Numerical Solution of Nonlinear Systems of Algebraic Equations

Considered in this paper are two basic methods of approximating the solutions of nonlinear systems of algebraic equations. The Steepest Descent method was presented as a way of obtaining good and sufficient initial guess (starting value) which is in turn used for the Broyden’s method. Broyden’s method on the other hand replaces the Newton’s method which requires the use of the inverse of the Jacobian matrix at every new step of iteration with a matrix whose inverse is directly determined at each step by up-dating the previous inverse. The result obtained by this method revealed that the setbacks encountered in computing the inverse of the Jacobian matrix at every step number is eliminated hence saving human effort and computer time. The obtained results also showed that the number of steps that is reduced when compared to Newton’s method used on the same problem.


Introduction
Numerical solution of algebraic equations is one of the main aspects of computational mathematics. Numerical computation involves numerical answers given numerical inputs. Such problem arises whenever mathematics is used to model the real world problems (See, De Cezaro, 2008).
It is by no means uncommon for systems of nonlinear equations to occur in practice. Systems of nonlinear equations or simply nonlinear equations are equations whose variables are either of degree greater than 1 or less than 1, but never 1.
Nonlinear equations arise in all branches of sciences and engineering. It is widely used in optimization problems and electrical circuits. There are few nonlinear equations that are readily solvable but these are definitely of the minority. Where no simple method exists for solving nonlinear equations, numerical methods are frequently employed and it is the purpose of this work to investigate some of these techniques. Gilberto (2004) presented methods for the solution of single non-linear equations as well as for systems of such equations using Newton-Raphson iteration method. Goh and McDonald (2015) discovered that, an exact line search at a point, far from the solution, may be counterproductive.
Numerical techniques often make use of a process known as iteration. Solutions to these systems of equations F x 0;i 1, 2, 3, 4, . . . , n , are a problem that is avoided when possible, because no simple method (analytical) is found yet for solving these systems of equations. Hence this work is concerned with the numerical solutions of systems of nonlinear algebraic equations by steepest Descent and Broyden's methods. Some simple examples are solved as a perquisite to each of these methods.
The methods for solving a nonlinear system of algebraic equations of the type, 0, , 1, 2, … , , dates back to the seminal work of Isaac Newton. Nowadays a Newtonlike algorithm is still the most popular; this is due to its easy numerical implementation (See Vincent and Grantham 1997, Barbashin andKrasovskii 1952 andPowell 1986). However, this type of algorithm is sensitive to the initial guess , , , . . . , ! of the solution and is expensive in the computations of the Jacobian matrix This work is aimed at closing the gap created by the problem of initial guess and the problem of finding the inverse of the Jacobian matrix at every iteration step encountered by Newton's methods. Though Steepest Descent methodsserve the purpose of providing sufficiently accurate initial guess, butit will always converge even for poor initial guess (See De Cezaro 2008, Huang 2011and Powell 1970 The idea behind Broyden's method is to compute the whole Jacobian matrix and its inverse only at the first iteration and to do a rank-one update at the other iteration steps while avoiding the inverse at each stage of the iteration (See LaSalle 1976, Gomes & Martinez 1992 and Goh 1994).

Methodology
Consider the solution of system of nonlinear algebraic equations for which we do not know a simple analytical technique of the form The methods of solution for one equation and one unknown will be adopted in this work.

Steepest Descent Method
This method is used to find sufficiently accurate starting approximations for the Newton-based and other techniques (See Fan and Yuan, 2005). The method determines a local minimum for a multivariable function of the form ): + → +. Although the method is valuable, quite apart from the application as starting method for solving nonlinear systems, the connection between the minimization of a function from + -. + and the solution of a system of nonlinear equation is due to the fact that the system of the form has the minimal value zero.
The method of Steepest Descent for finding a local minimum for an arbitrary function ) from + -. + can be intuitively be described as follows; 1. Evaluate ) at an initial approximation 2 = (2 , 2 , , . . . , 2 ) 3 2. Determine a direction from 2 that result in a decrease in the value of ).

Move an approximate amount in this direction and call
it the new vector 2 . 4. Repeat step 1 through 3 with 2 replaced by 2 To extend this result to multi-variable functions, we need the following steps and definition.

Brodyen's Method
This method is one of the most effective algorithms for solving nonlinear systems of equations when the number of equations and unknowns is very large memory less implementations of this method is frequently used (See Byrd and Nocedal, 2004) This method only requires n scalar functional evaluations per iteration and also reduces the number of arithmetic calculations to O( ). This method belongs to a class known as least change Secant updates that produces algorithms called quasi-Newton. This method replaces the Jacobian matrix in Newton's method with an approximation matrix that is updated at each iteration step (See Goh, 2010).
Suppose that an initial approximation 2 is given to the solution of ( ) = 0 . We calculate the next approximation 2 in the same manner as Nwetons method.

Conclusion
This work considered two basic methods of approximating the solutions of nonlinear systems of equations. The Steepest Descent method provided good and sufficient initial guess (starting value) for the Broyden's method. The starting values provided by Steepest Descent method were then used by Broyden's method to arrive at the approximate solution for each given systems of equations. This technique replaces the inverse of the Jacobian matrix [J] in Newton's method with a matrix 7 0O whose inverse is directly determined at each step by up-dating the previous inverse thereby eliminating many computational steps if the problem were to be solved using Newton's method.