1.1 Sampling

There is the question as to how many times, and for which arguments, the function is computed. Consider the following equation:

Computing q(x) generates the sample (x,q(x)), of the graph of y=q(x). Sampling, as we did before, generates the following table:

Table of q(x)

Surprisingly, the table matches our ealier one. It is not surprising that the plotted points match.

Continuing our procedure by rote, the same graph is generated.

By adding more samples to our table, we see that the previous graph is incorrect.

Richer Table of q(x)

Using our richer table, we again plot the points which we know satisfy our equation.

We have lost confidence in the line which connects the points. Without warning, it has failed us. There is hope that we may be able to predict its failure for polynomials, or other classes of functions, but we aim to graph general equations.

Although calculating a large number of samples guarantees to consume vast resources, it does not guarantee that a more reliable graph is generated. Consider the equation

Using over a million uniformly spaced samples of (x,r(x)), from [-2,2], results in the following graph:

The actual graph follows, which may be reliably computed using a handful of samples.

The actual graph is a very sharp hyperbola, and can not be generated by following our procedure, as the graph is composed of two disjoint curves.

Jeff Tupper

March 1996