• Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month
Page
  1. 1
    1
  2. 2
    2
  3. 3
    3
  4. 4
    4
  5. 5
    5
  6. 6
    6
  7. 7
    7
  8. 8
    8
  9. 9
    9
  10. 10
    10
  11. 11
    11
  12. 12
    12
  13. 13
    13
  14. 14
    14
  15. 15
    15
  16. 16
    16
  17. 17
    17
  18. 18
    18

maths pure

Extracts from this document...

Introduction

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.

Middle

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.

Conclusion

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)

    Using this formula, the next 5 values of x and the gradient are: 4 256 1280 5 625 3125 6 1296 6480 7 2401 12005 8 4096 20480 9 6561 32805 General proof - (x+h)5 - x5 = x^5 + 5x4h + 10x�h� + 10x�h� + 5xh4 +h5 - x5

  2. Marked by a teacher

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

    3 star(s)

    In other words, the absolute consumption function dose not predicts the consumption accurately during this period. The Keynesian consumption function can be changed slightly. This is to use logarithms of consumption and income and to estimate consumption by the form ct=c0+c1yt where ct is log (Ct)

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

    In addition this also proves that the mid-point rule is a second order polynomial. However, we should note that the mid-point rule is only an overestimate when the curve is convex. This is also the case for the curve On the other hand, the trapezium rule is an underestimate and

  2. Mathematics Coursework - OCR A Level

    then it is clear that the sign of the y values change here, hence the curve must cross the x axis. 0.91374 0.00022 0.91364 -0.00003 Therefore I am taking my second root as x=0.91365 Equation that does not work Sometimes this method does not work.

  1. Numerical Method (Maths Investigation)

    The more decimal place the more accurate the answer of the root is. X Y -0.69 -0.050563533 -0.68 -0.029629888 -0.67 -0.009507116 -0.66 0.009848393 -0.65 0.028477721 -0.64 0.046419535 -0.63 0.063710194 -0.62 0.080383854 -0.61 0.096472572 Tbl DS-03: Step 3 X Y -0.669 -0.007537736 -0.668 -0.005575983 -0.667 -0.003621818 -0.666 -0.001675197 -0.665 0.000263921 -0.664

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

    =(B15+C15)/2 =(D15-1)*(D15+2)*(D15-3)-1 =ABS(C15-B15)/2 =A15+1 =IF(E15>0,D15,B15) =IF(E15<0,D15,C15) =(B16+C16)/2 =(D16-1)*(D16+2)*(D16-3)-1 =ABS(C16-B16)/2 =A16+1 =IF(E16>0,D16,B16) =IF(E16<0,D16,C16) =(B17+C17)/2 =(D17-1)*(D17+2)*(D17-3)-1 =ABS(C17-B17)/2 =A17+1 =IF(E17>0,D17,B17) =IF(E17<0,D17,C17) =(B18+C18)/2 =(D18-1)*(D18+2)*(D18-3)-1 =ABS(C18-B18)/2 =A18+1 =IF(E18>0,D18,B18) =IF(E18<0,D18,C18) =(B19+C19)/2 =(D19-1)*(D19+2)*(D19-3)-1 =ABS(C19-B19)/2 =A19+1 =IF(E19>0,D19,B19) =IF(E19<0,D19,C19) =(B20+C20)/2 =(D20-1)*(D20+2)*(D20-3)-1 =ABS(C20-B20)/2 =A20+1 =IF(E20>0,D20,B20) =IF(E20<0,D20,C20) =(B21+C21)/2 =(D21-1)*(D21+2)*(D21-3)-1 =ABS(C21-B21)/2 In order to make sure there are always sign changes

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

    + log(Yt-1) + log(Yt-2)] /3 If this value is put back into our equation we obtain the following result: ct = 0.6181 + 0.9507ypt This approach weights all the previous years' income at the same level, however it is rational to assume that consumers will weight the more recent years

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

    The amount that the community spends on consumption depends (i) partly on the amount of their income, (ii) partly on other objective attendant circumstances, and (iii) partly on the objective needs and the psychological propensities and habits of the individuals comparing it...

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