# The method I am going to use to solve x−3x-1=0 is the Change Of Sign Method involving the Decimal Search method

Mathematics C3 Coursework

Numerical solutions of equations

Decimal Search Method --------------------------------------------p.1

Rearrangement Method ------------------------------------------- p.5

Newton-Raphson Method ----------------------------------------- p.9

Comparison ----------------------------------------------------------- p.12

Solving  x³−3x-1=0 using the “Change Of Sign” Method:

The method I am going to use to solve  x³−3x-1=0 is the Change Of Sign Method involving the Decimal Search method, which is that you are looking for the roots of the equations f(x)=0. This means that I want the value of x for which the graph of y=f(x) crosses the x axis. As the curve crosses the x axis, f(x)changes sign, so provided that f(x) is continuous function, once I have located an interval in which f(x) changes sign, I know that that interval must contain a root. Now, I have drawn the graph of x³−3x-1=0 by using the Autograph software, and the graph is shown below:

The point that the arrow pointing is the root I need to find.

From my graph above, I can see that the root of the equation is between x=0 and x=-1. The table of x values and f(x) values is shown below. I can work out the f(x) values by substituting the x-values into the equation.

x                      f(x)

-0.1                    -1.299

-0.2                    -0.408

-0.3                    -0.127

-0.4                     0.136

From the table above, there is a change of sign from negative to positive, so I am sure the root is between x=-0.3 and x=-0.4. Therefore, I can narrow there values down further to find another change of sign.

x                      f(x)

-0.31             -0.09979

-0.32             -0.07277

-0.33             -0.04594

-0.34             -0.01930

-0.35              0.00713

A

I can see there is a change of sign between x=-0.34 and x=-0.35. Therefore, I can narrow there values down further again to find another change of sign.

x                      f(x)

-0.349            4.491451 x 10^-3

-0.348            1.855808 x 10^-3

-0.347            -7.81923  x 10^-4

I can see there is a change of sign between x=-0.347 and x=-0.348. I can stop it as I just want a root correct to 3 decimal place only and the accuracy is reasonable as if I correct the 4 or even more decimal place, it will be too complicated to find as it need so many steps to repeat. The root now is

-0.3475 and the maximum error of this root would be 0.0005.

The “Change Of Sign” is very effective, however, sometimes the Decimal Search method would not work with some function. An example is 5x^4+x³−2x²−0.1x+0.1=0 . This would not work because the roots are so near. The graph of this function and the calculations are below:

I want the root that the arrow is pointing ,the root lies between x=0 and x=1.

x                        f(x)

0                        0.1

0.1                   0.0715         ...