Ostrowski type methods for solving systems of nonlinear equations

Four generalized algorithms builded up from Ostrowski’s method for solving systems of nonlinear equations are written and analyzed. A development of an inverse first-order divided difference operator for functions of several variables is presented, as well as a direct computation of the local order of convergence for these variants of Ostrowski’s method. Furthermore, a sequence that approximates the order of convergence is generated for the examples and it confirms in a numerical way that the order of the methods is well deduced.     

On the approximation of derivatives using divided difference operators preserving the local convergence order of iterative methods

A development of an inverse first-order divided difference operator for functions of several variables is presented. Two generalized derivative-free algorithms built up from Ostrowski’s method for solving systems of nonlinear equations are written and analyzed. A direct computation of the local order of convergence for these variants of Ostrowski’s method is given. In order to preserve the local order of convergence, any divided difference operator is not valid. Two counterexamples of computation of a classical divided difference operator without preserving the order are presented. A rigorous study to know a priori if the new method will preserve the order of the original modified method is presented. The conclusion is that this fact does not depend on the method but on the systems of equations and if the associated divided difference verifies a particular condition. A new divided difference operator solving this problem is proposed. Furthermore, a computation that approximates the order of convergence is generated for the examples and it confirms in a numerical way that the order of the methods is well deduced. This study can be applied directly to other Newton’s type methods where derivatives are approximated by divided differences.     

Ostrowski’s fourth-order iterative method speedily solves cubic equations of state

Pressure-volume-temperature (P-V-T) data are required in simulating chemical plants because the latter usually involve production, separation, transportation, and storage of fluids. In the absence of actual experimental data, the pertinent mathematical model must rely on phase behaviour prediction by the so-called equations of state (EOS). When the plant model is a combination of differential and algebraic equations, simulation generally relies on numerical integration which proceeds in a piecewise fashion unless an approximate solution is needed at a single point. Needless to say, the constituent algebraic equations must be efficiently re-solved before each update of derivatives. Now, Ostrowski’s fourth-order iterative technique is a partial substitution variant of Newton’s popular second-order method. Although simple and powerful, this two-point variant has been utilised very little since its publication over forty years ago. After a brief introduction to cubic equations of state and their solution, this paper solves five of them. The results clearly demonstrate the superiority of Ostrowski’s method over Newton’s, Halley’s, and Chebyshev’s solvers.     

Three-step iterative methods with optimal eighth-order convergence

In this paper, based on Ostrowski’s method, a new family of eighth-order methods for solving nonlinear equations is derived. In terms of computational cost, each iteration of these methods requires three evaluations of the function and one evaluation of its first derivative, so that their efficiency indices are 1.682, which is optimal according to Kung and Traub’s conjecture. Numerical comparisons are made to show the performance of the new family.     

Two new families of sixth-order methods for solving non-linear equations

In this paper, we developed two new families of sixth-order methods for solving simple roots of non-linear equations. Per iteration these methods require two evaluations of the function and two evaluations of the first-order derivatives, which implies that the efficiency indexes of our methods are 1.565. These methods have more advantages than Newton’s method and other methods with the same convergence order, as shown in the illustration examples. Finally, using the developing methodology described in this paper, two new families of improvements of Jarratt method with sixth-order convergence are derived in a straightforward manner. Notice that Kou’s method in [Jisheng Kou, Yitian Li, An improvement of the Jarratt method, Appl. Math. Comput. 189 (2007) 1816-1821] and Wang’s method in [Xiuhua Wang, Jisheng Kou, Yitian Li, A variant of Jarratt method with sixth-order convergence, Appl. Math. Comput. 204 (2008) 14-19] are the special cases of the new improvements.     

The self-validated method for polynomial zeros of high efficiency

The improved iterative method of Newton’s type for the simultaneous inclusion of all simple complex zeros of a polynomial is proposed. The presented convergence analysis, which uses the concept of the R-order of convergence of mutually dependent sequences, shows that the convergence rate of the basic third order method is increased from 3 to 6 using Ostrowski’s corrections. The new inclusion method with Ostrowski’s corrections is more efficient compared to all existing methods belonging to the same class. To demonstrate the convergence properties of the proposed method, two numerical examples are given.     

A new family of modified Ostrowski’s methods with accelerated eighth order convergence

Based on Ostrowski’s fourth order method, we derive a family of eighth order methods for the solution of nonlinear equations. In terms of computational cost the family requires three evaluations of the function and one evaluation of first derivative. Therefore, the efficiency index of the present methods is 1.682 which is better than the efficiency index 1.587 of Ostrowski’s method. Kung and Traub conjectured that multipoint iteration methods without memory based on n evaluations have optimal order 2 ^n − 1. Thus, the family agrees with Kung–Traub conjecture for the case n = 4. The efficacy of the present methods is tested on a number of numerical examples. It is observed that our methods are competitive with other similar robust methods and very effective in high precision computations.     

Numerical inversion of a general incomplete elliptic integral

We present a numerical method to invert a general incomplete elliptic integral with respect to its argument and/or amplitude. The method obtains a solution by bisection accelerated by half argument formulas and addition theorems to evaluate the incomplete elliptic integrals and Jacobian elliptic functions required in the process. If faster execution is desirable at the cost of complexity of the algorithm, the sequence of bisection is switched to allow an improvement by using Newton’s method, Halley’s method, or higher-order Schröder methods. In the improvement process, the elliptic integrals and functions are computed by using Maclaurin series expansion and addition theorems based on the values obtained at the end of the bisection. Also, the derivatives of the elliptic integrals and functions are recursively evaluated from their values. By adopting 0.2 as the critical value of the length of the solution interval to shift to the improvement process, we suppress the expected number of bisections to be as low as four on average. The typical number of applications of update formulas in the double precision environment is three for Newton’s method, and two for Halley’s method or higher-order Schröder methods. Whether the improvement process is added or not, our method requires none of the procedures to compute the incomplete elliptic integrals and Jacobian elliptic functions but only those to evaluate the complete elliptic integrals once at the beginning. As a result, it runs fairly quickly in general. For example, when using the improvement process, it is around 2–5 times faster than Newton’s method using Boyd’s starter (Boyd (2012) [25]) in inverting E(φ|m)$E\left(\phi \right|m)$, Legendre’s incomplete elliptic integral of the second kind.     

On interpolation variants of Newton’s method for functions of several variables

A generalization of the variants of Newton’s method based on interpolation rules of quadrature is obtained, in order to solve systems of nonlinear equations. Under certain conditions, convergence order is proved to be 2d+1, where d is the order of the partial derivatives needed to be zero in the solution. Moreover, different numerical tests confirm the theoretical results and allow us to compare these variants with Newton’s classical method, whose convergence order is d+1 under the same conditions.     

Solitary pattern solutions for fractional Zakharov-Kuznetsov equations with fully nonlinear dispersion

The fractional Zakharov-Kuznetsov equations are increasingly used in modeling various kinds of weakly nonlinear ion acoustic waves in a plasma comprising cold ions and hot isothermal electrons in the presence of a uniform magnetic field. This has led to a significant interest in the study of these equations. In this work, solitary pattern solutions of fractional Zakharov-Kuznetsov equations are investigated by means of the homotopy perturbation method with consideration of Jumarie’s derivatives. The effects of fractional derivatives for the systems under consideration are discussed. Numerical results and a comparison with exact solutions are presented.     

Third and fourth order iterative methods free from second derivative for nonlinear systems

In this work, we develop a family of predictor-corrector methods free from second derivative for solving systems of nonlinear equations. In general, the obtained methods have order of convergence three but, in some particular cases the order is four. We also perform different numerical tests that confirm the theoretical results and allow us to compare these methods with Newton’s classical method and with other recently published methods.     

Third order iterative methods free from second derivative for nonlinear systems

In this work we present a family of predictor-corrector methods free from second derivative for solving nonlinear systems. We prove that the methods of this family are of third order convergence. We also perform numerical tests that allow us to compare these methods with Newton’s method. In addition, the numerical examples improve theoretical results, showing super cubic convergence for some methods of this family.     

Sixteenth-order method for nonlinear equations

Modification of Newton’s method with higher-order convergence is presented. The modification of Newton’s method is based on King’s fourth-order method. The new method requires three-step per iteration. Analysis of convergence demonstrates that the order of convergence is 16. Some numerical examples illustrate that the algorithm is more efficient and performs better than classical Newton’s method and other methods.     

Hyman’s method revisited

The QR algorithm is considered one of the most reliable methods for computing matrix eigenpairs. However, it is unable to detect multiple eigenvalues and Jordan blocks. Matlab’s eigensolver returns heavily perturbed eigenvalues and eigenvectors in such cases and there is no hint for possible principal vectors. This paper calls attention to Hyman’s method as it is applicable for computing principal vectors and higher derivatives of the characteristic polynomial that may help to estimate multiplicity, an important information for more reliable computation. We suggest a test matrix collection for Jordan blocks. The first numerical tests with these matrices reveal that the computational problems are deeper than expected at the beginning of this work.     

Breaking the limits: The Taylor series method

This paper discusses several examples of ordinary differential equation (ODE) applications that are difficult to solve numerically using conventional techniques, but which can be solved successfully using the Taylor series method. These results are hard to obtain using other methods such as Runge-Kutta or similar schemes; indeed, in some cases these other schemes are not able to solve such systems at all. In particular, we explore the use of the high-precision arithmetic in the Taylor series method for numerically integrating ODEs. We show how to compute the partial derivatives, how to propagate sets of initial conditions, and, finally, how to achieve the Brouwer’s Law limit in the propagation of errors in long-time simulations. The TIDES software that we use for this work is freely available from a website.     

Uniform asymptotic methods for integrals

We give an overview of basic methods that can be used for obtaining asymptotic expansions of integrals: Watson’s lemma, Laplace’s method, the saddle point method, and the method of stationary phase. Certain developments in the field of asymptotic analysis will be compared with De Bruijn’s book Asymptotic Methods in Analysis. The classical methods can be modified for obtaining expansions that hold uniformly with respect to additional parameters. We give an overview of examples in which special functions, such as the complementary error function, Airy functions, and Bessel functions, are used as approximations in uniform asymptotic expansions.     

Chebyshev-type methods and preconditioning techniques

Recently, a Newton’s iterative method is attracting more and more attention from various fields of science and engineering. This method is generally quadratically convergent. In this paper, some Chebyshev-type methods with the third order convergence are analyzed in detail and used to compute approximate inverse preconditioners for solving the linear system Ax = b. Theoretic analysis and numerical experiments show that Chebyshev’s method is more effective than Newton’s one in the case of constructing approximate inverse preconditioners.     

An efficient and accurate lattice for pricing derivatives under a jump-diffusion process

Derivatives are popular financial instruments whose values depend on other more fundamental financial assets (called the underlying assets). As they play essential roles in financial markets, evaluating them efficiently and accurately is critical. Most derivatives have no simple valuation formulas; as a result, they must be priced by numerical methods such as lattice methods. In a lattice, the prices of the derivatives converge to theoretical values when the number of time steps increases. Unfortunately, the nonlinearity error introduced by the nonlinearity of the option value function may cause the pricing results to converge slowly or even oscillate significantly. The lognormal diffusion process, which has been widely used to model the underlying asset’s price dynamics, does not capture the empirical findings satisfactorily. Therefore, many alternative processes have been proposed, and a very popular one is the jump-diffusion process. This paper proposes an accurate and efficient lattice for the jump-diffusion process. Our lattice is accurate because its structure can suit the derivatives’ specifications so that the pricing results converge smoothly. To our knowledge, no other lattices for the jump-diffusion process have successfully solved the oscillation problem. In addition, the time complexity of our lattice is lower than those of existing lattice methods by at least half an order. Numerous numerical calculations confirm the superior performance of our lattice to existing methods in terms of accuracy, speed, and generality.     

Pryce pre-analysis adapted to some DAE solvers

The ODE solver HBT(12)5 of order 12 (T. Nguyen-Ba, H. Hao, H. Yagoub, R. Vaillancourt, One-step 5-stage Hermite-Birkho-Taylor ODE solver of order 12, Appl. Math. Comput. 211 (2009) 313-328. doi:10.1016/j.amc.2009.01.043), which combines a Taylor series method of order 9 with a Runge-Kutta method of order 4, is expanded into the DAE solver HBT(12)5DAE of order 12. Dormand-Prince’s DP(8, 7)13M is also expanded into the DAE solver DP(8, 7)DAE. Pryce structural pre-analysis, extended ODEs and ODE first-order forms are adapted to these DAE solvers with a stepsize control based on local error estimators and a modified Pryce algorithm to advance integration. HBT(12)5DAE uses only the first nine derivatives of the unknown variables as opposed to the first 12 derivatives used by the Taylor series method T12DAE of order 12. Numerical results show the advantage of HBT(12)5DAE over T12DAE, DP(8, 7)DAE and other known DAE solvers.     

A variational iteration method for solving Troesch’s problem

Troesch’s problem is an inherently unstable two-point boundary value problem. A new and efficient algorithm based on the variational iteration method and variable transformation is proposed to solve Troesch’s problem. The underlying idea of the method is to convert the hyperbolic-type nonlinearity in the problem into polynomial-type nonlinearities by variable transformation, and the variational iteration method is then directly used to solve this transformed problem. Only the second-order iterative solution is required to provide a highly accurate analytical solution as compared with those obtained by other analytical and numerical methods.