- Level: AS and A Level
- Subject: Maths
- Word count: 1884
Solving equations by numerical methods - The Interval Bisection method
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Introduction
Maxim Protsenko
Pure Mathematics 2 Coursework
Solving equations by numerical methods
The Interval Bisection method
The coursework on Numerical Methods shows the techniques that can be applied to find the solutions of equations, which have no algebraic solution. The first method is the Interval Bisection Method. It involves finding an interval of x in which f(x) changes sigh. If f(x) is a continuous function, it follows that it has a root within that interval. The equation for this method that I took is x³+2x²-x-3=0.
It is clearly seen on the graph that one of the roots of the equation is between 1 and 2. Within this interval f(x) changes the sigh from negative to positive. To find the root I built a table where a is the lower limit of the interval (in this case 1) and b is the upper limit of the interval (in this equation it is 2). The table looks like this.
The Interval Bisection Table
x³+2x²-x-3=0
f(x)<0 | f(b)>0 | |||
A | b | (a+b)/2 | f((a+b)/2) | possible error |
1 | 2 | 1.5 | 3.375 | 0.5 |
1 | 1.5 | 1.25 | 0.828125 | 0.25 |
1 | 1.25 | 1.125 | -0.169921875 | 0.125 |
1.125 | 1.25 | 1.1875 | 0.307373047 | 0.0625 |
1.125 | 1.1875 | 1.15625 | 0.06338501 | 0.03125 |
1.125 | 1.15625 | 1.140625 | -0.054592133 | 0.015625 |
1.140625 | 1.15625 | 1.1484375 | 0.004064083 | 0.0078125 |
1.140625 | 1.1484375 | 1.14453125 | -0.025346935 | 0.00390625 |
1.14453125 | 1.1484375 | 1.146484375 | -0.0106621757 | 0.001953125 |
1.146484375 | 1.1484375 | 1.1474609375 | -0.00330423657 | 0.0009765625 |
1.1474609375 | 1.1484375 | 1.14794921875 | 0.000378625351 | 0.00048828125 |
So the answer is 1.14794921 to (6 d. p.)
The possible error value represents the distance between the estimated root and the interval limits. The error bunds are a and b, so that x is between the two values. The error bounds for this root of the equation x³+2x²-x-3=0 are
1.1474609375 <x< 1.1484375
Middle
The equation I took for this method is x³-7x+1=0.
It looks like this:
The gradient of the tangent at (, f()) is f ‘(). The equation of the straight line can be written as
y-=m(x-),
the equation of the tangent is:
y-f()=f ‘()[x-].
This tangent cuts the x axis at (, 0), so:
0 -f()=f ‘()[-].
Rearranging this gives:
=-.
=> the Newton-Raphson iterative formula:
As my equation is x³-7x+1=0, with the root in the interval [2, 3], I can rewrite it as:
f(x)= x³-7x+1=0 and f ‘(x)=.
The iterative formula is therefore:
=
Starting with the point =3 gives
x(n) | f(x) | f '(x) | x(n+1) |
3 | 7 | 20 | 2.65 |
2.65 | 1.059625 | 14.0675 | 2.574676 |
2.574676 | 0.044678977 | 12.886864 | 2.571209 |
2.571209 | 0.00009 | 12.833342 | 2.571201 |
2.571201 | 0.0000000004 | 12.83323 | 2.571201 |
The Newton-Raphson method gives an extremely rapid rate of convergence.
The first root of the equation I found is 2.571201 with the error bounds of 2.571200<x<2.571202.
Conclusion
But despite of this it takes lots of time to obtain one root and would take ages to do it without any computer or graphic calculator aid. Also the initial search may miss one or more root, for example when the x-axis is a tangent to the urve or when several roots are very close together.
Newton Raphson method is the easiest way to estimate the root and takes much less time than to estimate the root using the first method. It doesn’t take that much time but it can go wrong very quickly and jump to another root.
And the last method rearranging is the hardest one. It takes plenty of time to rearrange the equation and then to estimate the root. And it’s very likely to do something wrong using it.
In conclusion I can say that the most efficient and easiest way for me to estimate the root, was the first one, because it takes not so much time if you get used to it. Very useful were PC programs such as Excel and Autograph. And at last the fact that in excel it is easier to change one equation on another and to get other roots, I can definitely say that it is the best method for me.
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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