The iterative processes used to solve linear and nonlinear equations are based on trial-and-error methods. A trial solution, consisting of numerical estimates for the variables used in the equation(s) is first generated. The numerical estimates of the variables is then substituted into the equation(s). The relative error of the variables(s) numerical values is then calculated based on the actual value obtained for the equation(s) with the substituted variables and the actual numerical value that the equation(s) is equal to. An new estimate of the numerical values of the variables is obtained using the error computed. The new trial solution is substituted again into the equations and the relative error calculated. The process is repeated until the error for the solution reaches a prescribed minimum.
Graphically, the iterative process requires plotting the graphs of each equation over regions that all the equations are most likely to intersect at, and then repeating the process over different regions until the common point that all the equations intersect at is found. The plot of the graphs of the equations over a region is started at one endpoint of the region and continued sequentially in one direction. If the plots of the graphs tend to create lines or curves that are headed continually toward each other (converge) at what appears to be the intersection point of the equations, further points on the equations are plotted until the intersection point is reached. If the lines or curves tend to diverge away from each other, a new region is selected for graphing and the points on the equations for the remainder of that region are not calculated.
Numerical iterative methods are used to find the point where the graph of an equation intersects the x-axis. These intersections points are also known as the roots, zeros or x-intercepts of the equation. There are numerous different iterative methods to find the roots of an equation. Different methods are used depending on the type of equation that needs to be solved. For example, an iterative method capable of finding the root of a polynomial will not converge to a solution with an equation that is not a polynomial.
Iterative methods are used to find the point of intersection of two or more linear (lines) or nonlinear (curves), also known as systems of linear and nonlinear equations. Different iterative methods have been developed and continue to be developed for systems of linear and nonlinear equations. Different techniques are used for solving linear systems and nonlinear systems. Different types of iterative methods are used to solve for different types of linear and nonlinear systems. That is because there is not one type iterative method that can be used to successfully and efficiently solve all types of linear and nonlinear systems. Engineers and programmers select the iterative method to use based on the characteristics of the system that needs to be solved.