In the solution of linear equations and systems, f(x) = 0, we had the choice of using either direct methods or iterative processes. A direct method in that setting was simply the application of an exact formula involving only the four basic operations: addition, subtraction, multiplication, and division. The issues with this method arise when cancellation occurs, mainly whenever sums and subtractions are present. Iterative methods, rather than computing a solution in a finite number of operations, calculate closer and closer approximations to the said solution, improving the accuracy with each step.

In the case of nonlinear equations, direct methods are seldom a good idea. Even when a formula is available, the presence of nonbasic operations leads to uncomfortable rounding errors. Let's see this using a very basic example.

Consider the quadratic equation ax² + bx + c = 0, with a = 10^–10, b = –(10¹⁰ + 1)/10¹⁰, and c = 1. These are the coefficients of the expanded...