# What is the time complexity of MATLAB's ode solvers?

Say I have a system that is $n$ differential equations that are all coupled together, over a time interval $[a,b]$ with timestep $t$. Is there a straightforward time complexity for MATLAB's built in differential equation solvers ode45, ode23, ode15s? Particularly I am interested in how ode15s grows as the size of $n$ increases. Any help is appreciated!

ode15s is "Variable-timestep, variable-order" method, so I have no idea how to determine the complexity

• Maybe you should check what algorithm it uses? It is publicly available info, then check algorithm itself? It starts with "R"... – Evil Aug 1 '18 at 17:38
• This has been flagged as off-topic, perhaps because it involves a specific piece of software. However, I disagree. It's asking about the asymptotic running time of some algorithm, even though it's not really clear what that algorithm is. If the algorithm used is public knowledge, I don't see why a computer scientist couldn't answer this. – David Richerby Aug 3 '18 at 10:51
• Its a "variable-timestep, variable-order" differential equation algorithm, which makes figuring out the runtime very hard in my eyes. No, it is not the Runge-Kutta method @Evil, no need to be rude. I looked at several research papers but I haven't taken an explicit runtime analysis class yet, so I couldn't figure it out on my own so I thought to ask for help – wjmccann Aug 3 '18 at 15:25
• I do not see any rudness, sorry. In any case, it was not intented. See: blogs.mathworks.com/cleve/2014/05/26/… I assume that creators of propertiary software know what they use. So, for the original post, I simply commented about algorithm, because some software internals are off-topic here. If you did research, it is somewhat expected to share crucial details, like it is RK method, or stating explicitly about ode15, which I do not know what it uses, if you know, it is better to state this in the post. – Evil Aug 3 '18 at 15:43
• Can you edit the question to incorporate as much information and context as you know? Can you describe/specify what the specific algorithm is, for each of those solvers? Can you describe what the "variable-timestep, variable-order" method is? Can you cite your sources for how you learned what its algorithm is? Is there any other relevant information you can share that might make it easier for others to answer? That might help us help you better. For example, there might be people here who know runtime analysis but who don't know what ode15s is. – D.W. Aug 3 '18 at 15:52