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We present a numerical method for computing the solution of a partial differential equation (PDE) for modeling acoustic pressure, known as an extra-wide angle parabolic equation, that features the square root of a differential operator. The differential operator is the negative of an indefinite Schrödinger operator with a piecewise constant potential. This work primarily deals with the 3-piece case; however, a generalization is made the case of an arbitrary number of pieces. Through restriction to a judiciously chosen lower-dimensional subspace, approximate eigenfunctions are used to obtain estimates for the eigenvalues of the operator. Then, the estimated eigenvalues are used as initial guesses for the Secant Method to find the exact eigenvalues, up to roundoff error. An eigenfunction expansion of the solution is then constructed. The computational expense of obtaining each eigenpair is independent of the grid size. The accuracy, efficiency, and scalability of this method is shown through numerical experiments and comparisons with other methods. [ABSTRACT FROM AUTHOR]

The computational efficiency of underwater acoustic positioning based on the ray tracing is mainly limited to a great amount of calculation of ray inverse problem. We propose two kinds of p-order secant methods to improve the efficiency of traditional method, and the proposed methods can be regarded as a generalization of the traditional secant method from two points to p points for rapidly solving the inverse problem. In the proposed methods, the calculation information in previous iterations is utilized to fit a polynomial model to speed up the algorithm convergence. In the first-kind method, the inverse problem is calculated by solving a polynomial equation approximating the function mapping from the emission angle to the radial distance of the ray. In the second-kind method, the inverse problem is however directly solved by approximating the function mapping from the radial distance to the emission angle. As the first-kind method needs to solve a p-order polynomial equation, the practicability of this method is limited to the complexity of solving the high-order equation, while the second-kind method can directly approximate the solution of the inverse problem, which is more practical and flexible. The proposed methods have been verified in deep-sea trial. It shows that, the proposed methods can precisely produce the solution of the acoustic ray inverse problem within one iteration, and the computational efficiency of proposed method is about 6 times faster than that of the traditional method. [ABSTRACT FROM AUTHOR]

The Jiles-Atherton model was widely used in the description of the system with hysteresis, and the solution for the model was important for real-time and high-precision control. The secant method was used for solving anhysteretic magnetization and its initial values were optimized for faster convergence. Then, the Fourth Order Runge-Kutta method was employed to solve magnetization and the required computation cycles were supplied for stable results. Based on the solving method, the effect of the nonzero initial magnetic field on the magnetization was discussed, including the commonly used linear model of the square of magnetization under the medium initial value. From computations, the proposed secant iteration method, with supplied optimal initial values, greatly reduced the iterative steps compared to the fixed-point iteration. Combined with the Fourth Order Runge-Kutta method under more than three cycles of calculations, stable hysteresis results with controllable precisions were acquired. Adjusting the initial magnetic field changed the result of the magnetization, which was helpless to promote the amplitude or improve the symmetry of magnetization. Furthermore, the linear model of the square of magnetization was unacceptable for huge computational errors. The proposed numerical solving method can supply fast and high-precision solutions for the Jiles-Atherton model and provide a basis for the application scope of typical linear assumption. [ABSTRACT FROM AUTHOR]

We investigate the root finding algorithm given by the Secant method applied to a real polynomial p of degree k as a discrete dynamical system defined on R 2 . We extend the Secant map to the real projective plane R P 2 . The line at infinity ℓ ∞ is invariant, and there is one (if k is odd) or two (if k is even) fixed points at ℓ ∞ . We show that these are of saddle type, and this allows us to better understand the dynamics of the Secant map near infinity. [ABSTRACT FROM AUTHOR]

This study aimed to develop a viscosity model and improve the existing density function solving method by using the BWRS equation of state. The analysis revealed that the density function increased monotonically, so the improved dichotomy for solving density was developed. On the basis of the similarity between P-V-T and T-μ-P relationship, a viscosity model was developed. The new algorithm for solving density has a wider application scope, its calculation accuracy can satisfy the requirements. The new viscosity model could well predict the viscosity of pure propane and carbon dioxide (ADD%: 1.75 and 2.49), and that of methane–propane (ADD%: 1.5). [ABSTRACT FROM AUTHOR]

This paper presents a new method for linear and nonlinear errors of pipelined analog to digital converters (ADCs) based on numerical analysis. The main contribution of this work is to use secant method for solving nonlinear equations. The proposed method is not required the derivative of function, compared to the previously Newton Raphson algorithm. To validate the effectiveness of the proposed method, circuit level simulations in the circumstance of the 12-bit 100 MS/s pipelined ADC in 90-nm CMOS technology are provided. Simulation results show that the signal-to-noise and distortion ratio (SNDR) and the spurious free dynamic range (SFDR) improved from 22dB/25 dB–68dB/74 dB, respectively, after applying secant method. The convergence time of the algorithm is approximately 1000 clock cycles which is faster than Newton Raphson. By using accurate analysis, the estimated power and area in the digital domain are 2.4 mW and 0.06 mm2, respectively. The corresponding figure of merit is 40 fJ/conversion-step. • The proposed Secant method corrected linear and nonlinear errors in pipelined ADCs. • In addition to initialization process which is introduced in [28] for calibration coefficients, the digital power and area are estimated with details. • The proposed calibration technique does not interrupt the normal operation of ADC. • The main advantages of the proposed method are minimized power and fast convergence time. [ABSTRACT FROM AUTHOR]

The high speed of xk \rightarrow x\ast \in R is usually measured using the C-, Q-, or R-orders: lim | x\ast xk+1| | x\ast xk| p0 \in (0,+\infty), lim ln | x\ast xk+1| ln | x\ast xk| = q0, or lim \bigm| \bigm| ln | x\ast xk| \bigm| \bigm| 1k = r0. By connecting them to the natural, term-by-term comparison of the errors of two sequences, we find that the C-orders---including (sub)linear---are in agreement. Weird relations may appear though for the Q-orders: we expect | x\ast xk| = \scrO (| x\ast yk| \alpha) \forall \alpha > 1 to imply "\geq" for the Q-orders of \{ xk\} vs. \{ yk\}; the contrary is shown by an example providing no vs. infinite Q-orders. The R-orders appear to be even worse: an \{ xk\} with infinite R-order may have unbounded nonmonotone errors: | x\ast xk+1|/| x\ast xk| \rightarrow +\infty. Such aspects motivate the study of equivalent definitions, computational variants, and so on. These orders are also the perspective from which we analyze the three basic iterative methods for nonlinear equations in R. The Newton method, widely known for its quadratic convergence, may in fact attain any C-order from [1,+\infty] (including sublinear); we briefly recall such convergence results, along with connected aspects (such as historical notes, known asymptotic constants, floating point arithmetic issues, and radius of attraction balls), and provide examples. This approach leads to similar results for the successive approximations method, while the secant method exhibits different behavior: it may not have high C-orders, but only Q-orders. [ABSTRACT FROM AUTHOR]

Grid-connected photovoltaic (PV) systems impose challenges like voltage fluctuations, low system inertia, and power quality issues. The need to tackle these challenges led to the introduction of flexible power point tracking (FPPT), where the PV power output is controlled by an energy management system, rather than solely operating the PV systems on the maximum power point. The requirement of fast transient response implies that algorithms such as the one proposed in this article are desirable. The proposed algorithm uses the secant method to achieve significantly improved results in comparison to the existing methods. The method also simplifies the prediction of variations during changes in the environment and power reference, hence, results in reduced oscillation around the set-point and faster convergence. Experimental validation is presented in this article for supporting the claims. The results in terms of accuracy, convergence rate, steady-state oscillations, and cumulative error are bench-marked against one of the most recent FPPT methods. [ABSTRACT FROM AUTHOR]

The variable projection method is a classical and efficient method for solving separable nonlinear least squares (SNLLS) problems. However, it is hard to handle the constrained SNLLS problems since the explicit form of the Jacobian matrix is required in each iteration. In this paper, we propose a secant variable projection (SVP) method, which employs a rank-one update to estimate the Jacobian matrices. The main advantages of our method are efficiency and ease of applicability to constrained SNLLS problems. The local convergence of our SVP method is also analyzed. Finally, some data fitting and image processing problems are solved to compare the performance of our proposed method with the classical variable projection method. Numerical results illustrate the efficiency and stability of our proposed SVP method in solving the SNLLS problems arising from the blind deconvolution problems. [ABSTRACT FROM AUTHOR]

Back-projection (BP) is a reliable synthetic aperture radar (SAR) imaging algorithm because of its high-resolution and strong adaptability. However, it is hard to implement because of its high computational complexity. Fast factorized BP (FFBP) is a new way to fix this problem. Like traditional BP, FFBP is compatible with arbitrary flight paths if the track deviations are measured within fractions of a wavelength. However, when the motion information is not accurate enough, autofocus become an important way to get well-focused images. In this paper, we present an effective autofocus method for FFBP to solve the imaging problem caused by platform’s motion errors. First, an image quality evaluation function with unknown phase error based on image sharpness for FFBP is established. Then, the phase error computation for autofocus is modeled as an optimization problem. Second, the coordinate descent (CD) and secant processing are introduced to the maximum image sharpness problem. The proposed method keeps the rapid imaging performance of FFBP and solves well the motion error compensation problem. In the end, simulated data and real data were used to verify the effectiveness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

Geomagnetically – Induced Currents (GIC) and monopole operation of high voltage direct current (HVDC) both can cause transformer DC bias, and Jiles-Atherton (JA) model is effective in transformer DC bias simulation while the influence of initial magnetization is neglected in existing papers. In this paper, the influence of initial magnetization and magnetizing current on simulation process are analysed based on the presented simulation model, and the conclusion that initial magnetization has evident influence while initial magnetizing current has not is reached. The trend of magnetizing current is valued by representative slope of envelop curve, then an improved secant method for getting proper initial magnetization is proposed basing on the slope. With the method, the dependence of proper initial magnetization on DC current and closing angle are presented, and the mechanization is analysed. [ABSTRACT FROM AUTHOR]

In this work, we propose new methods based on the root-finding methods in numerical analysis to calculate the inverse of an integer N modulo N for any positive integer N. We apply Newton's and secant methods with a power-reduction process and Newton's and secant methods together with the binary representation and Zeckendorf representation to determine the inverse of an integer N modulo N. The numerical results appear accurate when compared to analytical results. In addition, the two methods with the binary and Zeckendorf representations give better CPU times than Newton's and secant methods with power-reduction, for large N. [ABSTRACT FROM AUTHOR]

We present a new semilocal convergence analysis for Secant methods in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. Our analysis includes the computation of the bounds on the limit points of the majorizing sequences involved. Under the same computational cost on the parameters involved our convergence criteria are weaker and the error bounds more precise than in earlier studies. A numerical example is also presented to illustrate the theoretical results obtained in this study. [ABSTRACT FROM AUTHOR]

We present a local as well a semilocal convergence analysis of secant-like methods under g eneral conditions in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. The new conditions are more flexible than in earlier studies. This way we expand the applicability of these methods, since the new convergence conditions are weaker. Moreover, these advantages are obtained under the same conditions as in earlier studies. Numerical examples are also provided in this study, where our results compare favorably to earlier ones. [ABSTRACT FROM AUTHOR]

The front‐face temperature rise of Al alloy subjected to the high‐energy long continuing current (LCC) component in lightning is important but difficult to measure. Therefore, a numerical inversion model based on the control volume finite element method is proposed to analyse the front‐face temperature rise of Al alloy deduced from the rear‐face temperature rise measured in simulated LCC experiment. In this model, Al sample is divided into discrete control‐volume units with continuous medium by grids, forming internal nodes, surface nodes, edge nodes, and corner nodes. Temperature governing equations for different nodes are derived from Fourier's heat conduction law and solved with finite difference method in time and spatial domain. The secant method is used to make the optimal approximation to experimental data. The results show that subjected to LCC with amplitude 404 A and duration 520 ms, the highest front‐face temperature rise of Al alloy is 10,800 K. The damaged area calculated by the proposed model is 298.7 mm2. The relative error is 4.8% compared with the experimental result 314.0 mm2. The energy consumed in all processes is 6710.1 J calculated by this model, in contrast to the total energy 7140.9 J generated from the arc and Joule heat. [ABSTRACT FROM AUTHOR]

A direct series solution for the Falkner-Skan equation is obtained by first transforming the problem using the Crocco-Wang transformation. The transformation converts the third-order problem to a second-order two-point boundary value problem. The method first constructs a series involving the unknown skin-friction coefficient . Then, is determined by using the secant method or Newton's method. The derivative needed for Newton's method is also computed using a series derived from the transformed differential equation. The method is validated by solving the Falkner-Skan equation for several cases reported previously in the literature. [ABSTRACT FROM AUTHOR]

An understanding of traffic statics and dynamics is critical for developing effective and efficient control strategies for a signalized road network with turning movements, especially under oversaturated conditions. In this study, we first describe traffic dynamics in a signalized double-ring network with the link queue model, which is a space-continuous approximation of the network kinematic wave model, and rewrite it as a switched affine system, assuming a triangular traffic flow fundamental diagram. Then we define periodic density evolution orbits as stationary states in the network and introduce a Poincaré map in densities, whose fixed points correspond to stationary states. With short cycle lengths and identical green times and retaining ratios in both rings, we are able to derive the closed form of the Poincaré map, from which we can analytically solve stationary states and study their stability properties; it is found that a stationary state can be asymptotically stable, Lyapunov stable, or unstable. By defining the network flow-density relation in stationary states as the macroscopic fundamental diagram (MFD), we analytically derive an approximate closed-form formula for MFD with green ratios and retaining ratios as parameters. We confirm that in stationary states the network flow rate may not be uniquely defined, and the network can reach a gridlock state at relatively low densities. We also analyze the convergence patterns to asymptotically stable gridlock states with different retaining ratios and initial densities. With more general signal settings and retaining ratios, we develop a secant method to numerically solve the fixed points of the Poincaré maps and plot the corresponding MFDs. Furthermore, numerical simulations are used to study traffic statics in a homogeneous signalized grid network; simulation results reveal a high level of similarity in traffic patterns between the signalized double-ring and grid networks and validate analytical insights obtained from the former. This study provides a springboard for future analytical and numerical studies on traffic statics and dynamics in more general signalized road networks. The online appendix is available at [ABSTRACT FROM AUTHOR]

• An improved method of computing acoustic ray incident angle is proposed. • Apply initial value and root-containing interval to reduce the computational burden. • Demonstrate that the mathematical model of the incident angle is monotonic. • The positioning calculation time can be improved by 63%–68% in the sea-trail test. Constant gradient acoustic ray tracking is usually used in high-precision underwater acoustic positioning. However, it needs to calculate the incident angle of the acoustic ray, and the traditional incident angle computation method is very inefficient. To resolve this problem, we propose an improved acoustic ray incident angle iterative computation method. It firstly constructs the mathematical model for incident angle by using the function between horizontal propagation distance and incident angle based on secant method. The initial value of the incident angle is then determined by the incident angle calculated from the previous epoch and the last iteration. Meanwhile, the root-containing interval of the incident angle is reduced according to experimental parameters and coordinate correction values. Finally, the incident angle calculated in the last iteration is directly used for acoustic ray tracking and positioning solution until the end, when the coordinate correction values reach a threshold. Experiment conducted on the simulation cases demonstrates that the positioning calculation time of the proposed method is 77.7%, 77.1% and 88.1% lower than traditional methods in order of the dichotomy, secant method, and Newton method, respectively. As for the real experiment of the South China Sea, the computation efficiency improvement of the proposed method is about 62.9%, 67.6% and 66.1% compared with the dichotomy, secant method, and Newton method respectively. These results suggest that the proposed method can significantly improve the positioning calculation efficiency without loss of positioning accuracy compared with conventional methods. [ABSTRACT FROM AUTHOR]

One of the most common approaches for solving eigenvalue problems with eigenvector nonlinearities (NEPv) is the Self Consistent Field (SCF) method that uses a vector from the previous iteration to build a zeroth order approximation of the nonlinearity. This approach is often slow and unreliable, which is why most applications use mixing scheme extensions of SCF, that use a linear combination of multiple previous iterates to build the zeroth order approximation. In this paper, we present a method that uses multiple previous iterates to build an interpolating first order approximation of the nonlinearity. It can be shown that this method converges superlinearly to the desired eigenpair. [ABSTRACT FROM AUTHOR]

An integrated method is presented for the thin-walled axial compression member with open cross-section to calculate the elasto-plastic buckling load based on the relationship between strain and axial compression. The method puts forward the secant iterative method of buckling load for the flexural buckling, torsion buckling and flexural torsional buckling, which can be applied to the axial compression member with different opening sections, such as the H-type, T-type, L-type, cross shape, etc. The residual stress and elasto-plastic constitutive relation are considered in buckling load calculation. In the method the axial strain corresponding to the elastic buckling load without residual stress is taken as the initial value, the values of interval points are calculated according to the nonlinear function for different buckling modes and then the buckling load and other data are obtained based on the secant iterative method. The secant method offers significant improvements in accuracy, robustness and calculation amount of the solution, which can effectively solve the buckling load of thin-walled axial compression elasto-plastic member. [ABSTRACT FROM AUTHOR]

This paper presents an accurate initial pole position (IPP) estimation method for a surface mounted permanent magnet synchronous motor (PMSM) without the use of pole sensors. To estimate the IPP, the proposed method uses the concept of reference frame. In this method, three torque commands, separated by 45° electrically from each other, are applied to the PMSM. Using the reference torque commands, moving distances are measured and transformation matrix is applied to eliminate the load torque term. Finally, the IPP is determined using the secant method. The IPPs under no load and at specific loaded condition are measured. The result of the load condition is compared with the previously proposed two reference frame method. The proposed method gives better result under load condition, compared with the two reference frame method, which fails to give satisfactory result under load condition. [ABSTRACT FROM AUTHOR]

To understand the microstructural factors dominantly affecting the mechanical properties of metals with duplex microstructures on a mesoscale, the change in the 0.2% proof stress as a function of the hard austenite fraction of an Fe–Ni austenitic alloy having a duplex microstructure composed of soft and hard austenite was investigated in terms of the microstructural connectivity. The 0.2% proof stress continuously increased with increasing fraction of the hard austenite. However, the strengthening behavior changed when the hard austenite fraction exceeded a critical value; the 0.2% proof stress shifted from the lower limit stress to the upper limit one calculated by secant method at the critical fraction. This demonstrates that the matrix of duplex microstructure was switched from soft austenite to hard austenite at the critical fraction. On the other hand, it was also confirmed that the non-monotonic strengthening behavior has a good correlation with the microstructural connectivity of the hard austenite, that is, the percolation phenomenon. From these result, it was concluded that the strength of metals with duplex microstructure can be wholly predicted based on secant method while taking into account the matrix switching between soft and hard phases, and the percolation index has a possibility to be a useful parameter to distinguish between soft and hard phases which is the matrix of the duplex microstructure, although further investigation is needed on a degree of the hard phase connection. [ABSTRACT FROM AUTHOR]

By amalgamating a secant type method and a Newton type method, another method of order 4 has been derived. The method has been supported by examples and has been compared with existing similar methods. [ABSTRACT FROM AUTHOR]

This paper presents a class of stationary iterative processes with convergence order equal to the growth rate of generalized Fibonacci sequences. We prove that the informational and computational efficiency of the processes of our class tends to 2 from below. The paper illustrates a connection of the methods of the class with the nonstationary iterative method suggested by our previous paper, whose efficiency index equals to 2. We prove that the efficiency of the nonstationary iterative method, measured by Ostrowski–Traub criteria, is maximal among all iterative processes of order 2. [ABSTRACT FROM AUTHOR]

A fast and robust camera’s auto exposure (AE) technique is proposed in this work. It is achieved by modeling the luminance characteristics of the imaging sensor as a concave or convex function of a control parameter ( e.g. , exposure time or speed) and the optimal control parameter is computed using a modified secant algorithm with fast convergence. Furthermore, the proposed solution is able to adjust the control parameter automatically in the presence of erroneous exposure. Its superior performance is confirmed by experimental results. [ABSTRACT FROM AUTHOR]

Masonry walls are very sensitive to flexural effects due to low tensile strength which, in turn, greatly influences the load bearing capacity under compression. The main source of flexural effects may results from the eccentric loading at the ends of the wall or from any lateral loading like the wind action, the earth pressure, or the second order effect of the applied actions. Several analytical solutions were proposed in literature to solve the differential equation of the problem, but those solutions were limited to special conditions. In the current contribution, a general formulation for the non-linear stability problem has been formulated numerically based on the transfer-matrix method. Despite the method is out of professional use today and don’t possess the potential and flexibility of the finite elements but for the current addressed problem, it is still the most efficient. A relative form description has been introduced to formulate the stability theory of masonry walls. This description has been used to minimize the dimensions of matrixes in the transfer-matrix method and to produce the equations in a compact form. The algorithms of the method have been derived for general boundary and loading conditions with a user-defined non-linear material model. Algorithms and solution procedures have been explained and implemented into a computer code. The convergence of the iterative solution has been studied with clear definition for the cases at which the stability or material failures occur. The results of the developed solution procedure have been validated by comparing them with the existing solutions and the experimental results. The developed solution procedure provides a powerful tool to solve a wide range of problems related to stability of masonry walls and to check the existing empirical methods. [ABSTRACT FROM AUTHOR]

We present a new semilocal convergence analysis for the Secant and the Moser method in order to approximate a solution of an equation in a Banach space setting. Using the method of recurrent relations and weaker sufficient convergence criteria than in earlier studies such as Amat et al. (2014), Hernández and Rubio (2007), Hernández and Rubio (1999) and Hernández and Rubio (2002) we increase the convergence domain of these methods. The advantages are also obtained under less computational cost than in Amat et al. (2014), Hernández and Rubio (2007), Hernández and Rubio (1999) and Hernández and Rubio (2002). Numerical examples where the older convergence criteria are not satisfied but the new convergence criteria are satisfied are also provided in this study. [ABSTRACT FROM AUTHOR]

To reduce the dependence of trajectory tracking of reusable launch vehicle (RLV) on the aerodynamic parameters and initial conditions, and improve the robustness of traditional control method, a new entry trajectory tracking method based on the fuzzy variable structure is proposed from the perspective of longitudinal tracking guidance. Also, a method of on-line planning the point of the bank angle reversal is designed in the aspect of lateral trajectory control. The simulation results show that the longitudinal tracking control method which is based on the fuzzy variable structure can track the designed trajectory perfectly and stably, and has strong robustness in the condition of big initial state error and aerodynamic coefficient error. The online searching of the bank angle reversal point can be realized by using the proposed lateral guidance method, which can also improve the lateral guidance accuracy effectively. [ABSTRACT FROM AUTHOR]

Explicit iteration formulas were proposed for solving the equation f(x) ≡ 0 mod pk, when f was the polynomial axn-b. Speedy algorithms were formulated for lifting solutions of a polynomial congruence mod p, to polynomial congruence mod pk. This was done reasonably fast, using proposed algorithm. Polynomial time was k, which was about the best possible since the number of bits in the answer was in general proportional to k. The algorithm developed was instigated with an adaptation of secant method. For a polynomial f, with initial solutions x0 mod pk1 and x1 mod pk2 to f(x) ≡ 0 mod pk, haggled a solution x2 to f(x) ≡ 0 mod pk1+k2 with, x2 = x1-f(x1)(x1-x0)/f(x1)-f(x0)where the inverse was computed using the Euclidean algorithm in the ring of integers modulo pk. The proposed technique endeavored to keep the elucidation consistently a little low to give advantage in finding the solution of congruences by means of explicit iteration techniques which proved quite fast in finding these solutions. [ABSTRACT FROM AUTHOR]

We present a new semilocal convergence analysis for Secant method in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. Our analysis includes the computation of the bounds on the limit points of the majorizing sequences involved. Under the same computational cost on the parameters involved our convergence criteria are weaker and the error bounds more precise than in earlier studies such as (Amat and Busquier, 2003; Amat et al., in press; Argyros and Hilout, 2012; Argyros et al., 2014; Argyros and Magreñán, 2014, 2015; Dennis, 1971; Ezquerro et al., 2000; Ortega and Rheinboldt, 1970; Potra and Pták, 1984; Schmidt, 1978). Numerical examples are also presented to illustrate the theoretical results obtained in this study. [ABSTRACT FROM AUTHOR]

We present inexact secant methods in association with line search filter technique for solving nonlinear equality constrained optimization. For large-scale applications, it is expensive to get an exact search direction, and hence we use an inexact method that finds an approximate solution satisfying some appropriate conditions. The global convergence of the proposed algorithm is established by using line search filter technique. The second-order correction step is used to overcome the Maratos effect, while the line search filter inexact secant methods have superlinear local convergence rate. Finally, the results of numerical experiments indicate that the proposed methods are efficient for the given test problems. [ABSTRACT FROM AUTHOR]

We present a new convergence analysis, for the Secant method in order to approximate a locally unique solution of a nonlinear equation in a Banach space. Our idea uses Lipschitz and center-Lipschitz instead of just Lipschitz conditions in the convergence analysis. The new convergence analysis leads to more precise error bounds and to a better information on the location of the solution than the corresponding ones in earlier studies such as [2,6,9,11,14,15,17,20,22–26]. Numerical examples validating the theoretical results are also provided in this study. [ABSTRACT FROM AUTHOR]

We present a new technique for enlarging the convergence ball of the method of parabola in order to approximate a zero of derivatives. This approach also leads to more precise error estimates on the distances involved than in earlier studies such as Hua (1974), Ren and Wu (2009) and Wand (1975). These advantages are obtained under the same computational cost on the Lipschitz constants involved as in the earlier studies. Numerical examples are also given to show the advantages over the earlier work. [ABSTRACT FROM AUTHOR]

We use tighter majorizing sequences than in earlier studies to provide a semilocal convergence analysis for the secant method. Our sufficient convergence conditions are also weaker. Numerical examples are provided where earlier conditions do not hold but for which the new conditions are satisfied. [ABSTRACT FROM AUTHOR]

Abstract: We present a new semilocal convergence analysis for the Secant method in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. Our analysis is based on the weaker center-Lipschitz concept instead of the stronger Lipschitz condition which has been ubiquitously employed in other studies such as Amat et al. (2004) [2], Bosarge and Falb (1969) [9], Dennis (1971) [10], Ezquerro et al. (2010) [11], Hernández et al. (2005, 2000) [13,12], Kantorovich and Akilov (1982) [14], Laasonen (1969) [15], Ortega and Rheinboldt (1970) [16], Parida and Gupta (2007) [17], Potra (1982, 1984–1985, 1985) [18–20], Proinov (2009, 2010) [21,22], Schmidt (1978) [23], Wolfe (1978) [24] and Yamamoto (1987) [25] for computing the inverses of the linear operators. We also provide lower and upper bounds on the limit point of the majorizing sequences for the Secant method. Under the same computational cost, our error analysis is tighter than that proposed in earlier studies. Numerical examples illustrating the theoretical results are also given in this study. [Copyright &y& Elsevier]

Abstract: A local convergence analysis is presented for a fast two-step Newton-like method (TSNLM) for solving nonlinear equations in a Banach space setting. The TSNLM unifies earlier methods such as Newton’s, Secant, Newton-like, Chebyshev–Secant, Chebyshev–Newton, Steffensen, Stirling’s and other single or multistep methods. Numerical examples and a comparative study of these methods validating our theoretical results are also given in the concluding section of this paper. [Copyright &y& Elsevier]

Families of fourth-order methods are presented which are obtained by existing third-order methods applied in succession with the secant method. [ABSTRACT FROM AUTHOR]

Maximum-likelihood estimation technique is known to provide consistent and most efficient regression estimates but often this technique is tedious to implement, particularly in the modelling of correlated count responses. To overcome this limitation, researchers have developed semi- or quasi-likelihood functions that depend only on the correct specification of the mean and variance of the responses rather than on the distribution function. Moreover, quasi-likelihood estimation provides consistent and equally efficient estimates as the maximum-likelihood approach. Basically, the quasi-likelihood estimating function is a non-linear equation constituting of the gradient, Hessian and basic score matrices. Henceforth, to obtain estimates of the regression parameters, the quasi-likelihood equation is solved iteratively using the Newton–Raphson technique. However, the inverse of the Jacobian matrix involved in the Newton–Raphson method may not be easy to compute since the matrix is very close to singularity. In this paper, we consider the use of vector divisions in solving quasi-likelihood equations. The vector divisions are implemented to form secant method formulas. To assess the performance of the use of vector divisions with the secant method, we generate cross-sectional Poisson counts using different sets of mean parameters. We compute the estimates of the regression parameters using the Newton–Raphson technique and vector divisions and compare the number of non-convergent simulations under both algorithms. [ABSTRACT FROM AUTHOR]

Abstract: The linear complementarity problem is to find a vector in satisfying , ,, where and are given. In this paper, we use the fact that solving is equivalent to solving the nonlinear equation where is a function from into itself defined by . We build a sequence of smooth functions which is uniformly convergent to the function . We show that, an approximation of the solution of the (when it exists) is obtained by solving for a parameter large enough. Then we give a globally convergent hybrid algorithm which is based on vector divisions and the secant method for solving . We close our paper with some numerical simulations to illustrate our theoretical results, and to show that this method can solve efficiently large-scale linear complementarity problems. [Copyright &y& Elsevier]

This paper examines the application of various classical root-finding methods to digital maximum power point tracking (DMPPT). An overview of root-finding methods such as the Newton Raphson method, Secant method, bisection method, regula falsi method, and a proposed modified regula falsi method (MRFM) applied to photovoltaic (PV) applications is presented. These methods are compared among themselves. Some of their features are also compared with other commonly used maximum power point (MPP) tracking methods. Issues found when implementing these root-finding methods based on continuous variables in a digital domain are explored. Some of these discussed issues include numerical stability, digital implementation of differential operators, and quantization error. Convergence speed is also explored. The analysis is used to provide practical insights into the design of a DMPPT based on classical root-finding algorithms. A new DMPPT based on an MRFM is proposed and used as the basis for the discussion. It is shown that this proposed method is faster than the other discussed methods that ensure convergence to the MPP. The discussion is approached from a practical perspective and also includes theoretical analysis to support the observations. Extensive simulation and experimental results with hardware prototypes verify the analysis. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, a new method for solving nonlinear equations is presented. Analysis of the convergence shows that the asymptotic convergence order of this method is . Some numerical results are given to demonstrate its efficiency. [Copyright &y& Elsevier]

Abstract: We establish new sufficient convergence conditions for the Secant method to a locally unique solution of a nonlinear equation in a Banach space. Using our new concept of recurrent functions, and combining Lipschitz and center-Lipschitz conditions on the divided difference operator, we obtain a new semilocal convergence analysis of the Secant method. Moreover, our sufficient convergence conditions expand the applicability of the Secant method in cases not covered before (Dennis, 1971 , Hernández et al., 2005 , Laasonen, 1969 , Ortega and Rheinboldt, 1970 , Potra, 1982 , Potra, 1985 , Schmidt, 1978 , Yamamoto, 1987 , Wolfe, 1978 ). Numerical examples are also provided in this study. [Copyright &y& Elsevier]

Abstract: A shooting method is used to determine a solution to a third-order ODE modeling the steady profile of a non-Newtonian thin droplet. We compare a direct approach to an iterative approach using a secant method. We obtain a nonlinear relationship between the contact angle and the position of the contact line . From this nonlinear relationship we use curve fitting to obtain an empirical law of the form where is the power law coefficient and is a nonlinear function of . [Copyright &y& Elsevier]

Abstract: A secant approach is illustrated for the ultimate limit state (ULS) analysis of prestressed reinforced concrete sections subjected to axial load and biaxial bending in presence of softening stress–strain laws. The stiffness matrix and the resultant loads are evaluated analytically by a novel methodology, termed fiber-free, which represents a computationally efficient alternative to fiber approaches. Extensive computations of the ULS domains of benchmark test cases show that the robustness of the proposed algorithmic strategy is substantially unaffected by the amount of reinforcement, prestressing and softening, though localized non-convex regions have been occasionally experienced in presence of softening. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we present a new secant-like method for solving nonlinear equations. Analysis of the convergence shows that the asymptotic convergence order of this method is . Some numerical results are given to demonstrate its efficiency. [ABSTRACT FROM AUTHOR]

We provide a semilocal convergence analysis for a certain class of secant-like methods considered also in Argyros (J Math Anal Appl 298:374–397, , ), Potra (Libertas Mathematica 5:71–84, ), in order to approximate a locally unique solution of an equation in a Banach space. Using a combination of Lipschitz and center-Lipschitz conditions for the computation of the upper bounds on the inverses of the linear operators involved, instead of only Lipschitz conditions (Potra, Libertas Mathematica 5:71–84, ), we provide an analysis with the following advantages over the work in Potra (Libertas Mathematica 5:71–84, ) which improved the works in Bosarge and Falb (J Optim Theory Appl 4:156–166, , Numer Math 14:264–286, ), Dennis (SIAM J Numer Anal 6(3):493–507, , ), Kornstaedt (), Larsonen (Ann Acad Sci Fenn, A 450:1–10, ), Potra (L’Analyse Numérique et la Théorie de l’Approximation 8(2):203–214, , Aplikace Mathematiky 26:111–120, , , Libertas Mathematica 5:71–84, ), Potra and Pták (Math Scand 46:236–250, , Numer Func Anal Optim 2(1):107–120, ), Schmidt (Period Math Hung 9(3):241–247, ), Schmidt and Schwetlick (Computing 3:215–226, ), Traub (), Wolfe (Numer Math 31:153–174, ): larger convergence domain; weaker sufficient convergence conditions, finer error bounds on the distances involved, and a more precise information on the location of the solution. Numerical examples further validating the results are also provided. [ABSTRACT FROM AUTHOR]

We present a new quasi-Newton method that can solve systems of equations of which no information is known explicitly and which requires no special structure of the system matrix, like positive definiteness or sparseness. The method builds an approximate Jacobian based on input-output combinations of a black box system, uses a rank-one update of this Jacobian after each iteration, and satisfies the secant equation. While it has originally been developed for nonlinear equations we analyze its properties and performance when applied to linear systems. Analytically, the method is shown to be convergent in n+1 iterations (n being the number of unknowns), irrespective of the nature of the system matrix. The performance of this method is greatly superior to other quasi-Newton methods and comparable with GMRes when tested on a number of standardized test-cases. [ABSTRACT FROM AUTHOR]

In this paper, a class of generalized secant methods was investigated. The convergence of these methods was discussed, and their striking fractal patterns are generated to represent the chaotic behavior of these methods in the complex plane. [ABSTRACT FROM AUTHOR]