# Change of sign methods

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## Change of sign methods

When solving any complex equation we:

- Rearrange the equation into the form f(x) = 0
- Sketch the graph of this function (using a calculator or by plotting points.)

The solutions to our equation are where this graph crosses the x-axis. These solutions are called the **roots** of the equation.

To locate where a root is we have to find limits between which the solution lies. i.e. Find an x-value that has a positive value for f(x), and an x-value that has a negative value for f(x).

If the line is continuous then at some point in between these x-values the graph, (and value for f(x)), must = 0. **This is a solution to our equation:**

i.e. If f(a) < 0, and f(b) > 0, then the root lies in the interval a < x < b.

This idea is called a **change of sign**.

We will not be able to find the root exactly, but we will be able to 'home in' on the root until we have it to the desired degree of accuracy.

**Interval Bisection** is the simplest numerical method and uses the idea that if the root lies in the interval a < x < b, then a sensible estimate of the root is halfway between a and b.

For instance, at x = (a + b) ⁄ 2

Keeping a change of sign, replace one of the original limits with (a + b) ⁄ 2 and then repeat the process.

Each repetition is called an **iteration**.

After every iteration the interval containing the root will halve in width.

**Linear Interpolation** is very similar to interval bisection. To estimate the value of the root we join the y-values with a straight line and find where this crosses the x-axis.

This is our next estimate for the root, and replaces one of the original values, remembering to keep the 'change of sign'.

**The formula for the next estimate for the root is:**