Polynomial and Transcendental Equations

The complexity of solutions for polynomial and transcendental equations increases with increasing nonlinearity. Quadratic equations can be readily solved using the quadratic formula, provided such equations can be readily rearranged in the appropriate form. Formulas exist for obtaining roots of a cubic equation, but these are rarely used. No such easy formulas are available for solution of higher-order polynomials and transcendental equations.

These equations are typically solved by guessing a solution (root) and refining the value of the root on the basis of the behavior of the function. The principle of the Newton-Raphson technique, one of the most common techniques used for determining the roots of an equation, is represented by equation 4.16 [4]:

Here, x_n and x_n+1 are the old and new values of the root; f(x_n) and f’(x_n) are values of the function and its derivative, respectively, evaluated at the old root.

The calculations are repeated iteratively; that is, so long as the values of the roots do not converge, the new root is reset as the old root and a newer value of the root evaluated. It is obvious the new root will equal the old root when the function value is zero. In practice, the two values do not coincide exactly, but a tolerance value is defined for convergence. For example, the calculations may be stopped when the two values differ by less than 0.1% (or some other acceptable criteria).

The computations for this technique depend on not only the function value but also its behavior (derivative) at the root value. The initial guess is extremely important, as the search for the root proceeds on the basis of the function and derivative values at this point. Proper choice of the root will yield a quick solution, whereas an improper choice of the initial guess may lead to the failure of the technique.

The iterative successive substitution method can also be used to solve such equations [9]. The method involves rearranging the equation f(x) = 0 in the form x = g(x). The iterative solution algorithm can then be represented by the following equation:

Here, x_i+1 is the new value of the root, which is calculated from the old value of the root x_i. Each successive value of x would be close to the actual solution of the equation. The key to the success of the method is in the proper rearrangement of the equations, as it is possible for the values to diverge away from the solution rather than toward a solution.

Finding the roots of polynomial equations presents a particular challenge. An nth-order polynomial will have n roots, which may or may not be distinct and may be real or complex. The solution technique described previously may be able to find only a single root, irrespective of the initial guess. The polynomial needs to be deflated—its order reduced by factoring out the root discovered—progressively to find all the n roots. It should be noted that in engineering applications, only one root may be of interest, the others needed only for mathematically complete solution. For example, the cubic equation of state may have only one real positive root for volume, and that is the only root of interest to the engineer. A complex or negative root, while mathematically correct as an answer, is not needed by the engineer.