I have a question; well technically about calculus in general.

[maistral]

First off, flame-shield on.

I'm just curious. Why do I have to learn how to integrate, to differentiate, and to solve differential equations analytically if there are numerical techniques anyway? I'm not trying to be a ***** or whatever, I'm just as curious as hell. I mean, considering the overwhelming amounts of work needed just to get a clear-cut solution to some integrals, why do I have to?

Same goes with differential equations, why do I have to learn how to analytically solve them when there's the Runge-Kutta?

Someone guide my soul back to reality

 

[oli4]

Hi maistral, something solved analytically is infinitely more powerful than something 'solved' numerically.
The numerical answer will always be up to a certain precision, and errors will build up very fast for certain systems (like chaotic systems)
This is a funny question, I used to wonder about why we had to bother applying/learning numerical methods when it is so much more satisfying to solve the problems analytically (of course, the problem is that it's often much harder, and many times provably impossible)

 

[chiro]

Hi maistral, something solved analytically is infinitely more powerful than something 'solved' numerically.
The numerical answer will always be up to a certain precision, and errors will build up very fast for certain systems (like chaotic systems)
This is a funny question, I used to wonder about why we had to bother applying/learning numerical methods when it is so much more satisfying to solve the problems analytically (of course, the problem is that it's often much harder, and many times provably impossible)

This is true, but be aware that even in cases where an analytic solution is known sometimes numerical schemes are used over analytic due to the fact that the analytic form is way too complicated and computationally expensive in terms of calculations (even relative to a numerical scheme that has the required accuracy).

If you are interested, you should look into the applied mathematics/finance literature for examples of such situations.

 

[HallsofIvy]

"Runge-Kutta" does not give "qualitative" information about solutions, such as whether or not they are periodic for certain values of parameters, nor about the "end-behavior", whether the solution is bounded, or goes to positive or negative infinity as the independent variable goes to positive or negative infinity.

But your point is well taken- sure, if all you want to do is use "canned solutions", why both studying mathematics at all? I assume that you plan your career path so that you are only doing jobs on which the "solution", how to do the job, is well known and you only have to follow instructions, never being required to do any original thinking yourself.

 

[AlephZero]

Same goes with differential equations, why do I have to learn how to analytically solve them when there's the Runge-Kutta?

One good answer is "so you have some tools to judge whether the numbers you get from a computer program really are a solution to the equation, or nonsense."

Even if what you get isn't nonsense, it doesn't follow that it is the only solution, or the one that is relevant to the "real world" problem the equation represents.

 

[lavinia]

If you are only interested in calculations then computer models usually work better than analytical formulas.

If you are interested in the the nature of the underlying phenomemon or in understanding the properties of solutions of a differential equation, calculation will tell you nothing by itself. it is just an empirical tool that may help you find these properties.

As I remember, chaos was first noticed in computer simulations of weather but was not understood until analytical models were found.

 

[alexfloo]

As HallsOfIvy brings up, frequently in differential equations, especially with nonlinear or even chaotic systems, we're really not interested in specific solutions at all. Rather, we're interested in the effects of parameters, or general long-term behaviors ("Will the two species live in stable equilibrium, or will one ultimately kill of the other?")

These things *can* be explored computationally, but they usually can't be explored computationally in a consistent way. That is, there is no "algorithm" that can answer the question of the long-term behavior of a system. If you want to explore such a question using technology, you'd need still need to study and understand the system well enough to write the single-purpose program you'd need.

In any case, for extremely sensitive applications, one generally wants an exact expression anyway, before simulating (even if it's in terms of an integral that can't be evaluated exactly) because this makes it easier to carry out the calculation to arbitrary precision when needed.