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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.
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.
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.
a) Prove that can be rearranged to form .
b) Using the iterative formula and letting
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.
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).
A graph of
We know from rearranging that when . Therefore if we sketch the lines and , the intersection is the root of this equation.
An intersection between the lines and
Let's now plot our points for and .
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.
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.
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.
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 .
Intersection between the lines and ,
Now let's plot our points for and .
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.
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.
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.
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).
Why do we use iteration?
We mainly use iteration when we don't have clear methods to solve for the roots of an equation.
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.
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.
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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