• Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month
  1. 1
  2. 2
  3. 3
  4. 4

Solving Equations Using Numerical Methods

Extracts from this document...


Matt Coles        Centre Number – 40120        Candidate Number - 3135

Solving Equations Using Numerical Methods

For my coursework, I am going to solve equations that cannot be solved analytically. This is when the equation would have been solved by using the quadratic formula, completing the square or by factorising it. The equations used will be solved numerically and there are three methods that I will be using:

  • The Change of Sign method
  • The Newton-Raphson method
  • The Rearrangement method

Method 1: The Change of Sign Method

The first method that I will be using is the Change of Sign method. The equation that I will be solving is y=x3+3x2-2x-1. To do this, I will be drawing the function of                 f(x)=x3+3x2-2x-1.

Root α

Root β

Root γ

To solve this equation using the Change of Sign method, I will be using the Decimal Search on Microsoft Excel.Firstly, I started by entering the equation into Autograph to get a sketch of the graph.

...read more.


After that, I moved in closer, entering -3.5 to -3.49 (in 0.001’s) with the formula in the column next to it. This gave me the answer as to where the change of sign was – between -3.491 and -3.49.

Finally, I entered the numbers between -3.491 and -3.49 (in 0.0001’s) into Excel. This showed me that the change of sign was between         -3.4909 and -3.4908.

This is now up to 5 sig.fig. meaning that I have found the answer with the error bounds on the left hand side. The error bounds show that this works as there is a change of sign.


This method fails in two ways – when there is a repeated root or there are two roots within consecutive integers. To show how this method fails, I am using the equation x3+x2-5x+3=0. To do this, I will be drawing the function of f(x)= x3+x2-5x+3.

Method 2: The Newton-Raphson Method

Root α

Root β

...read more.


Root β. The red line is the function of f(x)=x3-7x2+x+3 and the blue lines are the sliding tangents.The third root I have chosen to use is Root γ. Below is the Excel spread sheet with f(xn) and its derivative, f’(x) shown.

Root γ

On the left is the equation that shows the sliding tangent of the Newton-Raphson Method in a graphical form forRoot y. The red line is the function of f(x)=x3-7x2+x+3 and the blue lines are the sliding tangents.  Root γ = 2.4023 (5 sig.fig.).This root has error bounds



Root γ

Method 3: The Rearrangement Method

The third method that I am using is the Rearrangement Method. For this method, I will be using a different function to the other two. The equation I will be using for the Rearrangement method is y=x3+2x2-2x-2. This means that the function I will be using is f(x)= x3+2x2-2x-2.

...read more.

This student written piece of work is one of many that can be found in our AS and A Level Core & Pure Mathematics section.

Found what you're looking for?

  • Start learning 29% faster today
  • 150,000+ documents available
  • Just £6.99 a month

Not the one? Search for your essay title...
  • Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

See related essaysSee related essays

Related AS and A Level Core & Pure Mathematics essays

  1. Marked by a teacher

    C3 Coursework - different methods of solving equations.

    5 star(s)

    As an example I have used to demonstrate it is the f(x) = x5 + 6x2 - x + 4 Below is how the graph looks like : When I re arrange the equation in the form of x= g(x), it turns out to be this: When I plot this

  2. Marked by a teacher

    The Gradient Function

    5 star(s)

    + (15x4 + 30x�h + 30x�h + 15xh3 +h4) = 4x + 15x4 = x (4 + 15x^3) This seems to follow the same pattern as the previous investigation; each term follows the correct pattern in accordance with naxn-1 (2x� and 3x5 have the respective gradient functions of 4x and 15x4).

  1. MEI numerical Methods

    the upper bounds and lower are two opposite signs the root must lie within the range of values of the interval. As K varies As we proved above, if K = 1 the root is 0.479731007. However what if K, was 2, 3 etc, could a pattern be distinguished, the

  2. Numerical solutions of equations

    I have now found the same root using the Decimal Search method, the Rearrangement method and the Newton-Raphson method. It is now possible to compare these three methods in terms of ease and the speed of convergence. I found that the Decimal Search method was the simplest out of the

  1. Mathematical equations can be solved in many ways; however some equations cannot be solved ...

    It is reasonable to start at x=2. When x = 2, the graph shows that the Newton Raphson method does not work as the starting point on the graph is a stationary point. As a stationary point, this means that the tangent never touches the x-axis hence the Newton Raphson method does not work.

  2. Solving Equations. Three numerical methods are discussed in this investigation. There are advantages and ...

    3x2+6x+0.5 6.549861 0.060656 0.676186 3 0.676186 0.018947 3x2+6x+0.5 5.9288 0.003196 0.67299 4 0.67299 5.13E-05 3x2+6x+0.5 5.89669 8.7E-06 0.672982 5 0.672982 3.8E-10 3x2+6x+0.5 5.896603 6.45E-11 0.672982 6 0.672982 0 3x2+6x+0.5 5.896603 0 0.672982 After 6 converge we can see that the 1st target root is to , with an error bound and solution bound.

  1. C3 COURSEWORK - comparing methods of solving functions

    At last, I can see that Newton Raphson converges to a value near to -3.5269. Error Bounds We need to use the change of sign method to trap the root between two solution bounds I evaluate f(-3.52695) and f(-3.52685): f(-3.52685)=-0.00051 f(-3.52695)=0.000193 This change of sign shows that the root must be between these two values.

  2. Evaluating Three Methods of Solving Equations.

    of x, a tangent is drawn and extended up to the x-axis. This tangent should intercept the x-axis at a point nearer to the root of our equation. We then draw another tangent to the curve at the point corresponding to this new value of x and continue this process

  • Over 160,000 pieces
    of student written work
  • Annotated by
    experienced teachers
  • Ideas and feedback to
    improve your own work