Roots of f (x) = 0
In this chapter, numerical solutions for a single equation, f (x) = 0, are sought. The function, f (x), can be a polynomial function or any nonlinear function of x.
The fundamental theorem of algebra states that an n-th order polynomial equation has n roots including complex roots. This does not mean that all the roots for polynomial equations can be obtained analytically in closed form. In fact, a fifth-order polynomial equation and beyond has no formula for their roots in closed form.1
Two important algorithms to numerically solve a single equation, f (x) = 0, are explained in this chapter. They are (1) the bisection method and (2) Newton’s method. The bisection method is guaranteed to obtain at least one root while ...