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Section 7.4 Limitations of the bisection method

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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)

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To illustrate the second limitation, consider the equation x2−2x+0.9999=0. 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.

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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 \([-1, 0]\text{.}\)