• 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
  5. 5
  6. 6
  7. 7
  8. 8
  9. 9
  10. 10
  11. 11
  12. 12
  13. 13
  14. 14
  15. 15
  16. 16
  17. 17
  18. 18
  19. 19
  20. 20

Investigate the three different numerical methods used to solve equations.

Extracts from this document...


Solving Equations by Numerical Methods Introduction In this coursework I am going to investigate the three different numerical methods used to solve equations. These include the change of sign method, Newton Raphson method and the Rearranging method. To carry out the investigation I will explain how each method works, using an example of a working equation in each case. I will also show when each of the methods will not work with other equations. I will then compare all three of the methods. Change of sign method Decimal search and Interval bisection are both ways of finding an interval where there is a change of sign. The change of sign can be found on a graph when the line crosses the x-axis. Wherever there is a change of sign there will be a root. These two methods find the interval where the root lies. Decimal Search To find the roots using decimal search, the y-values must be found, using the values of X from 0.1, 0.2, 0.3, all the way up to 1. When a change of sign occurs, this means there is a root lying here. Once the root interval to 1 decimal place has been found, it must then be found to 2 and 3 decimal places and so on to the required number of places. ...read more.


n Xn Xn+1 1 -2 -8.6 2 -8.6 877.8978571 3 877.8978571 -3417800597 4 -3417800597 1.99623E+29 5 1.99623E+29 -3.97741E+88 6 -3.97741E+88 #NUM! 7 #NUM! #NUM! After entering my formula into the spreadsheet, the table confirms what the graph displayed. The curve of the equation is too steep on both sides of the root. When a tangent is drawn at -2 it crosses the x-axis way past the root at -8.6 and so it is diverging away from the root. This confirms that the Newton Raphson method will not find the roots in this case. Rearranging Equations If you use any f(x) equation and rearrange it into the form x = g(x) the point on the graph f(x) where y = 0 (The root) is the same x-value as the point where y = g(x) crosses the line y = x. To find the roots, I need to find the point where the line y = g(x) crosses the line y = x. I will put in an estimate value of x (X0) and it will lead to a better estimate (X1). I have chosen the equation f(x) = x3-3x+1 This can then be rearranged into the form: 3x = x3+1 x = (x3+1) ...read more.


From the table it can be concluded that the Newton Raphson method is the quickest of the three as it shows convergence very quickly each time. The decimal search is the slowest by far of the three methods. It can take around 25 steps to home in on the root. At first the Rearrangement and Newton Raphson methods appear to be the most difficult to use, since the formulas must be calculated before hand. Once the formula has been calculated and inputted for the first estimate, after this it is very simple. When using a spreadsheet it can then just be dragged down until it produces convergence. Even including the calculations needed for the Newton Raphson and rearrangement method, they still prove to be far quicker than either of the change of sign methods. The main advantage of the change of sign method over the other 2 methods is that it is easy to find all of the roots to almost any equation. Overall I would say that the Newton Raphson method is the most efficient of the three. Although there is some preparation before using the method, it still ends up being the quickest. Pure Mathematics 2 Coursework Michael Hatt ...read more.

The above preview is unformatted text

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)

    Here is the proof: Newton Raphson method This method works by plotting the f(x) on a graph and visually looking at between what two points the root is (in single units such as 1 or 5 or 9). Then we draw a tangent at that point on the graph. E.g.

  2. Marked by a teacher

    The Gradient Function

    5 star(s)

    4.001 768.768288 768.5761 3 243 3.1 277.0563 357.492 3 243 3.01 246.256236 327.2508 3 243 3.001 243.324162 324.3241 2 48 2.1 58.3443 111.132 2 48 2.01 48.96722403 97.44721 2 48 2.001 48.09607202 96.14407 1 3 1.1 4.3923 15.972 1 3 1.01 3.12181203 12.36361 1 3 1.001 3.012018012 12.03604 x x4

  1. Marked by a teacher

    Estimate a consumption function for the UK economy explaining the economic theory and statistical ...

    3 star(s)

    Figure8 (a) shows that even there is still wide gap between the predict line and real line, but the gap closer than the before from 1950-1982. In addition, the Figure8 (b) also displays a random between the residuals. For the second reason the uncertainty leads the households save more and spend less.

  2. Am going to use numerical methods to solve equations that can't be solved algebraically

    I am confident that I can have this root to four decimal places giving error bounds. Therefore the root is x = 1.467443089= 1.46745 + 0.00005 Results from other two roots x0 = 0 x1 =0.334026333 x2 =0.615299538 x3 =0.348013325 x4 =0.334012999 x5 =0.334026363 x6 =0.334026364 x7 = 0.334026364 x1=

  1. C3 Mei - Numerical Methods to solve equations

    x f(x) 0 -0.61962 -0.1 -0.48442 -0.2 -0.35482 -0.3 -0.23682 -0.4 -0.13642 -0.5 -0.05962 -0.6 -0.01242 -0.7 -0.00082 -0.8 -0.03082 -0.9 -0.10842 -1 -0.23962 The table clearly shows that decimal search has failed to find the root as there is no sign change and f(x)

  2. C3 Coursework: Numerical Methods

    When differentiated, y=log(x+3)-x is This does not work as an equation so it is impossible to find the root of this equation using the Newton Raphson method. X=g(x) Method The x=g(x) method is another fixed point iteration method. This means that an estimate of the root is need as a starting point.

  1. MEI numerical Methods

    The difference between the two methods is that the secant method uses two approximations which are the most recent approximations of the root. The false position method also uses two approximations; it uses the most recent approximation of the root and the most recent root which has the opposite sign of the most recent approximation of the root.

  2. C3 COURSEWORK - comparing methods of solving functions

    This is because of the steep gradient where the graph crosses the x-axis, meaning that the resulting tangent is directed further away from the root.

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