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Iterative Methods

Iterative Methods

Solving an equation by an iterative method is carrying out a process to get closer to a value. We are given a rough approximated initial value, and then become more and more accurate. This gets us closer to the real value and is necessary when we do not have a straightforward way of solving for the roots of the equation.

Using the iterative method

An iterative method can be used to find a value of when . To perform this iteration we first need to rearrange the function.

The basis of this is we need to rearrange to . Therefore we need to make the subject of . However, it does not matter if we have terms remaining on the other side.

This is because we can perform , meaning we take our value of and iterate it.

Let's look at two key worked examples below.

Worked examples of iteration calculation

1.

a) Prove that .

b) Using the iterative formula , and letting , find values for . SOLUTION: 1a) 1b)

When iterating we use all the digits our calculator has given us to find our next value. You can do this easily by using the ANS button on your calculator

Let's look at a more difficult example.

2.

a) Prove that can be rearranged to form .

b) Using the iterative formula and letting

SOLUTION:

2a)

2 B)

How do we see iterative methods on a graph?

Iterative methods are all about getting closer and closer to a root of an equation. We use them when we cannot directly solve equations with any other methods.

The higher the value of in , the closer we are to the root of this equation as we are performing this process more and more times.

We can see this on a graph in two ways: a staircase diagram or a cobweb diagram.

Staircase diagrams

A staircase diagram works for a function that directly converges to a root meaning that from we are either directly increasing or decreasing towards our root with each iterative value.

Let's look at our initial example .

We will solely be focusing on the positive root (the one on the right).

Iterative Methods Graph used for iterative method  StudySmarter

A graph of

We know from rearranging that when . Therefore if we sketch the lines and , the intersection is the root of this equation.

Iterative methods Solving for intersection iteration StudySmarter

An intersection between the lines and

Let's now plot our points for and .

Values marked.

Alt-text: Values marked on graph.

This is a more zoomed-in version and we can see that these points are slowly converging to the intersection. If we add a few lines, we can see our staircase.

Iterative Methods Iteration staircase diagram StudySmarter

A staircase diagram attaching all of the aforementioned points.

If we continue and use more values we will get closer and closer to that intersection. We will never actually reach it but we can get it in view of high accuracy.

Cobweb Diagrams

A cobweb diagram is when we are converging on a root in more than one direction, meaning our values become both too high and too low around the root.

Our second example demonstrates this. We will be focusing on the negative root, the one to the left.

terative Methods Graph sketch for iteration StudySmarterImage Caption

A sketch of the graph

Once again, from rearranging we know that when . Therefore we can sketch the lines and , and their intersection is the solution to the equation .

Iterative Methods Intersection between lines before iteration. StudySmarter

Intersection between the lines and ,

Now let's plot our points for and .

Iterative Methods Values marked on graph StudySmarter

Points marked.

We can clearly see these points are not in order. Therefore we do not have a staircase, we have a cobweb and are converging on the value from different directions. This is what it looks like.

Iterative Methods Cobweb Diagram Iteration StudySmarter

A cobweb diagram attaching all marked points.

We can see this is like a cobweb, and is converging towards a value from values greater than and less than the intersection.

Iterative Methods - Key takeaways

  • Iterative methods can be used to find solutions to equations we cannot solve otherwise.
  • They give us formulas to help us converge on particular roots of equations.
  • We can use graphs to help us visualise how the roots are getting more accurate (the closer to the intersection, the more accurate our answer is).
  • A staircase diagram is one where we are moving in one direction towards the intersection.
  • A cobweb diagram is where we are moving around the intersection in more than one direction.

Frequently Asked Questions about Iterative Methods

An iterative method is one in which we use the previous answers to generate more accurate answers.

We construct an iterative formula by rearranging and then using this formula to generate more values.

We can sketch either cobweb or staircase diagram to help us visualise how more accurate roots are moving.

There are many applications to iteration especially in physics and engineering in calculating next values when we cannot solve.

Final Iterative Methods Quiz

Question

What is iteration?

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Answer

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?

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Answer

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?


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Answer

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?

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Answer

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?

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Answer

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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