Numerical Solution of Equations

2026 Syllabus Objectives

By the end of this topic, you should be able to:

  1. Locate approximately a root of an equation using graphical methods and/or by searching for a sign change (for example, finding a pair of consecutive integers between which a root lies)

  2. Understand the idea of, and use the notation for, a sequence of approximations which converges to a root of an equation

  3. Understand how a given simple iterative formula of the form x_{n+1} = F(x_n) relates to the equation being solved, and use a given iteration (or an iteration based on a given rearrangement of an equation) to determine a root to a prescribed degree of accuracy. You should also understand that an iteration may fail to converge.

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What is a Root of an Equation?

A root of an equation is a value of x that makes the equation true (equal to zero).

For example, if we have the equation x² - 5x + 6 = 0, the roots are x = 2 and x = 3, because:

  • When x = 2: 2² - 5(2) + 6 = 4 - 10 + 6 = 0 ✓
  • When x = 3: 3² - 5(3) + 6 = 9 - 15 + 6 = 0 ✓

Some equations are easy to solve exactly (like the one above). But many equations are much harder or even impossible to solve exactly. For these, we use numerical methods to find approximate roots (roots that are very close to the true value).

1. Locating Roots of an Equation

There are two main ways to locate where a root is:

Method A: Graphical Considerations

We can sketch the graph of a function and see where it crosses the x-axis. Each point where the graph crosses or touches the x-axis is a root.

Example: Consider the equation x³ - x - 1 = 0

We can think of this as the function f(x) = x³ - x - 1

By sketching the curve y = x³ - x - 1, we can see roughly where it crosses the x-axis. The x-coordinate of this crossing point is the root.

From a sketch, we might observe that the curve crosses the x-axis somewhere between x = 1 and x = 2.

Method B: Searching for a Sign Change

This is a very powerful method. The key idea is:

If a continuous function changes sign between two values, then there must be a root between those values.

In other words:

  • If f(a) is negative and f(b) is positive (or vice versa), then there is a root between a and b.

Why does this work? Imagine you're walking along the graph of a function. If you start below the x-axis (negative y-value) and end up above the x-axis (positive y-value), you must have crossed the x-axis at some point. That crossing point is a root!

Example: Find a pair of consecutive integers between which a root of x³ - x - 1 = 0 lies.

Let f(x) = x³ - x - 1

Test integer values:

  • f(0) = 0³ - 0 - 1 = -1 (negative)
  • f(1) = 1³ - 1 - 1 = -1 (negative)
  • f(2) = 2³ - 2 - 1 = 8 - 2 - 1 = 5 (positive)

There is a sign change between x = 1 and x = 2 (from -1 to +5), so there must be a root between 1 and 2.

Step-by-step process:

  1. Rearrange the equation to the form f(x) = 0
  2. Test consecutive integer values of x
  3. Calculate f(x) for each value
  4. Look for where the sign changes from positive to negative (or negative to positive)
  5. The root lies between these two values

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