I am working on a algorithm for acoustic scattering in two dimensions using MatLab, and one of the advantages of this algorithm is that Hankel functions can be replaced with log functions.
For example consider the following:
value = 0; tic; for t=1:100000, value = value + besselh(1,1,abs(1+1i)); end; toc
value = 0; tic; for t=1:100000, value = value + log(abs(1+1i)); end; toc
1) takes about 1.095455 seconds, whereas 2) takes about 0.043890 seconds.
Coming from a mathematics background I'm used to analyzing numerical methods in cases such as determining the order of convergence of an ODE/PDE solver or a numerical integration method, along with judging complexity in a program by counting operations.
However in this case, the number of operations is the same (on the surface anyway). So how can the improvement in efficiency be classified in this case? Are empiric results, such as showing that 2) takes much less time than 1), the best I can do in this situation?
Or is there a more formal/rigorous way to classify the improved efficiency?