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18: Iteration Diagrams and Convergence © Christine Crisp “Teach A Level Maths” Vol. 2: A2 Core Modules

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Iteration diagrams and Convergence Module C3 "Certain images and/or photos on this presentation are the copyrighted property of JupiterImages and are being used with permission under license. These images and/or photos may not be copied or downloaded without permission from JupiterImages"

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Iteration diagrams and Convergence We can illustrate the iteration process on a diagram. In the previous presentation, we found the approximate solution to Using the formula with we got The solution to the equation we called

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Iteration diagrams and Convergence So, is the x -coordinate of the point of intersection. For we draw and ( to 6 d.p. ) We’ll zoom in on the graph to enlarge the part near the intersection.

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Iteration diagrams and Convergence On the diagram we draw from to the curve... and we substituted to get. We started the iteration with

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Iteration diagrams and Convergence We started the iteration with On the diagram we draw from to the curve... and we substituted to get. which gives a y -value, x

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Iteration diagrams and Convergence On the diagram we draw from to the curve... By drawing across to y = x this y -value gives the point where x. This is. x We started the iteration with and we substituted to get. x which gives a y -value,

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Iteration diagrams and Convergence In the iteration, is now substituted into to give. On the diagram we draw from to the curve... giving the next y -value. The next line converts this y -value to and so on. x x x

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Iteration diagrams and Convergence In the iteration, is now substituted into to give. On the diagram we draw from to the curve... giving the next y -value. The next line converts this y -value to and so on. The lines are drawn to the curve and y = x alternately, starting by joining to the curve. x x x

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Iteration diagrams and Convergence The diagram we’ve drawn illustrates a convergent, oscillating sequence. This is called a cobweb diagram.

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Iteration diagrams and Convergence Looking at the original graph, we have

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Iteration diagrams and Convergence SUMMARY To draw a diagram illustrating iteration: Draw and on the same axes. Mark on the x -axis and draw a line parallel to the y -axis from to ( the curve ). Continue the cobweb line, going parallel to the x -axis to meet Continue the cobweb line, going parallel to the y -axis to meet Repeat

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Iteration diagrams and Convergence Iteration does not always give an oscillating sequence. We can also draw a diagram for sequences which iterate directly towards the solution. If you have a graphical calculator, before you look at my solution, you might like to have a go. You’ll need to zoom in to see the graphs very close to the solution, for example, from to... Copy this part of the graphs quite carefully onto paper, mark at and off you go. ( Remember you don’t need the values of etc. ) I am going to use an example from the previous presentation: (a rearrangement of )

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Iteration diagrams and Convergence Before we zoom in, the graphs look like this. The solution has 2 roots. We will find the one nearest to the origin.

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Iteration diagrams and Convergence So, to solve using with This is called a staircase diagram.

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Iteration diagrams and Convergence Exercise Copy the following graphs and sketch the diagrams which illustrate the convergence of the iterative process, showing Write on the names of the diagrams. (a)(b)

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Iteration diagrams and Convergence Solution: (a) A cobweb diagram.

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Iteration diagrams and Convergence Solution: (b) A staircase diagram.

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Iteration diagrams and Convergence If you have Autograph or another graph plotter you may like to try to find both roots before you see my solution. In the next example we’ll look at an equation which has 2 roots and the iteration produces a surprising result. The equation is. We’ll try the simplest iterative formula first :

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Iteration diagrams and Convergence Let’s try with close to the positive root. Let

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Iteration diagrams and Convergence Using the iterative formula, The staircase moves away from the root. Let’s try with close to the positive root. The sequence diverges rapidly.

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Iteration diagrams and Convergence Suppose we try a value for on the left of the root.

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Iteration diagrams and Convergence Suppose we try a value for on the left of the root.

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Iteration diagrams and Convergence Suppose we try a value for on the left of the root.

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Iteration diagrams and Convergence Suppose we try a value for on the left of the root.

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Iteration diagrams and Convergence The sequence now converges but to the other root ! Suppose we try a value for on the left of the root.

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Iteration diagrams and Convergence The equation was. With this gives the negative root: We can also arrange the equation as follows: Change to log form: giving The rearrangement we used was so the formula was. With, It is possible to use our iterative method to find in the previous example but not with the arrangement we had.

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Iteration diagrams and Convergence We will now look at why some iterative formulae give sequences that converge whilst others don’t and others converge or diverge depending on the starting value. Collecting the diagrams together gives us a clue. See if you can spot the important difference once you can see the 4 diagrams I’ve included the 4 th type of diagram that we haven’t yet met: a diverging cobweb.

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Iteration diagrams and Convergence Cobweb: converging Cobweb: diverging Staircase: converging Staircase: diverging The gradients of for the converging sequences are shallow

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Iteration diagrams and Convergence We write It can be shown that gives a convergent sequence if the gradient of... or, In practice, to test for convergence we use a value close to the root. Unfortunately since we are trying to find we don’t know its value and can’t substitute it ! The closer is to zero, the faster will be the convergence. is between 1 and +1 at the root.

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Iteration diagrams and Convergence e.g. By using calculus, determine which of the following arrangements of the equation will give convergence to a root near x = 3 and which will not. Solution: The sequence will converge.

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Iteration diagrams and Convergence e.g. By using calculus, determine which of the following arrangements of the equation will give convergence to a root near x = 3 and which will not. Solution: The sequence will not converge.

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Iteration diagrams and Convergence At, the gradient is greater than 1 ( so the curve here is steeper than y = x ): the iteration diverges. We can now see why we had the strange result when we tried to solve with At, the gradient is less than 1 ( so the curve here is shallower than y = x ): the iteration converges.

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Iteration diagrams and Convergence We can now see why we had the strange result when we tried to solve with Both iterations started close to but the one that converged to started on the left of.

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Iteration diagrams and Convergence SUMMARY To show that a formula of the type will give a convergent sequence, find show that, where x is close to the solution.

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Iteration diagrams and Convergence

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The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied. For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.

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Iteration diagrams and Convergence SUMMARY To draw a diagram illustrating iteration: Draw and on the same axes. Mark on the x -axis and draw a line parallel to the y -axis from to ( the curve ). Continue the line, going parallel to the x -axis to meet Continue the line, going parallel to the y -axis to meet Repeat 4 different diagrams are possible.

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Iteration diagrams and Convergence The gradients of for the converging sequences are between 1 and +1 Staircase: diverging Cobweb: converging Staircase: converging Cobweb: diverging

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