Roots of Nonlinear Equations

Abstract

A large number of engineering and scientific problems can be formulated in terms of finding the value, or values, of some variable x which results in a zero value of some function of that variable. Mathematically, this is represented by the equation (3.1)$$F\left( x \right) = 0,$$where F(x) is some given function of x. Examples are polynomial equations such as (3.2)$$F\left( x \right) = 5x^4 - 4x^2 + 2x - 3 = 0,$$or transcendental equations such as (3.3)$$F\left( x \right) = \tan \left( x \right) - 1/x.$$For the case of polynomial equations, the solution values of x which satisfy the equation are frequently called “Croots” of the polynomial. In general these may be real numbers or complex numbers. For the case of transcendental equations such as Eq. (3.3) the solution values are typically called “zeros” of the function. Mathematically the terms roots and zeros are used interchangeably.