Solution of equations by numerical methods

This investigation is to find equation solutions using three methods:

1. Change of sign using bisection, decimal search or linear interpolation.
2. Newton – Raphson.
3. Rearrangement of the equation f(x) = 0 into the form x = g(x).

The change of sign methods are systematic searches, which use the positive and negative signs of the f(x) solutions to find the location of the root within the intervals found using the graph of the function curve. The change of sign method I have chosen to use within my investigation is the decimal search method.

Fixed point iteration requires finding a single value or point as an estimation for the value of x, rather than establishing an interval as in the change of sign methods. The Newton – Raphson method and the rearrangement of equation f(x) = 0 into the form x = g(x) will be used to investigate this form of numerical equation solution.

Once each of these methods have been investigated I will compare each of them, in order to find the easiest method for equation solution. This comparison will also include the negative and positive points of each method, such as problems that result in an inability to find the correct root and the speed of convergence to the correct root.

Fixed Point Iteration – Newton – Raphson Method.

To start off using the Newton – Raphson method, we must first take an estimate of the root as a starting point. For a root f(x) we shall start with estimation x1 –we then draw a tangent to the curve y = f(x) at the point (x1, f(x1)).

From this we can see that there are three roots, within the intervals [-2,-1],[-1,0] and [1,2].

• one rearrangement: x = g(x) = x^5 - 2 which gives us the graph:

                                                        4

Taking x = 2.0 as the starting point to find the root within the interval [-1,0] – we gain the following set of results:

This rearrangement provides the basis for the iterative formula:

                        Xn+1 = x^5n - 2

                                   4

This gives the set of results:

This shows the root to be –0.50850 within the interval [-1,0], which can be written:

• -0.509 with a maximum error of ±0.0005, or
• -0.5 (to 1 d.p.)

Displayed graphically:

From which we gain the root as –0.5085 (to 4 s.f.) from the results box, which again gives the root in both the ways written above.

As         f(x) = x^5 – 4x – 2

And            g(x) = x = (x^5 – 2)/4

• g’(x) = (5x^4)/16

The magnitude of g(x) for the results gained above can be shown to be:

This shows that the magnitude is always less than one, and as the method will only succeed if        -1 < g’(x) < 1 , and as this is the case for all the values of x found, the method works.

Method Failure

We can see that this rearrangement of f(x) works when trying to find the root within the interval [-1,0], but I shall now try to find the root within the interval [1,2] using the same method.

The hardware and software that have been used in this investigation are a scientific calculator and a graph drawing program – Autograph. This allows us to find the roots of each graph without having to draw each of them out along with the tangents, and gives as accurate results for the roots with our own specification on the number of significant figures. This program proves little use for the decimal search method, other than establishing the intervals the roots lie in – the main body of work must then be done through calculations and recording of results, and each significant figure must be found one at a time.

The fixed-point iteration methods benefit greatly from the graph program, as they allow us to gain all the tangents and converging ‘staircases’ and ‘cobwebs’ simply by typing in the starting value of x. This gives us much faster convergence than calculating, and a graphical representation of what is happening, so that we can more fully understand the way each method works. Both are simple to use as long as the equation and the starting point for each interval are known.