Nonlinear Equation Solved using the Successive Substitution Method

Consider first a single nonlinear equation in one unknown.

f(x) = 0

One quick method for finding a root is to program a programmable calculator to evaluate f for a given x, and then you supply various x until f(x) = 0. Another alternative is to program a personal computer to make a table of x, f(x) and look for the root in the table, refining the table where needed. These methods can be employed when the problem is not too hard and you only want to solve it once or twice and go on, or when a computer program is not available.

Iterative methods are also employed to solve the equation by choosing a guess, getting a better value, and continuing. We denote the k-th iteration as x^k. The successive substitution method is written as

x⁰ = guessed

x^k+1 = x^{k+1 +} b f(x^k), k = 1,...,

If the constant b is chosen correctly these iterations will converge to a solution. The convergence is linear and may be slow, requiring many iterations, but the method is easy to program. This method is difficult to apply to systems of equations, but it is usefully applied in the solution of ordinary differential equations. In that case, the constant b is proportional to t, and t is reduced until the method converges.

As an example, consider the very simple problem

f(x) = x² ­ 2

with the solution x = 1.4142136. The successive substitution method is

The derivative of g is

It is clear that the absolute value of this will be greater than 1 unless 2bx is negative (this is a necessary condition for convergence (link)). Let's concentrate on the positive solution; then we need b negative. These iterations are done for an initial guess of x = 1.0 and b ­0.25, ­0.125, ­0.0625. The results are shown in the figure. Clearly the iteration is slower for smaller b. It was relatively

easy to find the appropriate b in this case, but it is usually harder, and may be very hard when there are several equations to be solved together. The Newton-Raphson method illustrated in the figure is described elsewhere (link).

Figure SS1. Successive Substitution and Newton-Raphson Methods Applied to Example, f(x) = x² - 2.

Take Home Message: The convergence of the successive substitution method is slower for smaller beta.