• 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
  19. 19
    19
  20. 20
    20
  21. 21
    21
  22. 22
    22
  23. 23
    23
  24. 24
    24
  25. 25
    25
  26. 26
    26
  27. 27
    27
  28. 28
    28
  29. 29
    29

decimal search

Extracts from this document...

Introduction

Introduction

In maths equations can be solved using various methods. A very common and efficient method in solving equations is algebraically. But not all equations can be solved algebraically; these equations must be solved using numeric methods.

I will study three specific numeric methods on different equations.

            ~ Change of sign, decimal search process.

            ~ Newton-Raphson method.

            ~ Re-arrangement method. 

I will be testing the numeric methods with separate equations which cannot be solved algebraically. I will also apply all of the methods to one of the equations and check if all the methods give me the same value for the root I want to find.  

Change of Sign, Decimal Search

To find the root of the equation f(x) = 0 means finding values of x for the graph          y = f(x). The change of sign method works on the bases that the y = f(x) graph changes signs when it crosses the x-axis.

image00.png


e.g.

image01.png                                                                                            y = f(x)

image10.png


The sketch above shows that there is a root between the interval [b , c] and the curve of y = f(x) crosses the x-axis and changes its sign from negative ( - ) to positive ( + ), and at the interval [a , b] f(x) curve crosses the x-axis changing its sign from positive  ( + ) to negative ( - ).

...read more.

Middle

image00.png


eg

image20.png


The tangent produces a closer estimate of the root. This eventually leads to the actual root. 

From this procedure the Newton-Raphson iterative formula is produced.

A chosen equation and an estimate root value are used with the iterative           Newton-Raphson formula to eventually converge to the actual root.

Failure of Newton-Raphson Method

The Newton-Raphson method will fail if the starting point is carelessly selected. If the initial value of x is not close to the root needed to be found, the Newton-Raphson formula may not converge to the root. Also if the initial value is selected close to a turning point then it will also cause the formula to either diverge or converge to another root.

image21.png


image02.pngeg

image03.png


The selected initial value lies to close to the turning point of the f(x) curve, this creates the tangent from the initial value to diverge outside the area of the f(x) curve. This subsequently renders the method to fail as no further estimates of the root can be obtained.

When trying to find the root between the interval [0 , 1] and using the initial value as 0.88.

image04.pngimage05.jpg

The tangent is forced to diverge from the f(x) curve, this illustrates the failure of the Newton-Raphson method.

Newton-Raphson Method in Practise

The equation to be used in finding a root using the Newton-Raphson method is,

                     y = xµ - 3x + 1

Sketch of       y = xµ - 3x + 1

Sketch of y = xµ - 3x + 1 with tangent line markings.

Initial value = 2 (x  )

...read more.

Conclusion

The Newton-Raphson method only required two iterations before converging to the required root.

The change of sign method had to go through six cycles before producing the required root correct to six decimal places.

The re-arrangement method went through four iterations to produce the root value.

Although the re-arrangement method only marginally lost in time efficiency when compared to the Newton-Raphson method; the re-arrangement method proves to be the most demanding. As the re-arrangement method relies on producing a re-arrangement to the form of y = g(x) and satisfies the fact y = g` (x) < y = x`.

The change of sign, decimal search method is a very straight forward method which can be applied easier then the other methods, but lacks in its rate of convergence.

So in an overall comparison I can state that the Newton-Raphson method has the fastest rate of convergence, and thus has an efficient process of finding the desired root of an equation that must be solved numerically.

The use of software packages and hardware devices make understanding the methods and application of the methods faster and more accurate.

Using Excel spreadsheet in displaying the calculations for the change of sign method proved to be efficient as it enabled repetitions of calculations to be performed much faster than by manual process.

Also using graphical displaying software such as Autograph allowed easier demonstration of the process of the different methods, and a more accurate interpretation of the methods.     

...read more.

This student written piece of work is one of many that can be found in our AS and A Level Decision 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 Decision Mathematics essays

  1. Marked by a teacher

    The Gradient Function

    5 star(s)

    However, here h tends to 0. All terms here contain an h value, apart from 3x�. So therefore it simplifies to... 3x� + (3 x 0 x x) + 0� = 3x�. Therefore, this is the gradient function. This also corresponds to the increment methods in the tables.

  2. Marked by a teacher

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

    3 star(s)

    = + 0.01767 + 0.3935*DLY + 0.04268*LS_1 - 0.001662*P + 0.001001*HP Some tests results come out followed in Table (9). The t-value tells us that the coefficient of LS_1 and U are insignificant as their absolute value of t-test are less than 2.

  1. MEI numerical Methods

    Fixed point iteration: If we apply the first order of convergence principles to this method we get the following: The first order of convergence for fixed point iterations remains relatively constant, this value is 0.786 however, bear in mind that the amount of iterations used here is once again loads relative to secant and method of false position.

  2. Decimal Search.

    0 = 2x5 - 2x5 = 2x = x = This method cannot work with this graph, as the gradient is smaller than -1, and the value for x consequently diverges. Comparison To compare the three methods, I shall use the equation used for the decimal search method: y =

  1. Portfolio - Stopping Distances

    Below is a graph showing the same graph as above but with an extended window frame. Graph 5. Quadratic model for Speed versus Braking distance with enlarged window frame Here we can see that the plots match well on the right side.

  2. Using Decimal search

    Therefore 0.375095 < X < 0.375105. To check for the different signs, f(0.375095) = x^3+3x^2+1.4x-1= -3.76959E-06 f(0.375105) = x^3+3x^2+1.4x-1= 3.69674E-05 - Graph 6 - For root indicated in interval (-2,-1), results are as follows: X -1 -1.25 -1.26106 -1.26113 Xn+1 -1.25 -1.26106 -1.26113 -1.26113 - Graph 7 - We can express this information as; the root

  1. Analyse the use of three methods which are called the: change of sign, Newton-Raphson ...

    -0.10555 1.0583 -0.00468 0 0 1.4 12.34464 1.04 -0.34391 1.054 -0.08668 1.0584 -0.00276 1 -1 1.5 18.84375 1.05 -0.16169 1.055 -0.06774 1.0585 -0.00083 2 84 1.6 27.10016 1.06 0.028133 1.056 -0.04872 1.0586 0.001093 3 549 1.7 37.40697 1.07 0.225736 1.057 -0.02962 1.0587 0.00302 4 2024 1.8 50.08608 1.08 0.431284 1.058

  2. Change of Sign Method.

    is less that 1, y=x is used as a 'barrier' and makes the sequence converge thus allowing the root to be found. The values that were obtained are as follows: Error Bounds Therefore the root to the equation between the interval [-1,0] is -0.9172560 correct to seven decimal places.

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