Rearrangement

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Rearrangement

This is an alternative method that rearranges the original question into two equations, a straight line and a curve, and then finds where these meet. It works as follows:

  1. Rearrange the equation to be solved into the form x = g(x). [g(x) is a different function].
  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. In this situation, we can guess a value x0 and hope that g(x0) will be a better guess. This can be iterated repeatedly.

Below is a diagram which shows that by starting at a value of x0 and repeating the iterations (i.e. finding x1, x2,...), you get closer and closer to the point where the two lines cross.

Rearrangement

Performing this iteration produces two possible results:

  1. It diverges (i.e. it gets further and further away from the start). This means the rearrangement has not worked.
  2. It converges (i.e. it homes in) to the root, and solves the equation.

Example:

Let's take the example used in the Newton Raphson Learn-It.

We want to find the roots to the following equation:

f(x) = ex − 3x,

This can be rearranged to x = ex ⁄ 3

Let's estimate that the root to this equation is 0.6.

Then taking the iteration equation above [xn+1 = g(xn)] and substituting our equation gives:

Example

Performing this iteration a number of times gives:

X1 X2 X3 X4 X5 X6 X7 X8 X9 X10
0.607 0.612 0.615 0.616 0.617 0.618 0.618 0.619 0.619 0.619

As the solution gives consistent values of 0.619, this suggests that a root of the equation f(x) = ex − 3x is 0.619 (to 3 decimal places).

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