- Level: AS and A Level
- Subject: Maths
- Word count: 2073
Am going to use numerical methods to solve equations that can't be solved algebraically
Extracts from this document...
Introduction
Greig Boyd
C3 Coursework
Solution of equations by numerical methods
Introduction
In this coursework I am going to use numerical methods to solve equations that can’t be solved algebraically, for example if the largest power of the equation was 3 or over , we would have to use the following methods: the change in sign method, Newton-Raphson method and the method of rearranging f(x) =0 in the form x = g(x).
I am going to use an equation which cannot be factorised or solved using the formula x = -b .
2a
Change of sign method
My equation is -6x3 +9x2 +5x-6.
My equation has four roots I am going to use change of sign to find one of them. For the change in sign method I am going to do a decimal search, i.e. I have taken the negative and positive y values that are in between zero. I did my first decimal search to two decimal places and carried on the same procedure all the way up to four decimal places.
x | Y |
0 | -6 |
0.1 | -5.416 |
0.2 | -4.688 |
0.3 | -3.852 |
0.4 | -2.944 |
0.5 | -2 |
0.6 | -1.056 |
0.7 | -0.148 |
0.8 | 0.688 |
0.9 | 1.416 |
1 | 2 |
The first thing I did was find out were the formula crosses the x-axis, which was 0.7 and 0.8.
Middle
Failure of “change of sign”
Failure will occur if:
- The curve touches the x-axis.
- There are several roots close together
- There is a discontinuity in f(x)
I am going to use decimal search to find the roots of another equation
-2x2-3x-1.12
I cannot use decimal search for this equation as it doesn’t cross the x-axis.
There are some cases where the roots cannot always be found. Reasons for this could be if the roots are too close together
Newton-Raphson method
X5-6x +2
The Newton raphson method is a fixed point estimation method. It is important to use an estimation of the root as a starting point.
I will give an estimation (x0) for a root of f(x) =0 I will then draw a tangent to the curve y=f(x) at the point (x0, f(x)). The point where the tangent cuts the x-axis gives a closer approximation for the root, and then the process is repeated
From this it is clear that D is a better estimation than B. We can continue this to get closer to A.
Conclusion
Comparing methods in terms of ease of use with hardware and software
With all three methods I used a graphical calculator as my hardware. With the decimal search method I only had to be aware of where the roots lie in the intervals to be able to use the method. With Newton-Raphson method I just had to be able to check on the graphics calculator what the roots were to know whether it was converging towards the root or diverging from the root. This made me use the arrow buttons to move the curser to where the roots lie. On the contrary, with rearranging I had to plot the y=x and the function on the same graph and for this I had to use Microsoft excel. In conclusion in terms of ease of use with involved hardware and software; the decimal search was the most efficient method to use, then Newton- Raphson and the most involved one was the rearranging method.
This student written piece of work is one of many that can be found in our AS and A Level Core & Pure Mathematics section.
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