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

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


Introduction
Extracting roots or finding solutions to equations is an important quest in mathematical computations. The roots of equations provide answers to many practical problems. Finding the most efficient numerical method for the purpose is very critical since accuracy of the result for most practical problems is so essential [3]. A problem becomes even more demanding if it requires solving systems of nonlinear equations after modelling.
Solving systems of nonlinear equations is one of the most important problems in numerical computations, especially for a diverse range of engineering applications, including applications in many scientific fields [13]. Several real -life problems can be reduced to solving systems of nonlinear equations, which is one of the most basic problems in mathematics [1]. Great efforts have been made by a lot of people and many constructive theories and algorithms are proposed to solve systems of nonlinear equations [12]. However there still exist some setbacks in solving such systems. For most traditional numerical methods such as Newton's method, the convergence and performance characteristics can be highly sensitive to the initial guess of the solution. However, it is very difficult to select a reasonable initial guess of a solution for most system of nonlinear equations [10]. The algorithm may fail or the results may be improper if the initial guess of the solution is unreasonable. Many different combinations of the traditional numerical methods and intelligent algorithms are applied to solve systems of nonlinear equations [14,15], which can overcome the problem of selecting a reasonable initial guess of the solution. But the algorithms are too complicated or expensive to calculate with when there are a number of nonlinear equations to solve simultaneously.
Among the classes of numerical methods for solving system of nonlinear equations, the Newton-Raphson scheme remains popular. However, the Newton-Raphson method is confronted with some drawbacks, a major one of which is the need to compute the inverse Jacobian matrix iteration by iteration. This makes it inefficient for large sized problems especially [20], which serves as the motivation for the current work.
The Broyden method, which is a quasi-Newton method, has seen significant modifications and improvements and these have motivated other researchers to develop new methods capable of solving efficiently nonlinear systems of equations [1]. Many authors continue to present different techniques which are Newton-like schemes [23,22,8,6,7], Mixed Free Secant methods, or Quadrature formulas. A leading trend of new methods developed for the computation of solutions of systems of nonlinear equations for the past few years has been to formulate using the quadrature rules. Some references in relation to developed methods using quadrature rules include [16][17][18][19][20] Newton Cotes quadrature rules are a group of formulas for numerical integration based on evaluating the integrand at equally spaced points. Named after Isaac Newton and Roger Cotes [7], they fit data to local order k polynomial approximants. The Newton-Cotes quadrature formulas approximate the integral of a function by evaluating the function at k nodes , , … , and weighting those nodes with n weights , , … , . The most common of Newton-Cotes quadrature formulas are the Mid-point, Trapezoidal and Simpson's rules. The general form of the Newton-Cotes formula is; The Newton's method can be derived from the Taylor's series expansion of a function (of a single variable) about the point : where , and its first and second derivatives are evaluated at . In the case of a multiple variable function : → , (2) can be shown [18], to equivalently give: The matrix of partial derivatives appearing in (3) is the Jacobian J, where ! is multiple integrals as in (4): The alternative approach is to treat the multiple integral as a nested sequence of one-dimensional integrals, and to use onedimensional quadrature rule with respect to each argument in turn [9]. Hence we can approximate ! with the weighted combination of quadrature formulas. The authors [6-8, 18-20, 23] and the references therein have proposed various methods for approximating the indefinite integral in equation (4) using Newton Cotes formulae of order zero to one. This study approximates the integral in Equation (4) by using the weighted combination of the Trapezoidal, Simpson -3/8 and Simpson -1/3 quadrature rules.
In this study the following objectives are achieved: (i) Broyden-like methods is developed using combined weights of the Trapezoidal, Simpson 3/8 and Simpson 1/3 quadrature rules; (ii) The new methods are analyzed by comparing the number of iterations and the CPU time with the existing Broyden-like methods using selected systems of nonlinear equations as test problems. In the rest of the paper, section 2.0 describes the general formula of Simpson 3/8 quadrature rule while section 3.0 gives details on how the Trapezoidal-Simpson 3/8 method was derived and the numerical schemes of the Trapezoidal -Simpson 1/3 method, with numerical tests and results well illustrated in section 4.0 and section 5.0 gives a summary conclusion on findings from the research.

The General Formula of Simpson 3/8 Quadrature Rule
Most (if not all) of the developed formulas for integration are based on a simple concept of approximating a given function f (x) by a simpler function (usually a polynomial function) f i (x), where i represents the order of the polynomial function. Simpsons 1/3 rule for integration was derived by approximating the integrand f (x) with a 2 nd order (quadratic) polynomial function f 2 (x) [2], given by: In a similar way, Simpson 3/8 rule for integration can be derived by approximating the given function f (x) with the 3 rd order (cubic) polynomial f 3 (x) given as The unknown coefficients a 0 , a 1 , a 2 , and a 3 in (6) can be obtained by substituting four known coordinate data points f ( )= + + ( + ( + ( ( ( The expression (7) can be put in matrix notation as: Expression (8) can symbolically be represented as Therefore: Substituting (10) into (6), we obtain (11): Furthermore Substituting the form of ( into I= , we have

Derivation of Trapezoidal-Simpson 3/8 Method (TS -3/8)
A Taylor's series expansion of a function (of a single variable) about a point given by where and its first and second derivatives, are calculated at , can be used to derive the Newton's method. For multiple vector variable function , an analogous expression for it [18], as in (4) where = , , … , and = , , … , . We assume that * is a simple root of the nonlinear equation = 0, an is sufficiently differentiable. We assume further that : L ⊂ → is a smooth mapping and has continuous second order partial derivatives on a convex open set L ⊂ and has a locally unique root in L. Taking into consideration the two quadrature rules that is: Trapezoidal quadrature rule Substituting (19) into (4) In (31) we have an implicit equation because of the presence of ? on both sides of it. To avoid its implicit nature we use the _ + 1 `F iteration of the Broyden's method on the right hand side of (31). Thus we have: and a = ! ?e ! Now replacing , b and a by c , c b and c a respectively and using the same procedure as prescribed in [5,4,9], we get Let c = 5c + 6c a + 5c b Hence we have the following method using initial matrix c + = : and an initial guess + . For a given + using initial matrix c + = : , an approximated solution for ? can be computed by the iterative schemes as in [18] where: a = ! ?e ! , _ = 0, 1, … In a similar way as in the above derivations, a weighted combination of the Trapezoidal -Simpson -1/3 quadrature rules the the numerical scheme as follows; Algorithm for the TS -3/8: 1. Given initial guess + , let _ = 0 and c + = : 2. Compute , f ≤ 10 " is satisfied stop, Else go to step

Convergence of the TS-3/8 Method
The properties of the local convergence for the proposed method are presented here with the following standard assumptions on the nonlinear function : 1. is differentiable in an open convex set L ∈ .
2. There exist * ∈ L ⊂ such that * = 0 and * is continuous for every ∈ L.

3.
is Lipschitz continuous of order 1 so that there exists a positive constant y such that z − h z ≤ yz − hz ∀ , h ∈ Definition 1.0 (q-super-linear convergence) [11] Let and * ∈ . Then → * is q -superlinear if lim →• z ? − * z z − * z = 0 Lemma 1.1 [21] Let : → be continuous and differentiable on an open convex set L ⊂ , ∈ L. If is Lipzschitz continuous with Lipscgitz constant y , then for any €, • ∈ L z • − € − • − € z ≤ yb <z€ − z, z• − zC. Moreover, if is invertible, then there exists ‚ and ƒ > 0 such that … z• − €z ≤ z • − € z ≤ ƒz• − €z for all €, • ∈ L for which yb <z€ − z, z• − zC ≤ ‚. Lemma 1.2 [21] Let ∈ , _ ≥ 0. If converges q-super-linearly to * ∈ , then Here, we present the main result which is a modified result in [16] to prove that the local order of convergence analysis is super-linear. Theorem 1.0 Let : → satisfy the hypothesis of Lemma 1.1 on the set D. Let c be a sequence of non-singular matrices in the linear space ‡ of real matrices of order n. Suppose for some + the sequence generated by (24) remains in D and lim →• = * for each ≠ * . Then the sequence < C converges q-super-linearly to * and * = 0 if and only if Where i = ? − and c = 5c + 3c a + 5c b .

Results and Discussion
Methods developed in this research (Trapezoidal -Simpson 3/8 and Trapezoidal -Simpson -1/3 methods) were compared with existing methods (Trapezoidal -Simpson and Trapezoidal -Simpson -Midpoint methods). The first and second results in Table 1, corresponds to the existing methods Trapezoidal -Simpson -Midpoint (TSM) and Trapezoidal -Simpson (TS) methods respectively, whilst the third and fourth methods are the results of the developed new methods namely the Trapezoidal -Simpson 1/3 (TS -1/3) and Trapezoidal -Simpson 3/8 (TS -3/8) methods.  Table 1 presents the results for solving the four benchmark problems with each of the four methods. The results indicate clearly that the proposed methods did not fail to meet the convergence criteria specified for all the selected benchmark problems. An observation from the table revealed that TSM method was unable to obtain solutions for problem four with values equal to 35, 65, 165, 365 665 and 1065, in addition, it recorded the highest number of iterations for most of the problems solved. TS and TS -3/8 methods recorded the lowest number of iterations for all four benchmark problems with each one of them recording the same number of iterations for each problem. Another important observation made from the data above shows clearly that the new developed TS -3/8 method required lesser CPU time to execute all four problems under consideration with values equal to 5, 35, 65, 165, 365 and 665 however in most cases for equal to 1065, TS method recorded a lesser CPU time compared to TS -3/8 method. Lastly, The TS -1/3 method also proposed in this research recorded a relatively high CPU time for all the problems compared with TS and TS -3/8 methods.

Conclusions
This paper has proposed and developed two new Broyden -like methods for solving system of nonlinear equations.