Change of sign methods

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Change of sign methods

When solving any complex equation we:

  1. Rearrange the equation into the form f(x) = 0
  2. 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:

Root

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

Interval Bisection

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.

Linear Interpolation

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:

Formula

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