converging to a particular solution?
Activities
Numerical methods of solving equations - sequence graphs and web plots
In the activity one
point iterative methods, it is seen that we cannot rely on an iterative
process
converging to a particular solution?
In this activity we will investigate how it depends on:
.Sequence graphs and tables on the TI-83
The TI-83 can be used to produce a graph or a table of values for a sequence. Consider the iterative process with a starting value of 4 (first discussed in the activity one point iterative methods):
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Web plots on the TI-83
The behaviour of a sequence can be demonstrated in a powerful way by a web plot (details).
Again, consider the same iterative
process for 
(an equation with two solutions).
A solution occurs at
a point of intersection of
The diagram shows two points of intersection and these points correspond to the two solutions. |
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The calculation of the next value in a sequence is represented by a pair of moves on a web plot:
.Will any starting value give the negative solution?
What starting values give the positive solution?
Investigate web plots for rearrangements 2 and 3.
Invent a problem - quadratic equation with no real solutions
Make up a quadratic equation with no real solutions.
Investigate web plots for iterative
processes of the form .
What happens?
Invent a problem - cubic equation
Make up a cubic equation (
)
with three real solutions.
Using web plots for iterative processes
of the form ,
find the values of the solutions accurate to 6 significant figures.
When will a sequence converge to a solution?
In each of the web plots below, the starting value is marked. Which of the plots converge to the solution represented by the point of intersection shown? What is the requirement for convergence?
It would be good to have an iterative method that works for all solutions and has a fast rate of convergence.
Newton's
method does just this. Well almost. It can be caught out.
.