Prerequisites for computational mathematics

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What are the mathematical prerequisites (essential and recommended courses) for advanced studies or research in the area of computational mathematics, particularly the field of numerical simulation and scientific computing as an undergraduate mathematics student?

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Depends on what sort of field you want to learn about. I would recommend taking an intro to "Numerical Analysis" course, which usually just requires a basic proofs/calculus background. Taylor series is pretty much all you need to know before getting into the basic error estimates for numerical analysis topics like ODE solvers, numerical differentiation, and numerical integration.

For research topics, there are many directions to pursue, and their prerequisites change based on the sub-genre of computational mathematics. Some (e.g., numerical PDE, finite-element methods, etc.) often require graduate-level functional analysis. Others (e.g., large-scale numerical linear algebra, sparse computations, randomized SVD, etc.) need more expertise in linear algebra, CS topics like Big-O notation, and potentially probability theory / stochastics.

But you don't need to master all of these topics to start; I'd recommend starting with some basic numerical analysis, and following/researching the topics from that class which interest you most!