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# Numerical Methods

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.

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.

## Numerical integration

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.

Trapezial method of numerical intregration

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.

## Root finding

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.

### How to locate a root

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).

Locating a root using numerical methods

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

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.

1. Show that can be rearranged to
2. Use the iteration with to find and to two decimal places
3. Continue this iteration to find the exact value of this root
1. If we continually do this iteration (using the 'ans' button on your calculator will help), you will reach a root of -1

### The Newton-Raphson method

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 - Key takeaways

• 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

A numeric method uses approximations to simplify a problem to allow an approximate answer to be reached.

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.

In numerical methods, convergence occurs when an iteration settles at a value.

Numerical methods are used to find approximate answers when other methods fail or appear incredibly time consuming and inefficient.

## Final Numerical Methods Quiz

Question

What is a root of an equation?

A root of the function f(x) is a value of x for which f(x) = 0.

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Question

Can an equation have multiple roots?

Yes

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Question

State whether the following statement is true or false: The location of roots theorem is used to calculate the exact root of an equation.

False

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Question

What is the Location of Roots theorem?

If the function f(x) is continuous in the interval [a,b] and f(a) and f(b) have opposite signs, then f(x) has at least one root, x, that lies between a and b ie. a < x < b.

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Question

For a given function, f(x),

f(1) = 0.5

f(2) = -1.5

Can we conclude that there is one root between x=1 and x=2?

No. Since there is a change of sign between f(1) and f(2), there is at least one root between x=1 and x=2. There may be more than one root as well.

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Question

For a given function, f(x),f(5) = 0.5f(10) = 1.5Can we conclude that there is no root between x=5 and x=10?

No

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Question

Can an equation have zero roots?

Yes

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Question

State whether the following statement is true or false: The location of roots theorem is used to calculate the exact root of an equation.

False

Show question

Question

For a given function, f(x),

f(1) = 0.5

f(2) = -1.5

Can we conclude that there is one root between x=1 and x=2?

No. Since there is a change of sign between f(1) and f(2), there is at least one root between x=1 and x=2. There may be more than one root as well.

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Question

Is there a root of f(x) = 5 cos x − 3x in the interval (0.5,1)?

Yes

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Question

Is there a root of f(x) = x³ + 3x − 7  in the interval (1,2)?

Yes

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Question

Is there a root of f(x) = x³ + 3x − 7  in the interval (150,200)?

No

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Question

Is there an odd number of roots of f(x) = x³ + 3x − 7  in the interval (0,-1)?

No

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Question

For a given function, f(x),

f(10) = 0.3

f(11) = -2.5

Can we conclude that there is one root between x=10 and x=11?

No. Since there is a change of sign between f(10) and f(11), there is at least one root between x=1 and x=2. There may be more than one root as well.

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Question

What is iteration?

Iteration is the process in which we start with an approximated value then using an iterative formula, we narrow in on a far more accurate value (closer to the real value).

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Question

Why do we use iteration?

We mainly use iteration when we don't have clear methods to solve for the roots of an equation.

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Question

What is a staircase diagram?

When we are converging to a value in the same direction (increasing or decreasing), then we can draw a staircase diagram between all our values we get.

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Question

What is a cobweb diagram?

A cobweb diagram is when we are converging on a value either above or below the real value. So we get a value a little too high then a little too low and eventually our margin for error becomes lower and lower. We get closer to our intersection.

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Question

Why can't we just solve iterative problems graphically?

As sometimes we can get irrational values that we can't just read off a graph to a good amount of accuracy.

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