3 ms·

What you say is correct but isn't it the case that numerical solutions necessarily involve approximations and the error in those approximations in some systems

What you say is correct but isn't it the case that numerical solutions necessarily involve approximations and the error in those approximations in some systems becomes too great over time. Thus the predictive power isn't there for numerical solutions that is present when exact solutions are known.

Someone 2y ago
…assuming that the functions used in the closed-form expression can be efficiently computed to arbitrary precision. What constitutes a closed-form expression depends on context/is up for debate: - https://en.wikipedia.org/wiki/Closed-form_expression#Alternative_definitions https://en.wikipedia.org/wiki/Closed-form_expression#Alterna...: “Changing the definition of "well known" to include additional functions can change the set of equations with closed-form solutions. Many cumulative distribution functions cannot be expressed in closed form, unless one considers special functions such as the error function or gamma function to be well known. It is possible to solve the quintic equation if general hypergeometric functions are included, although the solution is far too complicated algebraically to be useful. For many practical computer applications, it is entirely reasonable to assume that the gamma function and other special functions are well known since numerical implementations are widely available.” - https://mathworld.wolfram.com/Closed-FormSolution.html https://mathworld.wolfram.com/Closed-FormSolution.html: “For example, an infinite sum would generally not be considered closed-form. However, the choice of what to call closed-form and what not is rather arbitrary since a new "closed-form" function could simply be defined in terms of the infinite sum.” so that’s not a given.
openrisk 2y ago
That is true. Exact solutions have further advantages: they are expressed in terms of parameters (integration constants) which means that one can get deep insights into a system's behavior under changing parameters. In a numerical approach replicating such studies can be very cumbersome. But the irony is that in the broad scheme of things "interesting" exact solutions are very rare, almost freakish. The only systems we can reliably solve are linear :-). So while numerical approaches have weaknesses, they are the de-facto available tool in most cases.
BlueTemplar 2y ago
Which is exacerbated in simulations featuring chaotic behaviour, where changes to initial conditions smaller than what the numerical simulation is representing, result in wildly different outcomes after only a short amount of time.

Someone 2y ago

…assuming that the functions used in the closed-form expression can be efficiently computed to arbitrary precision. What constitutes a closed-form expression depends on context/is up for debate: - https://en.wikipedia.org/wiki/Closed-form_expression#Alternative_definitions https://en.wikipedia.org/wiki/Closed-form_expression#Alterna...: “Changing the definition of "well known" to include additional functions can change the set of equations with closed-form solutions. Many cumulative distribution functions cannot be expressed in closed form, unless one considers special functions such as the error function or gamma function to be well known. It is possible to solve the quintic equation if general hypergeometric functions are included, although the solution is far too complicated algebraically to be useful. For many practical computer applications, it is entirely reasonable to assume that the gamma function and other special functions are well known since numerical implementations are widely available.” - https://mathworld.wolfram.com/Closed-FormSolution.html https://mathworld.wolfram.com/Closed-FormSolution.html: “For example, an infinite sum would generally not be considered closed-form. However, the choice of what to call closed-form and what not is rather arbitrary since a new "closed-form" function could simply be defined in terms of the infinite sum.” so that’s not a given.

openrisk 2y ago

That is true. Exact solutions have further advantages: they are expressed in terms of parameters (integration constants) which means that one can get deep insights into a system's behavior under changing parameters. In a numerical approach replicating such studies can be very cumbersome. But the irony is that in the broad scheme of things "interesting" exact solutions are very rare, almost freakish. The only systems we can reliably solve are linear :-). So while numerical approaches have weaknesses, they are the de-facto available tool in most cases.

BlueTemplar 2y ago

Which is exacerbated in simulations featuring chaotic behaviour, where changes to initial conditions smaller than what the numerical simulation is representing, result in wildly different outcomes after only a short amount of time.