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

    Newton Raphson only takes 3 rows of calculations to find 2 results which match to 3 decimal places which is the specified degree of accuracy for the initial decimal search calculation. This is faster than the x=g(x) iteration as this method takes 4 rows of calculations to find 2 results that match to the specified degree of accuracy.

  1. Numerical solutions of equations

    Therefore, I have found a rearrangement of the same equation (Rearrangement 1) where the iteration has failed to converge to the required root. Comparison of methods I will now use the equation 0=x5+4x2-2 that I solved using the Rearrangement method and see if I can find the same root using

  2. This coursework is about finding the roots of equations by numerical methods.

    is negative and f(3) is positive tells us the graph must cross the x axis between 2 and 3. Hence there is a root in the range (2, 3) X y 2.56150 -0.00034 2.56151 -0.00028 2.56152 -0.00021 2.56153 -0.00015 2.56154 -8.2E-05 2.56155 -1.8E-05 2.56156 4.63E-05 This graph shows us how the root is finally trapped in the range (2.56155, 2.56156)

  1. C3 Coursework: Numerical Methods

    As the Newton Raphson method relies on the gradient of the points it is not possible to use the Newton Raphson method the find the roots of the equation y=log(x+3)-x. I shall now attempt to use the Newton Raphson formula to find the roots of the equation.

  2. Maths - Investigate how many people can be carried in each type of vessel.

    This is a mathematical concept that is quite new to me and it was discovered after browsing through a few mathematics websites during my search for knowledge upon the investigation. After reading a tutorial, I eventually realised that the use of matrices is perfect for solving this type of equation.

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

    Microsoft excel can be set up in order to reduce human error and increase efficient of calculation. A spread sheet is set up for n a f(a) B f(b) 1 0 =B2^3-5*B2^2+4*B2+2 2 =D2^3-5*D2^2+4*D2+2 2 =IF(G2>0,F2,B2) =B3^3-5*B3^2+4*B3+2 =IF(G2>0,D2,F2) =D3^3-5*D3^2+4*D3+2 3 =IF(G3>0,F3,B3)

  2. C3 COURSEWORK - comparing methods of solving functions

    or -0.54113([-1, 0]) or 0.82868([0, 1]). Hence g?(root) in the interval [0, 1]= -1.4617 As g?(x) < -1, we can expect that the iteration to diverge away from the root in a cobweb fashion. This can be shown on the graph above, and therefore this must be correct.

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