The numerical solution of equations

A helpful tool not very generally known among engineers is the use of approximation methods for the solution of equations which cannot be solved algebraically; the author describes and illustrates two such methods. IN engineering work many of the equations or systems of equations that are encountered may be solved explicitly for the unknowns. Some of these solutions are well known, such as the formula for the roots of a quadratic equation in one unknown and the use of determinants in solving a system of linear equations in any number of unknowns. If it is very difficult or impossible to solve explicitly for the unknowns of a system of equations, some approximation method may be used. The first and perhaps best known of these approximation methods is the graphical method,¹ which is easily understood but may be quite tedious and inaccurate, especially if more than one unknown is present. Horner´s method is well known and is used to approximate the real roots of algebraic equations. Two other approximation methods are Newton´s method^1,2 and the iteration process.¹ Although Newton´s method is really an iteration process, the two will be considered separately in this article, the purpose of which is to explain the use of these two methods and to illustrate by means of typical engineering examples their application to the numerical solution of equations.