Applications

Write a program to plot

The trouble is that, when , . We can overcome this problem with Taylor approximating polynomials:

and therefore

The Taylor approximation is well defined (even for ).

The second problem requires to accurately evaluate

for . We see that

Now, for , in single precision arithmetic,

and

Clearly this result is corrupted by a large error. The source is the subtraction of two almost equal numbers, and 1. Most of the significant digits will cancel out (in our case, the first 7 significant digits)

This is called a loss of significance error (or cancellation error). Now, we need to divide the small and inacurate number by the small number , which is in fact a multiplication by . The errors of order will become now errors of order .

Hence, by this division to a small number, the errors migrated from the digit after the decimal point to the first digit before the decimal point (which is 0, instead of 1).

Using a Taylor polynomial of order 5,

we can overcome the loss of significance error. Besides, this Taylor approximation is well defined (and equal to 1) when . For the Taylor approximation value is , correct in the first 7 digits.