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Section 7.4 Limitations of the bisection method

The main limitation of the bisection method are:

  • It does not apply to systems of more than one equation
  • It requires the knowledge of a bracketing interval
  • It requires a continuous function
  • Its speed of convergence is slow (linear)

To illustrate the second limitation, consider the equation \(x^2 - 2x + 0.9999 = 0\text{.}\) It has two roots, both of which are very close to 1. Outside of a small neighborhood of 1, the function is positive. So, in order to construct a bracketing interval one needs to place one of its endpoints very close to 1. This means one needs to have a good idea of where the roots are in order to start with the method.

Calculus methods involving the derivative \(f'\) can help us understand in what direction the function changes, improving our chance of finding a bracketing interval.

For the function \(f(x) = 2e^x + x^3 - 1\text{,}\) determine the number of roots and find a bracketing interval for each of them.

Solution

The derivative \(f'(x) = 2e^x + 3x^2\) is always positive. Therefore, the function \(f\) increases on the real line. Such a function either has no roots (if its graph never crosses the \(y\)-axis), or has one root. Since \(f(x)\to \infty\) as \(x\to \infty\) and \(f(x) \to -\infty\) as \(x\to-\infty\text{,}\) it follows that the graph crosses the \(y\)-axis. We need a finite bracketing interval, since for the bisection method to work, both \(a\) and \(b\) must be finite. Since \(f(0) = 2 + 0 - 1 = 1 > 0\) it remains to find a negative value. For example, \(f(-1) = 2e^{-1} - 1 - 1 = 2(1-e)/e \lt 0\text{.}\)

Answer: one root, with a bracketing interval \([-1, 0]\text{.}\)