented here. We noticed that the FDWFM approaches the root faster than any other existing method in the absence of the derivatives as an example, the popular nonlinear equation solver such as secant method (order of convergence is 1.618) in the absence of the derivative. And the FDWFM had three function evaluations and secant method had two function evaluations. By implementing FDWFM on nonlinear equations with complex roots and also on systems of nonlinear equations, we received very encouraging results. When...

Abstract:This paper used the Newton-type Methodfor estimating a single root of nonlinear equations. This method is iterative method and also known as one of the open methods. Open method are fast converging method as compared to closed method but the convergence is not guaranteed. Since open methods are fast convergence methods that is why they are widely used in Applied Mathematics. The proposed numerical technique is second order of convergence, and which is based on Newton Raphson method. The developed algorithm is compared with the well-known Newton Raphson Method and results show that our developed method is much better than the well-known method. Furthermore, examples are also give in order to give more detailed about the present work and reader will come to know that why developed method is much better as compare to well-known method.

Resumen. En este artículo desarrollamos una teoría general de convergencia de un método secante para resolver ecuaciones matriciales no lineales. Además, presentamos condiciones suficientes para que este método proporcione un algoritmo local y superlinealmente convergente.

Resumen. En este artículo desarrollamos una teoría general de convergencia de un método secante para resolver ecuaciones matriciales no lineales. Además, presentamos condiciones suficientes para que este método proporcione un algoritmo local y superlinealmente convergente.

In this work, we introduce a family of Least Change Secant Update Methods for solving Nonlinear Complementarity Problems based on its reformulation as a nonsmooth system using the one-parametric class of nonlinear complementarity functions introduced by Kanzow and Kleinmichel. We prove local and superlinear convergence for the algorithms. Some numerical experiments show a good performance of this algorithm.