Generalized Secant Method Research Articles

Application of a Generalized Secant Method to Nonlinear Equations with Complex Roots

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 {xn} of approximations to a real root of f(x), where, for n≥k, xn+1=xn−f(xn)/pn,k′(xn), pn,k(x) being the polynomial of degree k that interpolates f(x) at xn,xn−1,…,xn−k, the order sk of this method satisfying 1<sk<2. Clearly, when k=1, this method reduces to the secant method with s1=(1+5)/2. In addition, s1<s2<s3<⋯, such that limk→∞sk=2. 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.

Aitken type methods with high efficiency

In this paper, we study the iterative method of Aitken type for solving the non-linear equations, in which the interpolation nodes are controlled by variant of Newton method or by a general method of order p. By combining such methods with a generalized secant method, it is shown that the order of convergence can be increased to as high as desired and also in the limiting case efficiency of the method is 2. Several numerical examples are provided in support of the theoretical results.

Robustness Maximization of Parallel Multichannel Systems

Bit error rate (BER) minimization and SNR‐gap maximization, two robustness optimization problems, are solved, under average power and bitrate constraints, according to the waterfilling policy. Under peak power constraint the solutions differ and this paper gives bit‐loading solutions of both robustness optimization problems over independent parallel channels. The study is based on analytical approach, using generalized Lagrangian relaxation tool, and on greedy‐type algorithm approach. Tight BER expressions are used for square and rectangular quadrature amplitude modulations. Integer bit solution of analytical continuous bitrates is performed with a new generalized secant method. The asymptotic convergence of both robustness optimizations is proved for both analytical and algorithmic approaches. We also prove that, in the conventional margin maximization problem, the equivalence between SNR‐gap maximization and power minimization does not hold with peak‐power limitation. Based on a defined dissimilarity measure, bit‐loading solutions are compared over Rayleigh fading channel for multicarrier systems. Simulation results confirm the asymptotic convergence of both resource allocation policies. In nonasymptotic regime the resource allocation policies can be interchanged depending on the robustness measure and on the operating point of the communication system. The low computational effort leads to a good trade‐off between performance and complexity.

New Adaptive and Learning-Adaptive Control Techniques Based on an Extension of the Generalized Secant Method

ABSTRACTThis paper presents two new controllers and uses an extension of the generalized secant method of solving linear equations.1 The first controller is adaptive and operates in the time domain. The second one is a learning-adaptive controller which uses a similar technique to handle repetitive disturbances. This controller operates in the repetition domain. For both controllers, die formulation of the systems into a linear estimation problem is presented as well as the solution using the secant method. A proof of convergence is also given for both controllers. In addition, some practical modifications are presented to increase the robustness and stability of the systems. Furthermore, an analog network is proposed for increasing the speed of the controllers which is especially useful for the learning-adaptive controller since this network parallelizes most of the computation necessary for establishing the control, in hardware form. The adaptive controller is tested in simulations of two highly non-lin...

A finite element model for delamination propagation in composites

In this paper the theory and application of a modelling technique for three-dimensional planar delamination growth in laminated composites has been presented. The method is based on linear elastic fracture mechanics assumptions for delamination cracks and uses a strain energy release rate criterion. For a given component, strain energy release rate is considered to be a non-linear function of the location of the delamination front. Hence, satisfaction of the growth criterion reduces to the solution of a non-linear system of equations. A generalized secant method, by the Broyden's update method, is used to solve the system of non-linear equations. Several examples of the application of the technique are presented.

Efficient solution of Maxwell's equations for optical fibers with arbitrary refractive-index profiles

A straightforward method of solving vectorial wave equations in terms of electric fields for cylindrically symmetrical optical fibers is described. Input profiles are modeled by functions with continuous derivatives, so all fields are continuous. To iterate propagation constants, a generalized secant method is used. Numerical integrations are performed with a published FORTRAN code based on a fourth/fifth-order Runge-Kutta method. Techniques that help to make calculations efficient are presented. By using an IBM personal computer, propagation constants of actual single-mode fibers are found with 11-digit computational accuracy in 2 min each. Finally, the scalar wave equation is also solved efficiently.

An extension of the predictive Min-H method to multivariable control

A method is proposed for extending the interpolative algorithm of Gibson and Lowinger (1974) to problems with multivariable control. The suggested approach considers a linear interpolation of the control variables which in m-dimensional control space results in m+1 simultaneous equations, and may be solved using the generalized secant method )Wolfe 1959). The extended recurrence relationships are derived and a two-input plant example is computed.

Some iterations for factoring a polynomial. II a generalization of the secant method

This paper describes an iterative method for factoring a polymonial that bears the same relation to Bairstow's method as the secant method in a single variable bears to Newton's method. Like the secant method, the generalized secant method requires only one function evaluation for each iteration, and like the secant method it converges to a simple factor with order $${{\left( {1 + \sqrt 5 } \right)} \mathord{\left/ {\vphantom {{\left( {1 + \sqrt 5 } \right)} 2}} \right. \kern-\nulldelimiterspace} 2}$$ .

Comment on "Solution of Nonlinear Equations"

A recently proposed approach to the solution of non-linear equations gives the same computing formulas as the generalized secant method. The effect of rounding-off errors is shown to be less than might be feared. References are given to the quasi-Newton forn of the method.

“Some iterations for factoring a polynomial”

This paper describes an iterative method for factoring a polynomial that bears the same relation to Bairstow's method as the secant method in a single variable bears to Newton's method. Like the secant method, the generalized secant method requires only one function evaluation for each iteration, and like the secant method it converges to a simple factor with order (1+75)/2.

An Algorithm for Solving Non-Linear Equations Based on the Secant Method

This paper describes a variant of the generalized secant method for solving simultaneous nonlinear equations. The method is of particular value in cases where the evaluation of the residuals for imputed values of the unknowns is tedious, or a good approximation to the solution and the Jacobian at the solution are available. Experiments are described comparing the method with the Newton-Raphson process. It is shown that for suitable problems the method is considerably superior.