Is it more computationally strenuous to compute an exponential or a Bessel function?

Question

It is often desirable to obtain the solution to a mathematical problem in closed form, that is, as an expression that contains generally-accepted functions like polynomials, rational and irrational functions, roots, and exponentials and logarithms. One justification I often hear is that, when known functions are involved, it easier to visualize the behavior of the function. Another justification is that it is less computationally demanding to evaluate the function at a set of points. While I certainly agree with the first justification, is the second justification reasonable? For example:

Does it take a longer time to compute a modified Bessel function of the first kind and fifth order for 10 points than to compute an exponential?
Does it take a longer time to compute an exponential integral than to compute an exponential?

My intuition is that in all three cases, a Taylor series expansion around the desired point is formed, so it comes down to evaluating a polynomial, some other polynomial, or its antiderivative.

Doesn't seem to be related to a specific programming problem. Try cs or cstheory or programming stack. / Depends on the implementaton, of course. — user202729, Feb 06 '18 at 15:35
Remember to read their How to ask page before asking. I'm not familiar with any of those sites, so I don't know. — user202729, Feb 06 '18 at 15:36

score 0 · Answer 1 · answered Feb 15 '18 at 23:44

One justification I often hear is that, when known functions are involved, it easier to visualize the behavior of the function. Another justification is that it is less computationally demanding to evaluate the function at a set of points.

That's only if the function is "simple", not closed. You can construct closed forms which are arbitrarily complex and demanding computationally.

I can imagine two genuine advantages of closed form solutions:

Since most programming languages have support for sqrt, sin and the like, closed form solutions are easily representable in code.
If a solution has closed form then there will be a number of algebraic steps that you can follow to get to the solution – an algebraic (or perhaps trigonometric) solution algorithm. Those algorithms can also only contain "closed form" steps, so they'll be fairly easily implemented.

If you know that the solution of your problem probably doesn't have closed form, you'll have to reside to an entirely different approach for solving it. This can get quite tricky: The Babylonians were able to solve quadratic equations 2000 BC, and it took more than another 3000 years until roots of polynomials of arbitrary order could be solved – with numerics, not algebra.

Is it more computationally strenuous to compute an exponential or a Bessel function?

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