What is an Iterative Process?

An iterative process is a process where the same procedure is repeated over and over again to obtain a desired result.
Methods of showing an iteration or the iterative
process are as follows:

• Arrow diagrams such as "The Water Cycle"

• Tree diagrams.
• Recursive formulas
• Fractal Patterns (see video to the right)

• Flowcharts such as the one below.
  

Flowchart

A flowchart is a diagram or chart showing step-by-step progression through a procedure. The chart consist of a set of symbols to represent the activities, and arrows to show the direction of activity through the procedure. Examples of symbols that can be found in flowcharts are diamonds to represent a decision, and rectangles to represent the actions during the process.

Non-linear regression - Non-linear regression is the technique of finding a curve of best fit for a data that is not described by a straight line (linear regression). The following flowchart runs through the procedures in determining non-linear regression.

Note for above flowchart
In this example for argument's sake, consider that r², coefficient of determination, must be a minimum of 0.75 for the best choice for the model (curve of best-fit). With this being said, if r² is greater than 0.75, then the model is a perfect fit for the data. If r² is below 0.75, other options can be estimating a new equation for the curve of best-fit, or try performing linear regression analysis.


The equation for the coefficient of determination is:

• Where "Y-est" is the y-value generated by the chosen equation when the x-value of your data is plugged into it (the estimated equation).
• Where "Y-avg" is the is the mean of all the y-values.

Recursive Formula

For a sequence a1, a2, a3, . . . , an, . . . a recursive formula is a formula that requires the computation of all previous terms in order to find the value of an .

Examples

Questions from the Mathematics of Data Management text book.
Page 10.

1. Which of the following involve an iterative process?
a) The cycle of a washing machine.
b) Your reflection in two mirrors that face each other.
c) The placement of the dials on an automobile dashboard.
d) A chart of sunrise and sunset times.
e) Substituting a value for the variable in a quadratic equation.
f) A tessellating pattern, such as paving bricks that fit together without gaps.

Answers to the above questions.

a) The cycle of a washing machine is an iterative process because the motion of the washing machine is always repeated, every time it is used, it'll repeat the same process over and over again.
b) Creating reflections in two mirrors that face each other is an iterative process since the angle of the reflections will always be the same, and every time you look in the mirror at your own reflection, you're repeating the same process the previous days you looked in the mirror.
c) The placement of the dials on an automobile dashboard is non-iterative because the procedures in placing the dials will change from time-to-time, it will not always be the same.
d) A chart of sunrise and sunset times is an iterative process because the sun will always rise at the same time and it will always set at the same time, therefore if the times are recorded the sunrise and sunset times will be repeated.
e) Substituting a value for the variable in a quadratic equation is non-iterative because once you substitute the value in for the variable, you're basically done, there is no repetition of any sorts left.
f) The tessellating pattern is an iterative process because a pattern is basically a repetition of designs, therefore it is being repeated over and over again.

References

Canton, Erdman, et al. Mathematics of Data Management. Toronto: McGraw-Hill

Ryerson, 2002.

Col, Jeananda. "The Water Cycle." Enchanted Learning. 1999. Enchanted

Learning. 25 Oct. 2006. <http://www.enchantedlearning.com> Keyword: Water cycle.

"Gliffy: BETA." Gliffy. San Francisco, California, 2005. 16 Oct. 2006.

<http://www.Gliffy.com/>.