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Iteration π₯ π+1 =5β 1 π₯ π with π₯ 1 =2 π₯ 2 = π₯ 3 = π₯ 4 = π₯ 5 =
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Iteration Diagrams and Convergence
To begin to understand the conditions for convergence. To draw and interpret staircase and cobweb diagrams.
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Iteration Diagrams and Convergence
To begin to understand the conditions for convergence. To draw and interpret staircase and cobweb diagrams.
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Bilborough College Maths - Core 3 convergence diagrams (Adrian)
24/04/2017 In the previous lesson, we found the approximate solution to Using the formula with we got The solution to the equation we called a. We can illustrate the iteration process on a diagram.
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For we draw and So, a is the x-coordinate of the point of intersection. ( to 6 d.p. ) Weβll zoom in on the graph to enlarge the part near the intersection.
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We started the iteration with
and we substituted to get . On the diagram we draw from to the curve . . .
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y-value, We started the iteration with and we substituted to get .
On the diagram we draw from to the curve . . . x which gives a y-value,
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y-value, We started the iteration with and we substituted to get .
On the diagram we draw from to the curve . . . x x which gives a y-value, By drawing across to y = x this y-value gives the point where x This is .
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In the iteration, is now substituted into to give .
On the diagram we draw from to the curve . . . x x giving the next y-value. x The next line converts this y-value to and so on.
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In the iteration, is now substituted into to give .
On the diagram we draw from to the curve . . . x x giving the next y-value. x 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.
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The diagram weβve drawn illustrates a convergent, oscillating sequence.
This is called a cobweb diagram.
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Bilborough College Maths - Core 3 convergence diagrams (Adrian)
24/04/2017 Looking at the original graph, we have
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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 Repeat Continue the cobweb line, going parallel to the y-axis to meet
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Have a go⦠Textbook Page 134 B8 (a) ii and iv
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Bilborough College Maths - Core 3 convergence diagrams (Adrian)
24/04/2017 Iteration does not always give an oscillating sequence. We can also draw a diagram for sequences which iterate directly towards the solution. I am going to use an example from the previous presentation: (a rearrangement of ) If you have a graphical calculator, before you look at the solution, you might like to have a go. Youβll 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 donβt need the values of etc. )
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Bilborough College Maths - Core 3 convergence diagrams (Adrian)
24/04/2017 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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Bilborough College Maths - Core 3 convergence diagrams (Adrian)
24/04/2017 So, to solve using with This is called a staircase diagram.
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Bilborough College Maths - Core 3 convergence diagrams (Adrian)
24/04/2017 Exercise The following graphs are on your handout. Use them to illustrate the convergence of the iterative process, showing (a) (b) Write on the names of the diagrams.
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Bilborough College Maths - Core 3 convergence diagrams (Adrian)
24/04/2017 Solution: (a) A cobweb diagram.
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Bilborough College Maths - Core 3 convergence diagrams (Adrian)
24/04/2017 Solution: (b) A staircase diagram.
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Bilborough College Maths - Core 3 convergence diagrams (Adrian)
24/04/2017 We will now look at why some iterative formulae give sequences that converge whilst others donβt 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
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The gradients of for the converging sequences are shallow
Cobweb: converging Staircase: converging The gradients of for the converging sequences are shallow Cobweb: diverging Staircase: diverging
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It can be shown that gives a convergent sequence if the gradient of . . .
is between -1 and +1 at the root. We write or, Unfortunately since we are trying to find we donβt know its value and canβt substitute it ! In practice, to test for convergence we use a value close to the root. The closer is to zero, the faster will be the convergence.
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Bilborough College Maths - Core 3 convergence diagrams (Adrian)
24/04/2017 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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Have a go⦠Textbook Page 134 B8 (a) i and iii Page 137 Questions 7 and 11
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Have a go⦠Card sort Ordering convergence
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Bilborough College Maths - Core 3 convergence diagrams (Adrian)
24/04/2017 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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The gradients of for the converging sequences are between -1 and +1
Staircase: diverging Cobweb: converging Staircase: converging Cobweb: diverging
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Iteration π₯ π+1 =5β 1 π₯ π with π₯ 1 =2 π₯ 2 = π₯ 3 = π₯ 4 = π₯ 5 =
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