To the annoyance of many pure mathematicians, not all problems can be solved analytically, that is, by a method that uses known rules and logic to get to an exact solution. This is where a numerical method is used. A numerical method will approximate a solution, or at worst, bound where a solution would lie.
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We can use numerical methods in all areas of mathematics where we would otherwise struggle to find a solution. Generally, this will include Differential Equations, solving linear systems (Simultaneous Equations in many variables) and finding the derivative of a function at a point. However, at A-Level, we will focus on root finding and Finding the Area under curves.
Some Functions are not integrable, meaning that there is no antiderivative for that function. However, this doesn't mean that we cannot approximate the area underneath these Functions (ie find an approximate solution for a definite integral). We do this by splitting the area under the integral into smaller areas (or shapes that closely resemble the area of the integral), Finding the Area of each of these areas, and then summing these together to get an approximation.
At A-Level, we focus on a trapezoidal method. This is where we split the area into a series of trapeziums and then sum them. A sketch of how this happens is shown below.
The more trapeziums we add, the more accurate the approximation becomes.
Let's formalize this to obtain a formula. Suppose we have a function 
, and we want to approximate the integral of 
, with n equally spaced intervals. This means we need n + 1 data points. Let 
, and then
for 
. We then find the values of these data points evaluated on the function, so we have 
.
For any trapezium, the area is given as (width) * (average height of the uneven length sides). In this case, our width is given as 
. The average height for trapezium i is given as 
. This means that the area of trapezium i is given as 
. Summing all of these, we obtain the formula of 
. As each 
is counted twice apart from the two endpoints, we can simplify this to 
.
Find an approximation to 
using the trapezium rule, with four equally wide strips.
For four strips, we need 5 points. The points are 0, 0.5, 1, 1.5, 2.
The following table shows both 
and 
:
| 0 | 0.5 | 1 | 1.5 | 2 |
| 0 | 0.5 | 2 | 4.5 |
By the given formula, 
. This means that our approximation to the integral is given by 
.
If we were to evaluate this integral 'properly', we would obtain 
, which is close to 5.5, which shows this is a good approximation.
Not all Equations can be solved using algebraic methods. This is where using numerical methods to find roots comes in. Not all methods work in all cases, so sometimes we need to be selective about what method we use.
Suppose there is a function, 
and we think a root may be located between points a and b. If there is a single root, then the sign of 
will be different to that of 
. If the interval is too large between a and b, there may be multiple roots, which could mean that the signs stay the same, even with multiple roots (this happens if there are an even Number of roots).
The image above should allow you to understand how the change of sign indicates a root.
Show there is a root of 
between -1.5 and -1.4.
and 
. As there is a change in sign, there is a root of f between -1.5 and -1.4.
Iteration is the process of repeating a mathematical function, using the previous answer as the next input. For example, an iterative function could be as simple as 
. In this equation, we would start with a given and then use this to find 
. We can then continue this process to find as many 
as we require. This process can allow us to find roots of Equations so long as 
is close enough to the actual root.
can be rearranged to 
with 
to find 
and 
to two decimal places
If we continually do this iteration (using the 'ans' button on your calculator will help), you will reach a root of -1
This method can be derived by using maths you will not see at A-level (a Taylor expansion), but this is a type of iterative formula to find a root. Suppose we have a function 
, which is differentiable. The Newton-Raphson iteration is given as 
, with 
, and a suitable starting value 
.
Using the Newton-Raphson method, find (to 3 decimal places) a second approximation to a root of 
, taking the first approximation as 
. Let us first find 
which is given as 
.
THUS, 
.
Numerical methods are used when an answer cannot be found analytically.
The trapezium rule with n equal widths is given by 
, with 
If 
and 
, then there is a root between a and b
The Newton-Raphson formula is given as 
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What is a numerical method?
A numeric method uses approximations to simplify a problem to allow an approximate answer to be reached.
What is the difference between analytical and numerical methods?
An analytical method uses precise methods and techniques to reach a precise solution, whereas a numerical method uses approximations to get to an approximate answer.
What is convergence in numerical methods?
In numerical methods, convergence occurs when an iteration settles at a value.
Why are numerical methods used?
Numerical methods are used to find approximate answers when other methods fail or appear incredibly time consuming and inefficient.
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