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1.4 Numerical Round-Off

Given that a basic algorithm has failed us, it is reasonable to do a survey of our basic tools. We are most interested in finding a graphing algorithm which we may program a modern computer to perform. One immediate concern is the representation such machines use for real numbers. The representation often used is analagous to scientific notation, keeping a fixed number of digits for any given quantity. This can lead to further difficulties.

Consider graphing the equation

figure5214

by sampling n(x), limiting ourselves to three digits of precision. A transcription of the computations performed, while sampling n(x) at x = -1,0,1,2, and 3 follows:

figure5220

It is clear that for all x, our computations result in tex2html_wrap_inline31959 or tex2html_wrap_inline31961 , due to numerical round-off. It is equally clear that

figure5229

so that tex2html_wrap_inline31963 .

figure5235

Most calculations introduce some numerical round-off. With complicated equations, there will be long sequences of calculations, which allows numerical round-off to accumulate. For such equations, the generated graph may differ significantly from the actual graph.


next up previous notation contents
Next: 1.5 Computability Up: 1 Motivation Previous: 1.3 Relations
Jeff TupperMarch 1996