S-Cool Revision Summary

S-Cool Revision Summary

Numerical Methods are systems, or algorithms, for solving equations that cannot be solved using normal techniques.

Change of Sign Methods

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

  2. Sketch the graph of this function (using a calculator or by plotting points.)

  3. Find an x-value that has a positive value for f(x), and an x-value that has a negative value for f(x).

  4. (if the line is continuous) the roots of the equation will be between these two values of x.

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

This allows you to get a more accurate solution to an equation than just using the Change of Sign Methods alone.

  1. Find values x that change the sign (as above).

  2. Change one of the values to the average value of the two x values (i.e. )

  3. Repeat step 2 above (each repetition is called an iteration), until you get solution that is accurate (i.e. correct to 3 decimal places).

Linear Interpolation

Linear Interpolation is very similar to interval bisection; instead of taking the average of the two points (i.e. ), it estimates that the root is on a straight line between the two points.

Steps to solve an equation are exactly as the Interval Bisection method (shown above), but replace the equation with

The Newton Raphson method does not need a change of sign, but instead uses the tangent to the graph at a known point to provide a better estimate for the root of the equation. It works on the basis that an estimate for the root is found using the iteration:

.

Steps to find a solution:

  1. Rearrange the equation to the format =0 and differentiate this to give .

  2. Select an estimate of the root (i.e. the solution to the equation)

  3. Put this estimate into the Newton Raphson equation

  4. Perform step 3 until you get a consistent solution.

Rearrangement

  1. Rearrange the equation to be solved into the form x = g(x).

  2. The solution to this equation is to find where y = x meets y = g(x). This is where the coordinates on g(x) are the same.

  3. uess a value xo and hope that g(xo) will be a better guess.

  4. Take this solution and re-input it into the above step, until you get a consistent solution.

The solution may converge and provide you the solution OR it may diverge. In this case a solution will not be found.