• 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

maths pure

Extracts from this document...


MEI PURE 2 COURSEWORK SOLUTION OF EQUATIONS BY NUMERICAL METHODS 1. Change of Sign Method This method makes use of the fact f(x) changes sign at a root of an equation. f(x) must be a continuous function i.e. it must not have any asymptotes or other breaks in it. Once an interval in which f(x) changes sign is located, we know that that interval contains a root. It is best to sketch the diagram of f(x) first so that we can see how many roots the equation has and their approximate positions. Decimal Search The equation that will be investigated here is f(x) = 4x3+5, a diagram of which is shown below. From the graph we can see that there is only one root. Zooming in, as shown below, we can also see that this root lies between X=-1 and X=-2 Taking increments in x of 0.1 within the interval [-2, -1] and working out the value of the function f(x) = 4x3+5 for each one and then seeing where the sign changes will enable the narrowing down of the interval. x -2 -1.9 -1.8 -1.7 -1.6 -1.5 -1.4 -1.3 -1.2 -1.1 -1.0 f(x) -27 -22.4 -18.3 -14.7 -11.4 -8.5 -6.0 -3.8 -1.9 -0.3 1 The values calculated in the table above have only been given to 1 decimal place as we are only looking for a change in sign at this stage. ...read more.


will be very small. This will result in x2 not being close to the root. The values that are being computed may converge but it may be that they are converging towards a root other than the one we are trying to locate. 3. Rearranging f(x) = 0 in the form x = g(x) When trying to solve an equation f(x) = 0 by an iterative method, we first rearrange f(x) = 0 into a form x=g(x). The iteration formula is then? xn+1 = g(xn) The equation that will be looked at here is x3-3x-5=0, the graph of which is illustrated below. As can be seen above there is a root close to x=2. Rearranging the equation x3-3x-5=0 gives x = 3?(3x+5) Using the formula? xn+1 = 3?(3x+5) and starting with x0 = 2, the results for successive iterations are as follows: X0 2 X1 2.223980091 X2 2.268372388 X3 2.276967161 X4 2.278623713 X5 2.278942719 From this we can conclude that the root of the equation is 2.279 to 3 d.p. The convergence of a root such as this is often described as a staircase approach; why this is can be seen from the diagram below: The successive steps taken to approach the root is like that of a staircase with the values of xn approaching from one side. ...read more.


The rearranging method can be tedious in the sense that once the equation is rearranged, it has to be checked each time by differentiating and using an approximate value for the root. Again, similar to the Newton-Raphson method, if the rearranged function is a large, complex one a significant amount of time may be spent on the calculus part and with this method there is the added uncertainty that even when the function is differentiated it may not be suitable i.e. it may produce a diverging sequence. The use of SPA Software Omnigraph, Microsoft Excel and my Casio graphical calculator was beneficial for all three methods. Graphs could be easily plotted so that approximate values for boundaries could be seen and calculations were speeded up by the use of formulae functions. Although the use of software and hardware benefited all three methods it is perhaps the change of sign method that was speeded up most as virtually all the calculation with regards to this technique could be done by Excel/Graphical calculator. The other two techniques required differentiation which had to be done by hand. However, if the equation is a simple one that can be differentiated easily, the Newton-Raphson method, once programmed onto a computer is undoubtedly the quickest and reaches a high degree of accuracy in a relatively short amount of time. ...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

    The Gradient Function

    5 star(s)

    The general pattern here between the two values of x and the gradient, the value of the gradient function here is 4x�. Using this formula, the next 5 values of x and the gradient are: x x� 4x� 5 125 500 6 216 864 7 343 1372 8 512 2048

  2. Marked by a teacher

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

    3 star(s)

    It suggests that the equation dose quit well from 1950-1972, this can be showed in Figure 1(b), the scale (residual) in this diagram is quit small during the same period. The situation changed after mid-1970s, the gap in Figure1 (a)

  1. Numerical Differentiation

    x0 0 x1 0.8 0 1.6 x1 0.888888889 x2 0.92255409 1 -0.056343634 x2 0.960713297 x3 0.95157075 x3 0.961583416 x4 0.95897744 x f(x) x4 0.96158355 x5 0.96090272 0 1.6 x5 0.96158355 x6 0.96140552 0.1 1.421034184 x7 0.96153698 0.2 1.244280552 x8 0.96157137 0.3 1.069971762 x9 0.96158036 0.4 0.89836494 x10 0.96158272 0.5 0.729744254

  2. Mathematics Coursework - OCR A Level

    the root as -1.07375 with an error bound of + or - 0.00005 Rearranging Method Above the blue line is the line y=x, the red line is the rearrangement equation and the black line is showing the method get closer and closer to the solution.

  1. Estimate a consumption function for the UK economy explaining the statistical techniques you have ...

    That is ...Dc/Dy is positive and less than unity." (Keynes, 1936, p96) A simple 'Keynesian' Consumption Function: Keynes argued that on average men increase their consumption as their income increases, but not by as much as their income.

  2. Using Decimal search

    value is taken from the intersection point of graphs y = (x^3+1) / 5 and y = x, the y value diverges away to infinity. This method works when finding the root between the interval (0,1) but fails to find the other two roots.

  1. Numerical solution of equations, Interval bisection---change of sign methods, Fixed point iteration ---the Newton-Raphson ...

    Maximum Possible Error 1 0 1 =(B2+C2)/2 =(D2-1)*(D2+2)*(D2-3)-1 =ABS(C2-B2)/2 =A2+1 =IF(E2>0,D2,B2) =IF(E2<0,D2,C2) =(B3+C3)/2 =(D3-1)*(D3+2)*(D3-3)-1 =ABS(C3-B3)/2 =A3+1 =IF(E3>0,D3,B3) =IF(E3<0,D3,C3) =(B4+C4)/2 =(D4-1)*(D4+2)*(D4-3)-1 =ABS(C4-B4)/2 =A4+1 =IF(E4>0,D4,B4) =IF(E4<0,D4,C4) =(B5+C5)/2 =(D5-1)*(D5+2)*(D5-3)-1 =ABS(C5-B5)/2 =A5+1 =IF(E5>0,D5,B5) =IF(E5<0,D5,C5) =(B6+C6)/2 =(D6-1)*(D6+2)*(D6-3)-1 =ABS(C6-B6)/2 =A6+1 =IF(E6>0,D6,B6) =IF(E6<0,D6,C6) =(B7+C7)/2 =(D7-1)*(D7+2)*(D7-3)-1 =ABS(C7-B7)/2 =A7+1 =IF(E7>0,D7,B7)

  2. Numerical integration can be described as set of algorithms for calculating the numerical value ...

    Notice that f1, f2, f3 ect. to fn-1 do double duty as right-hand and left-hand body. Note that is the dependent on the number of trapezia. Thus T16 means that area under the curve is split in 16 rectangles whereas T32 means its split up in 32.

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